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This book examines the development of mathematics from the late 16th Century to the end of the 19th Century. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations., Aimed at students and researchers in Mathematics, History of Mathematics and Science, this book examines the development of mathematics from the late 16th Century to the end of the 19th Century. Mathematics has an amazingly long and rich history, it has been practised in every society and culture, with written records reaching back in some cases as far as four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries.This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary mathematics., 1. Beginnings ; 2. Fresh ideas ; 3. Foreshadowings of calculus ; 4. The calculus of Newton and of Leibniz ; 5. Early mathematical physics: Newton's Principia ; 6. Early number theory ; 7. Early probability ; 8. Power series ; 9. Functions ; 10. Making calculus work ; 11. Limits and continuity ; 12. Solving equations ; 13. Groups, fields, ideals and rings ; 14. Derivatives and integrals ; 15. Complex analysis ; 16. Convergence and completeness ; 17. Linear algebra ; 18. Foundations ; PEOPLE, INSTITUTIONS, AND JOURNALS ; BIBLIOGRAPHIES ; INDEX, Mathematics Emerging will provide a valuable resource for students of the history of mathematics, material for the enrichment of the studies of undergraduate mathematicians, and much enjoyment and fascination for anyone who loves mathematics... the author and publisher are to be congratulated on the care with which this beautiful volume has been produced. Tony Mann, Times Higher Education Supplement 'Beautifully and lovingly presented DS a joy to read... Anyone who enjoys mathematics will love it.' Tony Mann, Times Higher Education Supplement | 677.169 | 1 |
Algorithms, computation and mathematics. Student text by School Mathematics Study Group(
Book
) 226
editions published
between
1916
and
2015
in
English
and held by
1,536 WorldCat member
libraries
worldwide
This text contains material designed for about 18 weeks of study at grades 11 or 12. Use of a computer with the course is
highly recommended. Developing an understanding of the relationship between mathematics, computers, and problem solving is
the main objective of this book. The following chapters are included in the book: (1) Algorithms, Language, and Machines;
(2) Input, Output, and Assignment; (3) Branching and Subscripted Variables; (4) Looping; (5) Functions and Procedures; (6)
Approximations; (7) Some Mathematical Applications; and (8) Compilation and Some Other Non-Numeric Problems. Also included
is a discussion on future computer applications. (Rh)
Elementary functions. Students̕ text by School Mathematics Study Group(
Book
) 17
editions published
between
1961
and
1965
in
English
and held by
97 WorldCat member
libraries
worldwide
Essays on number theory by School Mathematics Study Group(
Book
) 2
editions published
in
1960
in
English
and held by
85 WorldCat member
libraries
worldwide
Concepts of informal geometry by School Mathematics Study Group(
Book
) 4
editions published
in
1960
in
English
and held by
79 WorldCat member
libraries
worldwide
Mathematics for junior high school by School Mathematics Study Group(
Book
) 11
editions published
between
1960
and
1961
in
English
and held by
76 WorldCat member
libraries
worldwide
A brief course in mathematics for elementary school teachers by School Mathematics Study Group(
Book
) 3
editions published
between
1962
and
1963
in
English
and held by
70 WorldCat member
libraries
worldwide
Designed to give teachers a good comprehension of elementary maths and methods of teaching it. Includes answers
Geometry by B. V Kutuzov(
Book
) 5
editions published
between
1960
and
1961
in
English
and held by
66 WorldCat member
libraries
worldwide
For high school teachers
Intuitive geometry by School Mathematics Study Group(
Book
) 1
edition published
in
1961
in
English
and held by
64 WorldCat member
libraries
worldwide
Prepared for elementary school teachers to develop sufficient subject matter competence in mathematics of the elementary school
program
Analytic geometry. Student's text by School Mathematics Study Group(
Book
) 9
editions published
between
1963
and
1965
in
English
and held by
56 WorldCat member
libraries
worldwide
Intermediate mathematics. Student's text : Part I-II by School Mathematics Study Group(
Book
) 12
editions published
between
1961
and
1967
in
English
and held by
23 WorldCat member
libraries
worldwide
This nineteenth unit in the smsg secondary school mathematics series is the teacher's commentary for Unit 17. First, a time
allotment for each of the chapters in Units 17 and 18 is given. Then, for each of the chapters in Unit 17, the goals for that
chapter are discussed, the mathematics is explained, some teaching suggestions are given, answers to exercises are provided,
and sample test questions are included. (Dt) | 677.169 | 1 |
Elementary Mechanics Using Matlab
4.11 - 1251 ratings - Source
This book a€" specifically developed as a novel textbook on elementary classical mechanics a€" shows how analytical and numerical methods can be seamlessly integrated to solve physics problems. This approach allows students to solve more advanced and applied problems at an earlier stage and equips them to deal with real-world examples well beyond the typical special cases treated in standard textbooks. Another advantage of this approach is that students are brought closer to the way physics is actually discovered and applied, as they are introduced right from the start to a more exploratory way of understanding phenomena and of developing their physical concepts. While not a requirement, it is advantageous for the reader to have some prior knowledge of scientific programming with a scripting-type language. This edition of the book uses Matlab, and a chapter devoted to the basics of scientific programming with Matlab is included. A parallel edition using Python instead of Matlab is also available. Last but not least, each chapter is accompanied by an extensive set of course-tested exercises and solutions.(a) A sprinter is accelerating along the track. Draw a free-body diagram of the
sprinter, including only horizontal forces. Try to make the length of the vectors
correspond to the relative magnitudes of the forces. Let us assume that the
sprinter isanbsp;...
Title
:
Elementary Mechanics Using Matlab
Author
:
Anders Malthe-Sorenssen
Publisher
:
Springer - 2015-06-01
ISBN-13
:
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Arihant CBSE All in One Mathematics For Class - 12 Book
Product Code : 9789351411208
Quick Overview
Arihant CBSE All in One Mathematics For Class - 12 Book
The first choice of teachers and students since its first edition, All in One for Mathematics has been designed for the students of Class XII strictly on the lines of CBSE Mathematics curriculum. The fully revised and updated edition has been authored by an experienced examiner providing all explanations and guidance needed for effective study and for ultimately achieving success in the CBSE Class XII examination. The present book for Class XII Mathematics has been designed to ensure Complete Study, Complete Practice and Complete Assessment. The whole syllabus has been divided into 13 chapters covering Relations & Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity & Differentiability, Application of Derivatives, Integrals, Application of Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Linear Programming and Probability. The book has been designed strictly in sync with the latest CBSE syllabus &Mathematics Class XII NCERT Textbook, with each chapter divided into individual topics for better understanding. Each individual topic contains detailed theory in notes form supported by illustrations, tables, flow charts, etc which will help in effective comprehension of the concepts. Questions given in each chapter have been grouped as Very Short Answer Type, Short Answer Type, Long Answer Type and HOTS. These Topicwise-Chapterwise questions cover NCERT Questions, Previous Years' Examination Questions as well as other important questions from the examination point of view. Solutions and explanations to all the questions have been given to facilitate easy learning and understanding. Chapter Assessment, a small test that has been given at the end of each chapter, to help the students in assessing their level of understanding. For thorough practice and to give students a real feel of the examination 10 Sample Question Papers have been given after the chapterwise study. CBSE Examination Paper 2016 (All India & Delhi) has been given at the end of the book with complete solutions to give the students an insight into the current exam pattern and types of questions asked therein. Also the book covers CBSE Sample Question papers which will give the students an idea about the types of questions which can be expected in the forthcoming examination and excerpts from topper's answer sheet have also been provided in the book. | 677.169 | 1 |
Roadmap to 6th Grade Math, Ohio Edition
4.11 - 1251 ratings - Source
The Roadmap series works as a year-long companion to earning higher grades, as well as passing the high-stakes6th Grade Math Ohio Proficiency Testthat reading charts and graphs, using fractions and decimals, and understanding basic geometry a€c 2 complete practice OPTsYou learned about word problems in Mile 9. But now youa#39;re going to spend the
last two miles reviewing them with some of the new skills wea#39;ve learned. The
Math OPT loves to ask about word problems, so you want to make sure that you
cananbsp;...
Title
:
Roadmap to 6th Grade Math, Ohio Edition
Author
:
James Flynn, Princeton Review
Publisher
:
The Princeton Review - 2002-01-15
ISBN-13
:
Continue
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Practical Problems in Mathematics for Electronic Technicians
4.11 - 1251 ratings - Source
Success in the electronics field requires a substantial background in mathematics. This updated book is written to provide beginning students with these needed skills. Practical, easy-to-understand problems help prepare students for the types of problems that professional electronic technicians face everyday. As part of the successful Practical Problems in Mathematics series, this fourth edition features expanded coverage of scientific notation, increased problems to be solved using a calculator, additional information on RLC circuits, and a new unit on simultaneous equations that includes coverage of Kirchoff's Law. Benefits: * trade-specific coverage develops an acute awareness of electronics symbols, basic circuits, and component terminology * a step-by-step approach to math concepts begins with basic arithmetic and progresses through algebra and trigonometry, helping students master each level of skill required by today's technicians * extensive multi-level word problems challenge students to use logical deduction, preparing them for situations like those to be encountered on the job * abundant example problems offer opportunities to practice math principles again and again * an Instructor's Guide is available to accompany the bookGiga is symbolized by the letter G. Microwave frequencies are often measured in
gigahertz (GHz). The average microwave oven operates at a frequency of about
2.45 GHz. which is 2.450.000, 000 hertz (Hz). The engineering unit above giga is
anbsp;...
Title
:
Practical Problems in Mathematics for Electronic Technicians
Author
:
Stephen L. Herman
Publisher
:
Cengage Learning - 2004
ISBN-13
:
Continue
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Mathematical Modeling and Simulation
4.11 - 1251 ratings - Source
This concise and clear introduction to the topic requires only basic knowledge of calculus and linear algebra - all other concepts and ideas are developed in the course of the book. Lucidly written so as to appeal to undergraduates and practitioners alike, it enables readers to set up simple mathematical models on their own and to interpret their results and those of others critically. To achieve this, many examples have been chosen from various fields, such as biology, ecology, economics, medicine, agricultural, chemical, electrical, mechanical and process engineering, which are subsequently discussed in detail. Based on the author`s modeling and simulation experience in science and engineering and as a consultant, the book answers such basic questions as: What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? What are simulation, parameter estimation, and validation? The book relies exclusively upon open-source software which is available to everybody free of charge. The entire book software - including 3D CFD and structural mechanics simulation software - can be used based on a free CAELinux-Live-DVD that is available in the Internet (works on most machines and operating systems). From the Contents: - Principles of mathematical modeling - Phenomenological and mechanistic models - Differential equation models (ODE's and PDE's) - Open-Source Software, e.g. for 3D CFD and structural mechanics - Introduction into CAELinux, Calc, Code-Saturne, Maxima, R, Salome-Meca Kai Velten is a professor of mathematics at the University of Applied Sciences, Wiesbaden, Germany, and a modeling and simulation consultant. Having studied mathematics, physics and economics at the Universities of GApttingen and Bonn, he worked at Braunschweig Technical University (Institute of Geoecology, 1990-93) and at Erlangen University (Institute of Applied Mathematics, 1994-95). From 1996-2000, he held a post as project manager and group leader at the Fraunhofer-ITWM in Kaiserslautern (consultant projects for the industry). His research emphasizes differential equation models and is documented in 34 scientific publications and one patent.This concise and clear introduction to the topic requires only basic knowledge of calculus and linear algebra - all other concepts and ideas are developed in the course of the book.
Title
:
Mathematical Modeling and Simulation
Author
:
Kai Velten
Publisher
:
John Wiley & Sons - 2009-02-02
ISBN-13
:
Continue
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III. VECTOR ALGEBRA :
(a) Algebra of vectors – angle between two non-zero vectors – linear combination of vectors – vector equation of line and plane (b) Scalar and vector product of two vectors and their applications c) Scalar and vector triple products, Scalar and vector products of four vectors.
V. COORDINATE GEOMETRY :
(a) Locus, Translation of axes, rotation of axes (b) Straight line (c) Pair of straight lines (d) Circles (e) System of circles (f) Conics – Parabola – Ellipse – Hyperbola – Equations of tangent, normal, chord of contact and polar at any point of these conics, asymptotes of hyperbola. (g) Polar Coordinates (h) Coordinates in three dimensions, distance between two points in the space, section formula, centroid of a triangle and tetrahedron. (i) Direction Cosines and direction ratios of a line – angle between two lines (j) Cartesian equation of a plane in (i) general form (ii) normal form and (iii) intercept form – angle between two planes (k) Sphere – Cartesian equation – Centre and radius | 677.169 | 1 |
MAT-1050 - Elements of Mathematics
Designed for students preparing to teach at the preschool and elementary level. Overview of mathematical systems, including sets, natural numbers, integers, rational and irrational numbers, algorithms and computational methods. 3 class/2 lab hours. | 677.169 | 1 |
Discrete Mathematics with Applications
This approachable text studies discrete objects and the relationships that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms, programming languages, compilers, databases, and computation. It covers all recommended topics in a self-contained, comprehensive, and understandable format for students and new professionals. It emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development and correctness, and numeric computations. It weaves numerous applications into the text. It helps students learn by doing with a wealth of examples and exercises: 560 examples worked out in detail; more than 3,700 exercises; more than 150 computer assignments; and more than 600 writing projects. It includes chapter summaries of important vocabulary, formulas, and properties, plus the chapter review exercises. It features interesting anecdotes and biographies of 60 mathematicians and computer scientists. This is the Instructor's Manual available for adopters. Student Solutions Manual is available separately for purchase (ISBN: 0124211828).
"synopsis" may belong to another edition of this title.
Review:
"A good source of topics for discrete mathematics, and many topics are covered in very good breadth and depth.", H.K. Dai, Oklahoma State University "This text is better than the current one that I am using in the sense that it has been written in a less pure mathematics text style, which will be much more receivable for the student outside of mathematical major. The plenty of exercises provided allow instructor to have more flexibility to choose. The author's easy going but interesting writing style will certainly make instructor's job easier.", Nan Jiang, University of San Diego.681882118031478870
Book Description Academic Press. Book Condition: New. 01242118211803
Book Description Book Condition: Brand New. New. Soft Cover International edition. Different ISBN and Cover image but contents are same as US edition. Customer Satisfaction guaranteed!!. Bookseller Inventory # SHAK13338509 | 677.169 | 1 |
Mathematics
(1) Applications Of Matrices And Determinants : Adjoint, Inverse –
Properties, Computation of inverses, solution of system of linear equations by matrix inversion method. Rank of a Matrix − Elementary transformation on a matrix, consistency of a system of linear equations, Cramer's rule, Non-homogeneous equations, homogeneous linear system, rank method.
(2) Vector Algebra :
Scalar Product – Angle between two vectors, properties of scalar product, applications of dot products. Vector Product − Right handed and left handed systems, properties of vector product, applications of cross product. Product of three vectors − Scalar triple product, properties of scalar triple product, vector triple product, vector product of four vectors, scalar product of four vectors. Lines − Equation of a straight line passing through a given point and parallel to a given vector, passing through two given points (derivations are not required). angle between two lines. Skew lines − Shortest distance between two lines, condition for two lines to intersect, point of intersection, collinearity of three points. Planes − Equation of a plane (derivations are not required), passing through a given point and perpendicular to a vector, given the distance from the origin and unit normal, passing through a given point and parallel to two given vectors, passing through two given points and parallel to a given vector, passing
through three given non-collinear points, passing through the line of intersection of two given planes, the distance between a point and a plane, the plane which contains two given lines, angle between two given planes, angle between a line and a plane. Sphere − Equation of the sphere (derivations are not required)whose centre and radius are given, equation of a sphere when the extremities of the given.
(4) Analytical Geometry :
Definition of a Conic − General equation of a conic, classification with respect to the general equation of a conic, classification
of conics with respect to eccentricity. Parabola − Standard equation of a parabola (derivation and tracing the parabola are not required), other standard parabolas, the process of shifting the origin, general form of the standard equation, some practical problems. Ellipse − Standard equation of the ellipse (derivation and tracing the ellipse are not required), x2/a2 + y2/b2 = 1, (a > b), Other standardform of the ellipse, eneral forms, some practical problems, Hyperbola −standard equation (derivation and tracing the hyperbola are not required), x2/a2 −y2/b2=1, Other form of the hyperbola, parametric form of conics, chords. Tangents and Normals − Cartesian form and Parametric form, equation of chord of contact of tangents from a point (x1, y1), Asymptotes, Rectangular
hyperbola – standard equation of a rectangular hyperbola.
Physical Chemistry
Unit 8 - Solid state II
Unit 9 - Thermodynamics - II
Review of I law - Need for the II law of thermodynamics - Spontaneous and non spontaneous processes - Entropy - Gibb's free energy - Free energy change and chemical equilibrium - Third law of thermodynamics.
Unit 10 - Chemical equilibrium II
Applications of law of mass action - Le Chatlier's principle.
Unit 11 - Chemical Kinetics -II
First order reaction and pseudo first order reaction - Experimental determination of first order reaction - method of determining order of reaction -temperature dependence of rate constant - Simple and complex reactions.
Physics
Unit – 1 Electrostatics
Frictional electricity, charges and their conservation; Coulomb's law – forces between two point electric charges. Forces between multiple electric charges – superposition principle.
Electric field – Electric field due to a point charge, electric field lines; Electric dipole, electric field intensity due to a dipole –behavior of dipole in a uniform electric field – application of electric dipole in microwave oven.
Electric potential – potential difference – electric potential due to a point charge and due a dipole. Equipotential surfaces – Electrical potential energy of a system of two point charges. | 677.169 | 1 |
Analyze this:
Our Intro to Psych Course is only $329.
*Based on an average of 32 semester credits per year per student. Source
Tutorial
The Order of Operations tells us how to complete a math problem in the correct order based on mathematical functions. It is also referred to as PEMDAS. This stands for Parenthesis, Exponents, Multiplication, Division, Addition and Subtraction. The slideshow below will further explain the Order of Operations. Be sure to explore the websites, videos, practice sites and games for extra help. Don't forget about your WSQ! Your WSQ will help you with our in class activity tomorrow when we go "shopping" with PEMDAS. | 677.169 | 1 |
sNet: Interactive Fractions to your Bookmark Collection or Course ePortfolio
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You can alter the geometric construction dynamically in order to test and prove (or disproved) conjectures and gain...
see more
You can alter the geometric construction dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. Requires Java Plug-in 1.3 or higher. Please be patient while the applet loads on your computer. If you are using a dial-up connection, it may take a few minutes but is well worth the waitge & d'Alembert Three Circles Theorem I with Dynamic Geometry to your Bookmark Collection or Course ePortfolio
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In the 21st century, airplanes are a normal part of everyday life. We see them fly over, or read about them, or see them on...
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In the 21st century, airplanes are a normal part of everyday life. We see them fly over, or read about them, or see them on television. Most of us have traveled on an airplane, or we know someone who has. Do you ever wonder how airplanes fly? What causes the lift that gets the airplane off the runway? How does a pilot control the movement of the airplane? Why are the engines on an airliner different from the engines on a fighter plane? How does aerodynamics affect the flight of a model rocket or a kite? The information at this site is provided by the NASA Glenn Educational Programs Office (EPO) to give you a better understanding of how aircraft work. Much of the material was originially developed for the Learning Technologies Project (LTP).Each page at this site describes a single topic related to basic airplane aerodynamics, propulsion, rockets, or kites. At the top of each page is a slide that illustrates the topic. The slide is accompanied by a caption that explains what the slide is all about and goes into some detail about the physics and math related to the subject of the slide. There are links and references to other slides and sites where you can find additional Beginner's Guide to Aerodynamics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material NASA Learning Objects: Beginner's Guide to Aerodynamics
Select this link to open drop down to add material NASA Learning Objects: Beginner's Guide to Aerodynamics The Case of the Shaky Quake to your Bookmark Collection or Course ePortfolio
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Excellent computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels....
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Excellent computer-based mathematical manipulatives and interactive learning tools at elementary and middle school levels. This is an NSF supported project that began in 1999 to develop a library of uniquely interactive, web-based virtual manipulatives or concept tutorials, mostly in the form of Java applets, for mathematics instruction (K-12 emphasis). Free for Mathematics to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material National Library of Virtual Manipulatives for Mathematics
Select this link to open drop down to add material National Library of Virtual Manipulatives for Mathematics discussionsThere is today a greater awareness that elementary mathematics is rich in important ideas and that its instruction requires far more than simply knowing the "math facts" and a handful of algorithms. Mathematics courses for teachers must reflect the intellectual depth and challenge of the elementary school curriculum. The Conference Board of Mathematical Sciences (CBMS) recommends that the preparation of mathematics teachers include courses that develop a "deep understanding of the mathematics they teach," that are designed to "develop careful reasoning and 'common sense' in analyzing conceptual relationships, . . . that develop the habits of mind of a mathematical thinker and that demonstrate flexible, interactive styles of teaching" (CBMS, 2000, pp. 7-8). Judy Sowder, Larry Sowder, and Susan Nickerson recognize and accept the challenge of presenting mathematics to teachers in a manner that addresses these recommendations. In doing so they provide instruction that will lead teachers of mathematics to "reconceptualize" the mathematics they often think they already know, thus allowing them to develop a deeper understanding of the mathematics they will teach. The authors believe that teachers must know mathematics differently than most people do. Teachers need to know the mathematics they teach in a way that allows them to hold conversations about mathematical ideas and mathematical thinking with their students. A persistent pursuit of explanation is a hallmark of a classroom in which learning is taking place. A common axiom is that teachers teach the way they were taught. Prospective teachers are unlikely to demonstrate flexible, interactive styles of teaching unless they have experienced mathematics taught this way. Instructors of the "Reconceptualizing Mathematics" courses, however, may not have experienced such instruction themselves. Thus the authors provide many forms of instructional assistance to help instructors better understand the mathematics their prospective teachers need to know, to begin to model teaching strategies that these prospective teachers will be expected to use in their own classrooms, and to assist them in many ways throughout the Reconceptualizing Mathematics to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Reconceptualizing MathematicsThis site was developed by CATEA, the Center for Assistive Technology and Environmental Access at Georgia Tech. The site is...
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This site was developed by CATEA, the Center for Assistive Technology and Environmental Access at Georgia Tech. The site is designed specifically to train high school math and science teachers to be more effective instructors for students with disabilities. The project includes research, instruction for teachers on how to make their coursework, classrooms and labs more accessible, and information resources for teachers including assistive technology for their SciTrain to your Bookmark Collection or Course ePortfolio
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will make you the mathematical genius when it comes to unit conversion. They'll only ease your burden and spare you the blushes of failing a conversion test or task. Utilizing unit conversion tools It should be noted that unit converters are of two types namely expanded style and compact style. The former converts a single unit into several other different units while the latter only converts from one single unit to another single unit. Locating the units on the software is very easy once it has been installed into the system. You can find them without any hassle as they are mostly arranged in alphabetical order. You'll therefore find units of measurement such as 'acres' on top of the list while units such as 'yards' will feature towards the bottom of the list. The main advantage of using this tool is that it will save you a lot of time. This will in the long run increase your productivity should you be working with limited time. The chances of an error are zero. The slim possibility of an error being reported will basically be blamed on human error. The software is flawless in the way it's designed so the only error that might surface is if the user inputs and uses the wrong figuresSuper calculator It add as many functions of this kind as possible to suit your common requests.
Such as trigonometry, statistics, convert, logical and string functions. Includes all bits operations for IT researches.Such as bit wise AND, bits shift, maximum and minimum i
Math bootCamp The early stages of learning math require the memorizing a lot of detailed information and this can be quite frustrating for some small children. Experts know that young children learn at a faster pace while they are having fun. That is especially true fo
ClinTools ClinTools is a statistical suite of evidence-based decision making tools. It augments traditional statistical software by conducting specialized analyses that are required for evidence-based practice, whether that be in the health system, education system,matGeom matGeom is a geometry library for Matlab in 2D and 3D. You are able to create, transform, manipulate and display geometric primitives (points, lines, polygons, and planes). Typical operations involve creation of shapes, computation of intersections, transf | 677.169 | 1 |
Refresher in Basic Mathematics
4.11 - 1251 ratings - Source
This text provides exactly what the title promises. With clear explanations and plenty of examples and practice questions, this book takes you step-by-step through: fractions, decimals, percentages, powers, simple linear equations, quadratic equations, elementary algebra, coordinates and graphs, simultaneous equations, the straight line, use of calculators and rounding. This edition comes with an easy-to-use CD-Rom to demonstrate all these functions with extra clarity.This text provides exactly what the title promises.
Title
:
Refresher in Basic Mathematics
Author
:
R. N. Rowe
Publisher
:
Cengage Learning EMEA - 2001-12
ISBN-13
:
Continue
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Beginning and Intermediate Algebra: A Guided Approach
4.11 - 1251 ratings - Source
The new edition of BEGINNING a INTERMEDIATE ALGEBRA is an exciting and innovative revision that takes an already successful text and makes it more compelling for todaya€™s instructor and student. The authors have developed a learning plan to help students succeed and transition to the next level in their coursework. Based on their years of experience in developmental education, the accessible approach builds upon the booka€™s known clear writing and engaging style which teaches students to develop problem-solving skills and strategies that they can use in their everyday lives. The authors have developed an acute awareness of studentsa€™ approach to homework and present a learning plan keyed to Learning Objectives and supported by a comprehensive range of exercise sets that reinforces the material that students have learned setting the stage for their success. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.In this exercise, you will explore Blooma#39;s Taxonomy of learning at the first three
levels. ... 1. Given a table of data, find the slope of the line it represents. 2. Write
the formula used to calculate the slope of a line between two points. 3. ... Check
your answers to the previous three statements against the descriptions you find.
... Solving Quadratic Equations using the Square-Root Property and by
Completing Editorial review has deemed that any suppressed content does not
materially affectanbsp;...
Title
:
Beginning and Intermediate Algebra: A Guided Approach
Author
:
Rosemary Karr, Marilyn Massey, R. Gustafson
Publisher
:
Cengage Learning - 2014-01-01
ISBN-13
:
Continue
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Multivariable Calculus (saylor.org)
Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables.
The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results.
The structure of this course very much resembles the structure of Single-Variable Calculus I and II. We will begin by taking a fresh look at limits and continuity. With functions of many variables, you can approach a limit from many different directions. We will then move on to derivatives and the process by which we generalize them to higher dimensions. Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals.
The goal of Multivariable Calculus is to provide you with the tools you need to handle problems with several parameters and functions of several variables and to apply your knowledge of their behavior. But a more important goal is to gain a geometrical understanding of what the tools and computations mean | 677.169 | 1 |
Basic College Mathematics a Real-World Approach
Basic College Mathematics will be a review of fundamental math concepts for some students and may break new ground for others. Nevertheless, students of all backgrounds will be delighted to find a refreshing book that appeals to all learning styles and reaches out to diverse demographics. Through down-to-earth explanations, patient skill-building, and exceptionally interesting and realistic applications, this worktext will empower students to learn and master mathematics in the real world. | 677.169 | 1 |
Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer. Author : James Stewart ISBN : 0840068794 Language : English No of Pages : 464 Edition : 6 Publication Date : 6/22/2011 Format/Binding : Paperback Book dimensions : 9.9x8x1 Book weight : 0.01 | 677.169 | 1 |
U Can: Basic Math and Pre-Algebra For Dummies
4.11 - 1251 ratings - Source
Math is hard, and it's important for students to build a solid foundation in their mathematics education on which their future coursework will later expand. Lucky for students, Dummies makes it easy. Students will find all of the classic no-nonsense, how-to content they need, paired with practical examples and practice problems they want, PLUS access to the 1, 001 more Basic Math a Pre-Algebra practice problems online. In U Can: Basic Math a Pre-Algebra For Dummies, lessons and practice are fully integrated, creating a product that combines the qhow-toq with the qdo itq to form one perfect resource for students.Practice. Answers. 1. x = 3. To get all constants on the right side of the equation,
add 2 to both sides: 9 2 6 7 2 2 9 6 9 x x x x To get all x terms on the left side,
subtract 6x from both sides: 9 6 9 6 6 3 9 x x x x x Divide by 3 to isolate x: 3 3 9 3
3 x xanbsp;...
Title
:
U Can: Basic Math and Pre-Algebra For Dummies
Author
:
Mark Zegarelli
Publisher
:
John Wiley & Sons - 2015-07-27
ISBN-13
:
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Oxford GCSE Maths for Edexcel
4.11 - 1251 ratings - Source
This book has been specifically written for the new two-tier Edexcel linear GCSE specification for first examination in 2008. The book is targeted at the B to A* grade range in the Foundation tier GCSE, and it comprises units organised clearly into homeworks designed to support the use of the Higher Plus Students' Book in the same series. Each unit offers: * A review test focusing on prior topics for continual reinforcement * Two sets of questions that relate directly to individual lessons in the unit, providing ample practice * A synoptic homework that covers the whole unit, so students consolidate the key techniques * Full answers in the accompanying teacher book It forms part of a suite of four homework books at GCSE, in which the other three books cater for grade ranges G to E, E to C and D to B.This book has been specifically written for the new two-tier Edexcel linear GCSE specification for first examination in 2008.
Title
:
Oxford GCSE Maths for Edexcel
Author
:
Clare Plass
Publisher
:
Oxford University Press, USA - 2006
ISBN-13
:
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Descripción:
Sobre este título:
In a Liberal Arts Math course, a common question students ask is, ?Why do I have to know this?? A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and whythis is important. The Ninth Edition further emphasizes this with the addition of new ?Why This Is Important? sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills.
Angel
Note: This is a standalone book, if you want the book/access card please order the ISBN listed below:
Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy.
Christine Abbott received her undergraduate degree in mathematics from SUNY Brockport and her graduate degree in mathematics education from Syracuse University. Since then she has taught mathematics at Monroe Community College and has recently chaired the department. In her spare time she enjoys watching sporting events, particularly baseball, college basketball, college football, and the NFL. She also enjoys spending time with her family, traveling, and reading
Dennis Runde has a BS degree and an MS degree in Mathematics from the University of Wisconsin--Platteville and Milwaukee respectively. He has a PhD in Mathematics Education from the University of South Florida. He has been teaching for more than fifteen years at State College of Florida?Manatee-Sarasota and for almost ten years at Saint Stephen's Episcopal School. Besides coaching little league baseball, his other interests include history, politics, fishing, canoeing, and cooking. He and his wife Kristin stay busy keeping up with their three sons--Alex, Nick, and Max.
Descripción Pearson. Estado de conservación Nº de ref. de la librería ING-2940321837535
Descripción Pearson. Hardcover. Estado de conservación: Good. 0321759664 Good condition with normal wear. Supplemental materials such as CDs or access codes may NOT be included regardless of title. May have bookstore stickers on cover. Expedited shipping available (2-4 day delivery)! Contact us with any questions!. Nº de ref. de la librería Z0321759664Z3
Descripción Pearson. Hardcover. Estado de conservación: Very Good. 0321759664 Great condition with light wear! Supplemental materials such as CDs or access codes may NOT be included regardless of title. Expedited shipping available (2-4 day delivery)! Contact us with any questions!. Nº de ref. de la librería Z0321759664Z2
Descripción Pearson, 2012. Hardcover. Estado de conservación: Used: Acceptable 315471
Descripción Pearson, 2012. Estado de conservación: Used. This Book is in Good Condition. Clean Copy With Light Amount of Wear. 100% Guaranteed. Summary: In a Liberal Arts Math course, a common question students ask is, "Why do I have to know this?" A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and whythis is important. The Ninth Edition further emphasizes this with the addition of new "Why This Is Important" sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills. Angel Note: This is a standalone book, if you want the book/access card please order the ISBN listed below: 0321837533 / 9780321837530 A Survey of Mathematics with Applications plus MyMathLab Student Access Kit Package consists of: 0321431308 / 9780321431301 MyMathLab/MyStatLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321759664 / 9780321759665 Survey of Mathematics with Applications, A. Nº de ref. de la librería ABE_book_usedgood_0321759664
Descripción Pearson. Hardcover. Estado de conservación: Very Good. 0321759664 Item in very good condition and at a great price! Textbooks may not include supplemental items i.e. CDs, access codes etc. Nº de ref. de la librería Z0321759664Z2 | 677.169 | 1 |
What is a Derivative? Interactive Notebook to Explore on your own!
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What is a Derivative? Interactive Notebook to Explore on your own!
This interactive mathematica notebook allows the student to explore the concept of the derivative as the slope of the tangent line to a curve at a point. Visit in the calculus section to download the interactive notebook!
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(Original post by 34908seikj)
They've completely killed off D1 btw now. Since they moved to linear exams from from modular, they combined all the modules together except for the decision modules which are like, gone, dead, you get the point...
And "S" is statistics.
is statistics like fractions and all that chance stuff? i hate that T_T but then i also hate physics
(Original post by 34908seikj)
eh, fractions is sort of a general maths thing lmfao. But yes, statistics is that chance stuff. It doesn't really have anything to do with physics, at least compared to mechanics.
No I mean I hate mechanics cos it's physics and I also don't like stats but what do you think is more useful stats or mechanics.
I just finished year 12 and did Biology and Chemistry too - chemistry was a killer and was nearly as hard as history (and chemistry and physics are infamous for being the hardest A-Levels) you're doing geography as well, aren't you? Apparently it's super easy but can get boring. Awww, haha, it'll become second nature in no time although a lot of things will throw you off in core 2 if that's the case (logs)
Sorry, D1 is Decision 1 and it's basically just loads of algorithms, so like set methods of doing something and you just repeat it over and over again but with different numbers. That's why it's so easy and it's not like they can make it hard. Ooh, mechanics is hard although I'm not familiar with either M or S - you're getting ahead of me already! XD and I wanna do further maths next year so this isn't looking good.
(Original post by 34908seikj)(Original post by ihatePE)Actually Maths A-Level is only going linear next year, so it won't apply for your year group anywayIt's definitely allowed. The regulations specifically prohibit algebraic integration/differentiation/etc - this calculator can only perform numeric integration/differentiation/etc so it can check definite integrals but won't give you answers to indefinite integrals. These very subtle differences is what changes it from prohibited to permitted.
(Original post by JT423)That calculator is allowed for edexcel exams, I can't believe they actually took it off you. You would think that the people running the exam would know. I would actually RKO the person who took it off me. And then educate them so that I could still use the calculator.
I mean it shouldn't make a huge difference what calculator you use, but if you're prone to little mistakes like me, when you check a definite integral and it's different to what the calculator says, you get back and check it until both answers match. | 677.169 | 1 |
Keyport, WA ACT MathNamrata P.
...It allows one to perform numerical calculations much faster compared to programming languages like C, C++ and Java. We can visualize the results using graph plotting. I have been using Matlab software to solve mathematical problems such as Linear equations, non-linear equations, First order differential equations, second order differential equations.
Valentine W | 677.169 | 1 |
In this paper, we analyze the influence of applying the double strand discrete-continuous on College students' conceptual understanding of central notions in analysis such as limit and derivative. The research is done in the context of a course on differential equations. Our aim is to analyze how using Mathematica, while iterating the simplest mathematical expression such as quadratic functions, the students are introduced to the great beauty of the field of dynamical systems and to a better understanding of convergence problems. | 677.169 | 1 |
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using GeometerInstructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) withoutTo ifferentiate logarithmic functions with bases other than e, use 1 1 To ifferentiate logarithmic functions with bases other than e, use log b m = ln m ln b 1 To ifferentiate logarithmic functions withMathematics Learning Centre Introduction to Differential Calculus Christopher Thomas c 1997 University of Sydney Acknowledgements Some parts of this booklet appeared in a similar form in the booklet Review The rent control agency of New York City has found that aggregate demand is Q D = 100-5P. Quantity is measured in tens of thousands of apartments. Price, the average monthly rental rate, is measured
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternativeTitle: Do These Systems Meet Your Expectations Brief Overview: This concept development unit is designed to develop the topic of systems of equations. Students will be able to graph systems of equations | 677.169 | 1 |
Did you know that a graph of a function f is not the same as the function itself? It might seem like there is clearly a difference, but sometimes it's hard to articulate into words. We have spoken about the definition of a function. Simply put, it's a rule that transforms one real number into […]
The October 22nd ACT is upon us! Just a few more days of studying before you can put your energy into planning your Halloween costume and catching up on fall TV. The October ACT is very important for seniors applying Early Action or Early Decision. Get your last minute resources here.
In Tricky Questions on the AP Calculus Exam we discussed the derivative of a function; recall that a function f(x) is differentiable exactly when the limit exists for every x in the domain of f(x). The Geometry of the Derivative The derivative of a function f at a point x in the domain of […]
Warm-up: Keeping with the theme of volume of solids of revolution, try this multiple choice question, which should need only understanding of the geometric meaning of the definite integral as a measurement of volume, discussed in Calculus in a Nutshell: Part II. Question: A solid is generated by revolving the region enclosed by the function […]
The school year is in full swing. Before the fun of Thanksgiving and Winter Break, it's time to take the new SAT this Saturday, October 1, 2016. Have you prepared for the New SAT? You've hopefully already been doing some SAT prep… but more prep materials can't hurt. Magoosh has you covered! First, drop by the […]
If you think college is just one boring lecture after another, think again. Unlike the curriculum high school offers, college has a much more flexible and broader set of courses you can take. More importantly, these are courses you can choose on your own and put together based on your own interests. While every college […] | 677.169 | 1 |
Mathematics for Engineers and Scientists, Sixth Edition
4.11 - 1251 ratings - Source
Since its original publication in 1969, Mathematics for Engineers and Scientists has built a solid foundation in mathematics for legions of undergraduate science and engineering students. It continues to do so, but as the influence of computers has grown and syllabi have evolved, once again the time has come for a new edition. Thoroughly revised to meet the needs of today's curricula, Mathematics for Engineers and Scientists, Sixth Edition covers all of the topics typically introduced to first- or second-year engineering students, from number systems, functions, and vectors to series, differential equations, and numerical analysis. Among the most significant revisions to this edition are: Simplified presentation of many topics and expanded explanations that further ease the comprehension of incoming engineering students A new chapter on double integrals Many more exercises, applications, and worked examples A new chapter introducing the MATLAB and Maple software packages Although designed as a textbook with problem sets in each chapter and selected answers at the end of the book, Mathematics for Engineers and Scientists, Sixth Edition serves equally well as a supplemental text and for self-study. The author strongly encourages readers to make use of computer algebra software, to experiment with it, and to learn more about mathematical functions and the operations that it can perform.A sixth edition of this book has become necessary because of the many changes
to syllabuses that have occurred since the fifth edition was produced, and also
because of the ... Material in some chapters has been simplified for ease of
understanding, and supplementary background material has been added
wherever it seems likely to be helpful. ... like differentiation, integration and
finding the general solution of a differential equation, and then to present the
results in symbolic form.
Title
:
Mathematics for Engineers and Scientists, Sixth Edition
Author
:
Alan Jeffrey
Publisher
:
CRC Press - 2004-08-10
ISBN-13
:
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Basic Mathematics for Economists
4.11 - 1251 ratings - Source
Economics students will welcome the new edition of this excellent textbook. Mathematics is an integral part of economics and understanding basic concepts is vital. Many students come into economics courses without having studied mathematics for a number of years. This clearly written book will help to develop quantitative skills in even the least numerate student up to the required level for a general Economics or Business Studies course. This second edition features new sections on subjects such as: matrix algebra part year investment financial mathematics Improved pedagogical features, such as learning objectives and end of chapter questions, along with the use of Microsoft Excel and the overall example-led style of the book means that it will be a sure fire hit with both students and their lecturers.This second edition features new sections on subjects such as: matrix algebra part year investment financial mathematics Improved pedagogical features, such as learning objectives and end of chapter questions, along with the use of ...
Title
:
Basic Mathematics for Economists
Author
:
Mike Rosser
Publisher
:
Psychology Press - 2003
ISBN-13
:
Continue
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LOOK INSIDE the book at 4jc-tutoring.com---About the book: ------Objective ---As an avid teacher, I am constantly identifying student training needs for the SAT exam, evaluating effectiveness of resources, and developing strategies and tools for the students. I believe that having a good book is the first step toward success! This is an excellent and ready-to-use tool for students to use as a workbook to self study, or for teachers to run a SAT math class review.The unique strengths of this book: 1. During math review, no student wants to be overloaded with too much text reading. Rather, he or she wants to learn how to solve problems as quickly as possible, with clear directions. Therefore, I base my book on Teaching through Examples, which is a quick and clear way of learning2. Instead of focusing too much on strategies that teach guessing and eliminating, yet lightly touch upon the problem-solving skills, I focus on teaching the important and foundational math skills that give students confidence in solving problems and answering correctly3. "Practice makes perfect," is crucial to math review, hence the well thought-out structure and exhaustive practices and solutions in every chapter. The best strategies to solve problems are also taught and demonstrated in the lessons and practice solutions.A wise proverb says, "I see, I remember; I do, I understand." This is the ob used books,books Books, 4JC Tutoring | 677.169 | 1 |
Calculus wasn't discovered all at one time. The majority of the moment, the Calculus is simpler than the Algebra. The calculus has the ability to handle the natural situation where the car moves with changing speed. Differential calculus was applied to numerous questions that aren't first formulated in the language of calculus.
Fortunately, an individual can do a great deal of introductory physics with only a few of the fundamental methods. Mathematics is a language that is utilised to spell out abstract concepts. Since Math is a tough subject for the majority of children, a teacher cannot enable the students just stick to a textbook, as it is only part of the info necessary to fix the Math problems correctly.
Calculus is also utilized to acquire a more precise comprehension of the disposition of space, time, and motion. Calculus covers an extensive region of modern mathematics. Calculus is an extensive subject, and it forms the foundation for much of modern mathematics. Calculus is a Latin word that is a small stone that's used for counting. Calculus plays an essential part in modern mathematics. There are some good documentaries online about the history of calculus too although you might need to use a VPN if in certain locations, here's a good one – How to Bypass the Netflix Block.
Calculus may be used together with other mathematical disciplines and provides tools to fix many troubles. Calculus is all about changes, so using a single variable calculus is the secret to the overall problem too.
There are some other normal topics in such a training course. By doing this, you might be able to address a question on the last exam, even when you do not fully understand why it is you are solving it the way you're. With the integral, you're going to be given lots of problems to solve, but there's no algorithm. After that, identify just what you should see in the issue. Men and women who are considering solving problems employing ancient, theory-based methods are encouraged to have a calculus program or two while trying to find online college math courses. Also, although in our simple instances, it doesn't be a lot of difference, in higher dimensions, understanding the derivative during the linear approximation is a lot more relevant. It's important that you know the difference whenever you are working these issues.
The above mentioned graph needs to be familiar to anybody who has studied elementary algebra. There are a number of methods for solving these equations to discover an explicit sort of the function. There are lots of approaches to work out this equation. As soon as an explicit solution to a differential equation isn't possible, the slope field stipulates a means to fix the equation graphically. Polynomials are differentiated utilizing the rule. There's an algorithm for doing this. On occasion the computation could be long and complicated.
The process of discovering the derivative is known as differentiation. Put simply, you spend the derivative of the very first factor feature and multiply it by the second factor feature, then spend the derivative of the second factor feature and multiply it by the very first factor feature, and add them together. The operation of locating a derivative is known as differentiation. Specific derivatives, like the derivatives of logarithms, exponents, sums, quotients, goods, and trigonometric functions, are taught, and implicit differentiation.
In the event the variables can't be separated directly, then other methods have to be utilised to fix the equation. In the event the function isn't continuous, the limit could differ from the worth of the function at that point. It won't execute the function. It is possible to integrate discontinuous functions and they're going to come out continuous. It isn't possible to have a system which is described this equation. Otherwise the integration method is like rectangular coordinates. In addition, there are tons of unique applications for differential Calculus!
As one would anticipate, Euler was not just a gifted mathematician he was also good at all types of things. Euler realised he could use the exact same formula for to receive a similar outcome. Euler was an extremely gifted mathematician, not just in terms of what he accomplished, but in addition with respect to his methods. On account of the simple fact that Euler lived in Switzerland, a land-locked nation, he wasn't given the chance to see ocean-going ships. It is sufficient to apply the prior process to Euler's polynomials to seek out result. But algebra was not merely a highly effective tool for engineers. This equation is known as the second type of Euler's equation.
His genius was admired by some best contemporaries of his time. His mathematical genius was proved in subjects of of infinitesimal calculus and graph theory. He's recognized among the most extraordinary mathematicians in the country. This became used by biologists all around the Earth, and he is called the `father of contemporary taxonomy'.
His works include things like finding various computation methods to determine volume and area of many shapes, for instance, conic section. But first, I wish to insist an additional time on the spectacular splendor of Euler's identity. There's certainly no limit to the collection of such amazing men and women, whose works created the platform for other people to create seminal works in mathematics. His work is used in many ways to this day, some of the functions he discovered are important in this connected world of security and encryption, which are used to often to filter content like the way the Netflix VPN ban was implemented.
We only receive a one solution and will require a second solution. It can likewise be demonstrated utilizing an elaborate integral. But this dilemma is an excellent instance of the way to approach problems with constraints. This is an issue with a few of the equations on the website unfortunately.
The use of math principles allows scientists to acquire a deeper comprehension of their various studies and preforming calculations for practical use. It appears natural to rate the limit by viewing its modulus and argument separately. Alternatively, you may choose to argue that the investment rate ought to be computed monthly.
The complicated exponential forms developed by Euler are often utilised in electrical engineering and physics. As stated by the forms of materials that arrive in contact, there are several kinds of friction. We thus need to be able to use the second kind of EulerOs equation to work out this issue. Both materials in touch with one another may be a liquid and a good, a gas and a good, or possibly a gas and a liquid in touch with one another. It's the simplest of all of the varieties of friction to analyze.
Diffie-Hellman can likewise be applied as a member of public key infrastructure today. This is the reason Diffie-Hellman is employed together with an extra authentication method, generally digital signatures. The so-called Diffie-Hellman method stipulates a manner. Authenticated Diffie-Hellman is also called Unified Diffie-Hellman. In this instance, an eavesdropper would need to fix the overall Diffie-Hellman problem to get the shared key which is regarded extremely hard to accomplish.
Inside this scheme, the public key is utilized to stop main-in-the-middle attacks. Neither reaction is correct, Bellovin argues. It must be a main concern of all online users. The issue is called the discrete logarithm issue. Another problem is that initial full disk encryption sometimes takes a very long time (based on the quantity of data to encrypt). These are a few of the issues that you want to get concerned with prior to encrypting your device. Although the topic needs to be presented initial, it is visited after the initial two examples so there's a frame of reference.
Unfortunately, a decent amount of detail wasn't readily apparent (or was lost), and the loss could lead to a system which is insecure overall. The example below is offered in dh-init. Lastly, The sample below is offered in dh-unified, which is particularly useful in network enabled cryptography applications like VPNs such as this.
Step 7 when the function call is done, the Diffie-Hellman public key will be prepared to use. As it's a sizable and apparently random number, an expected hacker has almost no possibility of correctly guessing x, despite the aid of a highly effective computer to conduct millions of trials. There are a lot of methods to implement PAKE. It is among the first practical examples of Key exchange implemented within the area of cryptography. Possible solutions incorporate the usage of digital signatures and other protocol variants.
Encryption is a somewhat essential measure. It is extra effort of an extra layer to protect your data. As more become aware of the generally unsecured nature of the Internet, it will undoubtedly become increasingly popular. Asymmetric encryption is extremely slow. It's very slow in comparison to Symmetric Encryption. Symmetric encryption is extremely fast. This may be remedied with a vital confirmation protocol.
The entire program can be found in dh-param. Its the type of thing that may subtly earn a system insecure, though the system was constructed from secure components. Most encryption methods provide a choice between them instead of combining them. If you would like to share your gadget among more than 1 gadget remember, data encrypted on a single gadget isn't going to be readable on another gadget. After you encrypt your device, there isn't any turning back. You could also encrypt your devices with the assistance of android apps. Both of them are public and can be employed by all the users in a system.
The functions have these signatures. Utilizing the Windows Crypto API functions may be the alternate. This will assign the worth of P. The worth of p could be large but the worth of q is usually tiny. At this time the values of both G and P need to be sent to the intended recipient together with the essential when conducting an essential exchange. It's a sort of key exchange. After the preceding examples, we're finally prepared to carry out key exchange and arrive at a shared secret.
In regard to Riemann Integration, Riemann Integration may be used to decide on the accuracy of the Fourier Series used. Additionally, This is called Riemann integral. Here we are going to try out the approach of Riemann. Riemann integral was made by Bernhard Riemann. Riemann's integral cannot take care of this function.
When modeling real-world troubles, it's easy to compose expressions involving derivatives. In humans, a genetic mutation usually means this sugar isn't present in any cell within the body. They might also play part in disease susceptibility.
Area isn't yet properly defined, you can access a documentary on some TV channels in Europe although you'll probably need a residential VPN service. There is a multitude of techniques to attempt to ascertain the region. Indeed, the area below the similar bit of the given parabola is always precisely the same, whatever letter we write near the horizontal axis.
Integration may be used to discover areas, volumes, central points and lots of helpful things. It is a main topic in calculus. It is a way of adding slices to find the whole. Nonetheless, in this scenario, it is possible to utilize Riemann Integration to discover the area below the curve, and thus the distance the object has traveled.
A huge value for the mesh is supposedly coarse, though a little mesh is supposed to be fine. So if we opt to use a different variable in precisely the same formula, the form and thus the integral stay an identical. The integrated function is occasionally known as the integrand. There are different functions that are non-integrable too. We've been doing Indefinite Integrals to date. Classical multiple integrals are wholly covered via this approach.
In this kind of situation, the integration operation is needed to discover the function, which gave the specific derivative. Now we'll make this procedure precise. There are lots of approaches, here we use the one which is simplest to manage. On the opposite hand, the case of Dirichlet function demonstrates that if there's too many points of discontinuity, the function isn't Riemann integrable.
All about the way that it works and more. To start with, you can imagine this integral using almost the exact same picture. However, the time wasn't yet ready for measure theory. It's all an issue of interpretation in the end. This is a rather crucial question. And this matter is to turn into central to the notion of integral.
His proof demands a monotonicity of f. It is founded on an easy observation that the area of a rectangle is not hard to calculate. Now we must choose their heights. The snaky shape is known as the integration sign, it's in fact an extremely elongated S (for sum). The very first pattern is known as altriciality. This breaking pieces are known as the partition.
As you may be expecting, the geometric mean can become very complicated. Geometric Mean may be the square root of the item of both numbers. The geometric mean is really not the arithmetic mean and it's not a straightforward average. Now take a glance at The Mean Machine.
For instance, the typical percentage sum of growth in a financial institution account per year employs the geometric mean since the development each year is dependent upon multiplying the amount within the bank account by the proportion development. We'd utilize the geometric mean when we would like to find out the ordinary rate of growth in the event the growth rate is dependent upon multiplication. It's likewise called average. It's used to figure out the typical rate of growth once the growth depends upon multiplication as in the instance of annual proportion growth of the bank account.
Biologists utilize this calculation for quantifying average population development prices, which are also known as the intrinsic rate of development" for early phases of population development where there are not any density dependent facets controlling populations. Usually, this problem arises when it's desired to figure out the geometric mean of the percent change in a population or perhaps a financial return, including negative numbers.
In addition, They are natural for summarizing ratios. Instead, as described within this tip, you ought to utilize Excel's GEOMEAN function to figure the geometric mean of the range of numbers. In these instances, you ought to utilize Excel's GEOMEAN function to compute the typical growth rate, given the effect of compounding.
To calculate geometric mean in these types of situations, you need to utilize Method 2. This dilemma wants the Altitude Rule. Understanding the issue.
The following step is in fact solving these undesirable boys. The geometric mean employs multiplication and roots. The arithmetic mean is used while the growth depends upon addition. Also, study the formula and the manner to use it.
Perhaps some insight is provided by the graph in the right. For instance, in the easy function GeoMean" is provided to figure out the geometric mean of a number of data. This easy example can be achieved in your head. Look carefully at the diagram to learn what is given.
Aybeesee's height could be the short side of a single baby triangle and also the lengthy side of the other baby triangle. The altitude is additionally the lengthy side of the bottom triangle, and the more compact piece of the hypotenuse may be the brief side.
On occasion, you might need to figure out the mean of the range of numbers. The data ought to be divided into 4 different types. Nominal category utilizes some labels. Negative numbers could cause imaginary results based on how many negative numbers are really in a set.
In case you are multiplying eight numbers with each other, then you'll take the eighth root. Do not forget that the capital PI symbol method to multiply a number of numbers. For instance, for the item of two numbers, we'd take the square root. The AM-GM for just two positive numbers are sometimes a beneficial tool in examining some optimization difficulties.
Two matrices with precisely the same dimensions may be added utilizing the procedure for matrix addition. A matrix is just a rectangular or square selection of numbers. An echelon matrix is utilized to solve a method of linear equations. Nonsingular matrix is, in addition, called Invertible Matrix.
An universe is made within a second. In a sequence of numbers, the following expression within the series is figured by means of a formula, which uses previous expressions within the similar series. A limit test for divergence is actually a convergence test that's based upon the truth that the conditions of the convergent series needs to have a limit of zero. Similarly according to a lot of theologists the Pi collection, the trigonometric collection, the exponential collection etc represent each a singular facet of the truth that nature beholds.
To begin with, the fundamental formulas are listed, that can help you solve problems. Find a lot of fractional exponent issues and begin solving. Using graphical techniques to figure out the mathematical troubles. A way of solving a method of linear equations.
Mathematical induction is employed to prove complex troubles. Analytical methods are utilized to figure out the problems by assistance from algebraic and numeric methods.
Algorithm is a basic step by step to reach the solution of any issue. Graphic calculators are utilized to solve an issue graphically. Here may be the work for this particular equation. Alternate strategy to answer the n power problem is provided below.
Relative minimum is actually a point within the graph, which is at the bottom point for that specific section. Vertical shrinking of the geometrical figure is known as vertical compression. There is a variety of factors which go into calculating a yield. Based on this theorem, there's always a minumum of one absolute maximum and one absolute minimum for absolutely any continuous function on a closed interval.
There's a specific formula you can employ if you want to change the base of the logarithmic function. The y coordinate of the point is normally known as the ordinate. This kind of arbitrary point is known as focus of the parabola. Unless an explanation isn't proved correct for an expression, it's always a field of examination and debate.
Relative maximum is just a point within the graph, which is at the maximal point for that specific section. The highest point of the role or relation over the whole domain is known as absolute maximum. It's used to try whether a relation is really a function.
Mathematical expressions are derived for quite a few classes of reactions. Functions involving absolute value are likewise a great case of piecewise functions. The aforementioned examples offer some insight into the complete process of simplifying exponents. There is a variety of factors contributing to the infrequent usage of declarative languages.
Geometric mean is really a method of discovering the average of specific pack of numbers. Mathematically, a scalar is supposed to be any actual number or some quantity that may be measured using an individual actual number. The response lies within this period, where in fact the initial quarks and anti-quarks were formed.
While this primary formula is simple, there are several variables that could factor in to this formula. The measure of the closeness of the value to the true value of the result is known as accuracy. The independent quantity within an algebraic expression is known as variable. The magnitude could be the absolute value of the quantity, there are numerous scientific calculators that can work these out for you – even one as an app on the iPad.
Absolute value is, in addition, known as a mod value. Magnitude is actually a value, and it could never become a negative number.
Or how not to get a date, ever – but seriously here are a few other hyperbolic function properties. The hyperbolic secant arises within the profile of the laminar jet. The hyperbolic functions might be defined concerning the legs of the appropriate triangle covering this sector. A trigonometric function is really a periodic function, however a hyperbolic function isn't so. Basically you're supposed in order to differentiate any function. Hyperbolic functions could be differentiated and integrated. Since the hyperbolic functions are defined regarding the pure exponential function, it's not surprising that their inverses might be expressed concerning the organic logarithm function. Now think about the hyperbolic functions.
Recall the inverse of the pure exponential function may be the pure logarithm function. Rainbow phenomena could also be seen within the droplets generated by lawn sprinklers and hose nozzles, or any additional wellspring of water droplets.The function is a whole analytical use of that is described over the entire complex plane and doesn't have branch cuts and branch points. A function with a bounded selection. An unit length ought to be chosen freely. In the hyperbolic geometry it truly is allowable for at least one line to be parallel to the very first (meaning the parallel lines won't ever meet the very first, however far they're extended).
If you have trouble watching the above video it might be due to some stupid region locking, the same reason Netflix block proxies and stop me watching my favorite sci-fi shows !
This one involves utilizing the slope of the function at 0, just how the sine and also the cosine did. The very first and second derivative tests are generally utilised to get the absolute maximum of the function. Based on this theorem, there's always a minumum of one absolute maximum and one absolute minimum for absolutely any continuous function on a closed interval.
The absolute most striking distinction is that the hyperbolic functions aren't periodic. The hyperbolic functions are defined regarding the organic exponential functionex. The most important utilization of these functions will be to integrate common and easy functions with less computation as well as the other utilization of these functions could be observed within the models of real-life difficulties. Just as the hyperbolic functions themselves could possibly be expressed regarding exponential functions, so their inverses could possibly be expressed regarding logarithms.
Analytical methods are utilized to answer the problems by the aid of algebraic and numeric methods. It's likewise called arithmetic sequence. Segment of the circle is any internal region of the circle, bounded by means of an arc or possibly a chord.
An integration at which bounds of integration has discontinuities within the graph. The hyperbolic functions might be expressed concerning exponentials. These hyperbolic identities may be verified. In the old times, mathematicians had a tough time locating the equation of the curve.
In a sequence of numbers, the following expression within the series is figured by means of a formula, which uses previous expressions within the identical series. Sometimes natural data are in a form of an asymptotic curve like Eq. Recall the first derivative is known as y prime and also the second derivative is known as y double-prime. So allow me to explain to why the all-natural log is the one that's all-natural for economics.
In the aforementioned applet, there's a pull-down menu at the very top to select which function you want to examine. Visualization isn't yet complete, however. So this is really typical of mathematics.
Its equations are usually given within the polar coordinates. Any equation that is certainly reflexive, symmetric, and transitive. The in the very first formula is actually a hyperbolic-angle and also a parameter. This isn't a very useful formula.
Digital data is discrete instead of continuous, and thus, the signal really needs to be sampled at fixed intervals. Additionally, It has got the automatic measurements you'd expect in addition to FFT. The Fourier series may be used to locate a function which will excite the greatest number of frequencies possible. The DFT, such as the Fourier collection, implies a periodic extension of the first function.
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As an example, planetary probes can generate loads of image data, also a big challenge is always to send large quantity of data back to Earth. Maxwell also proved mathematically, that this kind of phenomenon of the cloud contracting into a planet couldn't occur. It can't be done within the physical universe.
To fully grasp how Battery Life Saver works, it really is first required to comprehend how batteries work. There are several chemical reactions which take place within the battery. A battery is really a system that stores electricity by way of chemical changes within the battery. With Battery Life Saver, the battery is not going to slow down or die due to lead sulfate, the most familiar reason behind battery failure.
The only distinction is that, it gives the signal amplitude within the frequency domain, as the oscilloscope supplies the signal amplitude within the time domain. The 2 electrons within the P subshell, being within the outer subshell, combine more freely than both electrons within the S subshell. In fact it really is useless to do this for stationary signals. Although the logarithmic scale stipulates a broader array of frequencies, it doesn't offer the absolute financial value of the signal.
Each layer got two interfaces. The gradient is figured utilizing the derivative of the Gaussian filter. Thermocouple is just a system which uses conduction for a manner of heat transfer. This system is, in addition, known as the vector signal analyzer.
To realize more compression, the image's quality must be compromised. If there is loss of information involved within the data encoding process, the initial array of data symbols doesn't even need to be encoded as-is. It achieves compression by using a combination of quite a few different algorithms. Besides compression and archiving, it truly is additionally effective at achieving error recovery.
Due to the way the laser light is reached, it becomes highly focused and intense. The technology is extremely close but doesn't make use of a visible light. Lasers work as an outcome of resonant results. They are one of the most significant inventions developed during the 20th century.
Both substantial sensitivity and superior directionality can be accomplished using a huge telescope in the receiver end. The mid-frequency of the band is automatically tuned within the device, since the range changes. It always has 1 side that is certainly dark, precisely the same side. The output of the laser is actually a coherent electromagnetic field.
Upon examination, the testing wasn't done in agreement with any recognised specifications like SAEJ2185, but instead testing very similar to what was done by Don Plisko. In the last few ages there's been considerable interest in the growth of neural network based pattern recognition apparatus due to their capacity to classify data. It is even feasible to set up short-range optical data connections with no direct field of sight. Moreover, it does not result in interference between different data links, so it generally does not require a license to be operated, which is superior when it comes to data security, since it's more challenging to intercept a tightly collimated laser beam when compared to a radio link.
Isosceles Triangle is among the special sort of the triangle. Search for isosceles triangles.
Consequently, triangle ADB is really a 30-60-90 triangle. An ideal triangle could possibly be scalene or isosceles. An isosceles triangle got two sides which are congruent. An isosceles triangle got two equal sides.
The triangle inequality theorem states the sum of any 2 sides of the triangle has to be greater in relation to the length of the 3rd side. Inside this lesson, you are going to find out how an isosceles triangle's sides and angles allow it to be unique. Isosceles triangles got two equal sides. Since 2 sides are congruent, additionally, it means that both angles opposite those sides are congruent.
The distance between the opposite sides of the parallelogram is known as altitude of the parallelogram. To solve a triangle method to know all 3 sides and all 3 angles. It states the length of the side of the triangle is regularly less in relation to the sum of the lengths of both of the other sides. Though this picture looks like two squares stacked in addition to one another, it certainly is a Right Triangle.
Its equations are by and large given within the polar coordinates. To start with, a theorem is actually a statement that may be proved. Here is our very first theorem. This theorem proves that for triangles to be similar, it really is sufficient they be equi-angular in dimensions.
There are only two different kinds of isosceles triangle dependent on the measure of the angles. All appropriate angles are congruent. In the event the angles in a triangle are given as algebra (usually with regard to x), and you're asked to recognize the size of each and every angle, then you can definitely follow these 3 simple things to do to get every one of the angles. An acute angle is under a suitable angle.
The diagram in the appropriate shows a perfect triangle with representations for just two angles. The purpose of intersection of three altitudes of the triangle is known as orthocenter. The intersection point of the 3 medians of the triangle.
There are some great examples and sample problems to practice with online, check with any of the online colleges and Universities to pick up some courses and examples , you might need to hide your location using an online IP changer like this.
The longest chord of the circle is known as diameter. The sides of the geometrical figure are frequently known as dimensions. The perimeter of the circular figure.
Two matrices with exactly the same dimensions could be added utilizing the procedure for matrix addition. For instance, a hexagon could also be called 6-gon.
A matrix is really a rectangular or square selection of numbers. An echelon matrix is utilized to solve a method of linear equations. Non-singular matrix is, in addition, called the Invertible Matrix. A way of solving a method of linear equations.
There are a lot of people varieties of triangles on the planet of geometry. As a standard polyhedron, all of its own faces are equal, and every vertex has an identical degree. A normal right prism is one whose bases consist of right polygons Right Pyramid is just a pyramid where base is really a regular polygon and also the apex is directly in addition to the middle of the bottom of polygon.
The measure of every base angle within the triangle is 54 degrees. The distance between both bases of the prism is called the altitude of the prism. Just Take a set of opposite sides.
Thus a System of Equations could have several equations and lots of variables. In regards to trigonometry, you should have to know how to solve trigonometric equations. Solving for Three Simultaneous Equations The procedure is very much like solving for just two equations. To begin with, arrange all 3 equations in standard form.
Vedic Mathematics is an ancient kind of Mathematics that's spreading its wings across the educational systems and also the knowledge centers. Now we must solve these simultaneous equations. Up to now we've solved equations with just one unknown variable. Simultaneous equations may also be solved graphically.
Finite element way is easily available to a lot of disciplines and companies, mainly because of the power and very low expense of modern computers. Two equations with two unknowns don't always have an exceptional solution. Software technology in this period was really primitive rather than the machines we have today. This ancient kind of mathematics was born within the Vedic Age but the system in addition to the significance of the form was buried deep below the centuries and ages.
Learning trigonometry is really not a difficult topic since most folks think. Geometry is, in addition, near trigonometry and also the areas you must focus on include problems involving circles. We will likewise show that a method of simultaneous equations might be solved graphically. Equations are many times utilized to solve practical difficulties.
To begin writing the JS application, you first specify both numbers whose AGM you would like to compute. These examples assist you to understand the way to solve simultaneous equations utilizing the graphical method. Add or subtract both equations so the variable with precisely the same coefficient will cancel out. Now repeat the procedure, but just for the previous 2 equations.
The period of foresight is the way far you are able to make predictions in advance. Remember there are other 3 ratios you have to comprehend. Now a number of the scientific calculators are simpler to learn than many others. Aside from learning about the angles, you are required to understand the 3 ratios.
It really can be noted the answers are precisely the same utilizing the substitution method along with the elimination method to answer the exact same simultaneous equation. Only the very first solution works within the context of the issue as the formula is simply accurate for pieces of no less than a specific size and also the side length of the paper can't be negative. It then provides a general way of rapid multiplication as well as a special two finger method. They're the substitution method along with the elimination method.
As inconceivably massive variety of calculations got to analyze a huge structure, a computer is required. Start by multiplying both equations so one particular variable has an identical coefficient in both equations (ignoring sign). Some would say ability to do simple calculations within your head may be mental mathematics. The row addition operation doesn't change the worth of the determinant."
There is but one point both equations cross. For this particular pack of equations, there is but one mixture of values for and that can satisfy both. Both of These terms will cancel if added together, therefore we will bring the equations to get rid of y. To bring the equations, bring the left side expressions and the correct side expressions separately. Sometimes both equations must certainly be modified as a way to cancel a variable. | 677.169 | 1 |
Who is it for?
Course Description
This course will help teachers engage students in advanced math lessons and activities that encourage them to explore math principles that are complex and fun to use. The book contains an amazing collection of quirks, illustrations, problems and surprises that engage learners. The book's many problems use attention-getters, motivators, and enrichment activities that keep students exploring mathematical mysteries. Teachers will write an essay on learning theory and develop lessons or activities for classroom applications. As well as to help students use the book content to advance their mathematical inquiry, problem solving and creativity | 677.169 | 1 |
Understanding Pure Mathematics
4.11 - 1251 ratings - Source
A classic single-volume textbook, popular for its direct and straightforward approach. Understanding Pure Mathematics starts by filling the gap between GCSE and A Level and builds on this base for candidates taking either single-subject of double-subject A Level.Understanding Pure Mathematics starts by filling the gap between GCSE and A Level and builds on this base for candidates taking either single-subject of double-subject A Level.
Title
:
Understanding Pure Mathematics
Author
:
A. J. Sadler, D. W. S. Thorning
Publisher
:
Oxford University Press, USA - 1987
ISBN-13
:
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Beginning and Intermediate Algebra
4.11 - 1251 ratings - Source
The wide-ranging debate brought about by the calculus reform movement has had a significant impact on calculus textbooks. In response to many of the questions and concerns surrounding this debate, the authors have written a modern calculus textbook, intended for students majoring in mathematics, physics, chemistry, engineering and related fields. The text is written for the average student -- one who does not already know the subject, whose background is somewhat weak in spots, and who requires a significant motivation to study calculus.The authors follow a relatively standard order of presentation, while integrating technology and thought-provoking exercises throughout the text. Some minor changes have been made in the order of topics to reflect shifts in the importance of certain applications in engineering and science. This text also gives an early introduction to logarithms, exponentials and the trigonometric functions. Wherever practical, concepts are developed from graphical, numerical, and algebraic perspectives (the qRule of Threeq) to give students a full understanding of calculus. This text places a significant emphasis on problem solving and presents realistic applications, as well as open-ended problems.... function, 954 scientific notation on, 271 Scientific notation, 267-271, 300, 999
applications of, 267, 270 on calculator, 271 ... 967 Set(s) definition of, 34
intersection of definition of, 600-601 solving, 601-603 symbol for, 600 interval
notation for, anbsp;...
Title
:
Beginning and Intermediate Algebra
Author
:
Julie Miller, Molly O'Neill
Publisher
:
McGraw-Hill Science Engineering - 2006
ISBN-13
:
Continue
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Although Mathematica 7 is a highly advanced computational platform, the recipes in this book make it accessible to everyone -- whether you're working on high school algebra, simple graphs, PhD-level computation, financial analysis, or advanced engineering models.
Learn how to use Mathematica at a higher level with functional programming and pattern matching
Delve into the rich library of functions for string and structured text manipulation
Learn how to apply the tools to physics and engineering problems
Draw on Mathematica's access to physics, chemistry, and biology data
Get techniques for solving equations in computational finance
Learn how to use Mathematica for sophisticated image processing
Process music and audio as musical notes, analog waveforms, or digital sound samples
Sal Mangano has been developing software since the days Borland Turbo C and has worked with an eclectic mix of programming languages and technologies. Sal worked on many mission-critical applications, especially in the area of financial-trading applications. In his day job, he works mostly with mainstream languages like C++ and Java so he chooses to play with more interesting technology whenever he gets a chance.
Sal's two books (XSLT Cookbook and Math Mathematica Cookbook) may seem to be an odd pair of technologies for a single author but there is a common theme that reflects his view at what makes a language powerful. Both Mathematica and XSLT rest on the idea of pattern matching and transformation. They may use these patterns in different ways and transformations to achieve different ends but they are both good at what they do and interesting to program in for a common reason. Sal's passion for these languages and ideas comes through in both these cookbooks. He also likes to push technologies as far as they can go and into every nook and cranny of application. This is reflected in the wide mix of recipes he assembled for these books.
Sal has a Master's degree in Computer Science from Polytechnic University. 2010. Paperback. Condizione libro: New. 191mm x 43mm x 235mm. Paperback. "Mathematica Cookbook" helps you master the application's core principles by walking you through real-world problems. Ideal for browsing, this book includes recipes for work.Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 800 pages. 1.415. Codice libro della libreria 9780596520991
Descrizione libro Oand#8242;Reilly596520991 | 677.169 | 1 |
Calculus by and for Young People (Ages 7, Yes 7 and Up). By Donald Cohen (1989) teen-adult. "A description of how young people, Don and some mathematicians, solved problems which involve infinite series, infinite sequences, functions, graphs, algebra, +, - important mathematical ideas. Also available, the Worksheets (some say all you need)..."
3.
How Math Works: 100 Ways Parents and Kids Can Share the Wonders of Mathematics.By Carol Vonderman (Putnam, 192 pp, 1999) (Ages 12 +) "Fascinating explanations, activities, profiles of history's most noted mathematical thinkers, and experiments introduce young readers to the world of mathematics."
HOW-TO: SCIENCE
1.
Practical Electronics for Inventors. By Paul Scherz. (McGraw-Hill/TAB, 604 pp, 2000) HS-Adult. "This book gives you easy-to-use, hands-on instructions on how to turn your ideas into workable electrical gadgets. Hand drawn illustractions help this crystal-clear, learn-as-you go guide show you what a particular device does, what it looks like, how it compares with similar devices, and how it is used in applications. Includes the basic passive components: resistors, capacitors, inductors, transformers, as well as discrete passive circuits such as current limiting networks, voltage dividers, filter circuits. Topics also include diodes, transistors, integrated circuits, amplifiers, and integrated circuits."
2.
Homemade Lightning. By R.A. Ford. (McGraw-Hill/Tab, 257 pp, 2001) HS-Adult. "This book is perfect for beginning electrical experimenters or those with an interest in advanced electrostatics. You will find complete descriptions of several types of high-voltage generators, including a Van de Graaf generator, electroscopes, cold light, electric tornadoes, and much more."
MATHEMATICS – GENERAL
1.
The Enjoyment of Math. By Hans Rademacher and Otto Toeplitz (Dover, 216 pp, 1966/1990) Teen-Adult. "What is so special about the number 30? How many colors are needed to color a map? Do the prime numbers go on forever? Are there more whole numbers than even numbers? These and other mathematical puzzles are explored in this delightful book by two eminent mathematicians. Requiring no more background than plane geometry and elementary algebra, this book leads the reader into some of the most fundamental ideas of mathematics, the ideas that make the subject exciting and interesting. Explaining clearly how each problem has arisen and, in some cases, resolved, Hans Rademacher and Otto Toeplitz's deep curiosity for the subject and their outstanding pedagogical talents shine through."
2.
The Fourth Dimension: Toward a Geometry of Higher Reality. By Rudy Rucker (Houghton Mifflin, 228 pp, 1984) HS-Adult. "Superb! It will hurt your brain if you don't know what you're getting into. On the other hand, if you know what to expect from Science Fact based text then you should be extremely pleased. The Plato's cave story is exceptional, and the tale of Flatland and the contemplation of a 2-D creature seeing/fathoming a 3-D creature is thought provoking. MUST READ."
3.
From Zero to Infinity : What Makes Numbers Interesting. By Constance Reid (MAA, many editions) Teen-Adult. Interesting "A classic of popular mathematical literature (since 1955) that combines the mathematics and the history of number theory with descriptions of the mystique that has, on occasion, surrounded the numbers even among great mathematicians." | 677.169 | 1 |
GrafEq (pronounced 'graphic') is an intuitive, flexible, precise and robust program for producing graphs of implicit relations. GrafEq is designed to foster a strong visual understanding of mathematics by providing reliable graphing technology.
MaTris is a nice program for practicing the basic operations of arithmetic. The calculation method is preselectable. It includes simple counting exercises, addition with symbols, addition/subtraction, multiplication and division.
Learning mathematics can be a challenge for anyone. Math Flight can help you master it with three fun activities to choose from! With lots of graphics and sound effects, your interest in learning math should never decline.
This is an advanced expression and conversion calculator. Vast array of built-in functions, constants and confersion operations that can be extended with your own user-defined functions. Now with graphs.
GraphiCal is a programmable graphics calculator which lets you visualize expressions and formulas as graphs in a chart. Creates animated video clips from a sequence of graphs. Built-in functions (>50) include integration, root finding .. | 677.169 | 1 |
Popular in Mathematics (M)
Reviews for MATH302,Introduction to Mathematical Thinking Learn how to think the way mathematicians do a powerful cognitive process developed over thousands of years. About the Course NOTE: Coursera encountered difficulties in converting my course to run on the new platform. Working together, we have found a way to modify the course to circumvent the missing platform features, without losing too much of what made the course work. Completing that work will involve considerable time and effort, and I am unlikely to have much time to look at this until the summer. This means that the earliest Session 8 could run is Fall 2016. Please check back here in August. Sorry about this. Keith Devlin, 1/25/2016 (modified 4.21.2016) The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years. Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think insidethebox. In contrast, a key feature of mathematical thinking is thinking outsidethebox – a valuable ability in today's world. This course helps to develop that crucial way of thinking. The course is offered in two versions. The eightweeklong Basic Course is designed for people who want to develop or improve mathematicsbased, analytic thinking for professional or general life purposes. The tenweeklong Extended Course is aimed primarily at firstyear students at college or university who are thinking of majoring in mathematics or a mathematicallydependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course. Subtitles for all video lectures available in: Portuguese (provided by The Lemann Foundation), English Course Syllabus Instructor's welcome and introduction 1. Introductory material 2. Analysis of language – the logical combinators 3. Analysis of language – implication 4. Analysis of language – equivalence 5. Analysis of language – quantifiers 6. Working with quantifiers 7. Proofs 8. Proofs involving quantifiers 9. Elements of number theory 10. Beginning real analysis Recommended Background High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course. A good way to assess if your basic school background is adequate (even if currently rusty) is to glance at the topics in the book Adding It Up: Helping Children Learn Mathematics (free download), published by the US National Academies Press in 2001. Though aimed at K8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the TwentyFirst Century (which was the National Academies' aim in producing it). Suggested Readings There is one reading assignment at the start, providing some motivational background. There is a supplemental reading unit describing elementary set theory for students who are not familiar with the material. There is a course textbook, Introduction to Mathematical Thinking, by Keith Devlin, available at low cost (US base price $10.99) from Amazon, in hard copy and Kindle versions, but it is not required in order to complete the course. For general background on mathematics and its role in the modern world, take a look at the five week survey course on mathematics ("Mathematics: Making the Invisible Visible") Devlin gave at Stanford in fall 2012, available for free download from iTunes University (Stanford), and on YouTube (1, 2, 3, 4, 5), particularly the first halves of lectures 1 and 4. Course Format The Basic Course lasts for eight weeks, comprising ten lectures, each with a problembased work assignment (ungraded, designed for group work), a weekly Problem Set (machine graded), and weekly tutorials in which the instructor will go over some of the assignment and Problem Set questions from the previous week. The Extended Course consists of the Basic Course followed by a more intense two weeks exercise called Test Flight. Whereas the focus in the Basic Course is the development of mathematicallybased thinking skills for everyday life, the focus in Test Flight is on applying those skills to mathematics itself. FAQ Will I get a certificate after completing this class? The course does not carry Stanford credit. If you complete the Basic Course with more than a minimal aggregate mark, you will get a Statement of Accomplishment. If you go on to complete the Extended Course with more than a minimal mark, you will receive a Statement of Accomplishment with Distinction. What are the assignments for this class? At the end of each lecture, you will be given an assignment (as a downloadable PDF file, released at the same time as the lecture) that is intended to guide understanding of what you have learned. Worked solutions to problems from the assignments will be described the following week in a video tutorial session given by the instructor. Using the worked solutions as guidance, together with input from other students, you will selfgrade your assignment work for correctness. The assignments are for understanding and development, not for grade points. You are strongly encouraged to discuss your work with others before, during, and after the selfgrading process. These assignments (and the selfgrading) are the real heart of the course. The only way to learn how to think mathematically is to keep trying to do so, comparing your performance to that of an expert and discussing the issues with fellow students. Is there a final exam for this course? No. The Test Flight exercise in the final two weeks of the Extended Course is built around a Problem Set similar to those used throughout the course, and your submission will be peer evaluated by other students, but the focus is on the process of evaluation itself, with the goal of developing the ability to judge mathematical arguments presented by others. Whilst not an exam, Test Flight is an intense and challenging capstone experience, and is designed to prepare students for further study of university level mathematics. How is this course graded? In the Basic Course, grades are awarded for the weekly Problem Sets, which are machine graded. The aggregate grade is provided in the cover note to the Statement of Accomplishment, with an explanation of its significance within the class. In the Extended Course, additional grades are awarded for a series of proof evaluation exercises and for the Test Flight Problem Set (peer evaluated). The aggregate grade is provided in the cover note to the Statement of Accomplishment with Distinction, with an explanation of its significance within the class.
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Reviews for MATH BIA VERY BRIEF LINEAR ALGEBRA REVIEW 5485 Introduction to Mathematical Biophysics Fall 2006 Introduction Linear Algebra also known as matrix theory is an important el ement of all branches of physics and mathematics Very often in this course we study of the shapes and the symmetries of molecules Motion of 3D space which leave molecules rigid can be described by matricesi Brie y mentioned in these notes will be quantum mechanics where matrices and their eigenvalues have an essential ro elabi These tools make multiplication of matrices very easy and they work with complex numbers The main difference between Maple and Matlab is that Maple can work symbolically that is you can use letter as well as numbers for entries When using numbers Matlab is often fasteri Below we give a review of a few basic ideas that will be used in the course 2 1 A is a matrix with 2 rows and 2 columns ie a 2 X 2 matrix A matrix with m rows and n columns is called an m X n matrix A matrix with the same number of rows and columns is called a square matrix 3 X 3 square matrix Matrices Example 3 1 7 B 71 2 0 0 1 5 3 X 2 matrix 2 0 C 79 10 1 14 A 1 X 1 matrix is the same as a number or scalar 3 Vectors Matrices with 1 row are called row vectors and matrices with 1 column are called column vectors are column VeCtOI Si 2 are row vectors Usually we will assume vectors are column vectors A row vector can be converted into a column vector or vice versa by the transpose operation Which changes rows to columns Example 1 1 71 0t 71 0 Complex Numbers Complex numbers can be used in matrices A number 2 a bi Where a and b are real numbers and Where i fl is called a complex number The number a is called the real part a 992 and b the imaginary part b 32 If 2 a bi Where a and b are real numbers then the complex conjugate of 2 is E a 7 bi The number 2 is real if b 0 or equivalently 2 2 Basic operations With matrices are 2 0 addition 0 scalar multiplication o multiplication Matrix Addition Add matrices by adding corresponding entries 1 11 2112 1 Scalar Multiplication Every matrix entry is multiplied by the scalar 2i2 1 4i 2 fl 3 722 62 Matrix Multiplication Multiplication AB can be done only if the number of columns of A is the same as the number of rows of B Each entry of the product is the dot product of a row of the rst matrix With a column of the second 2 13 1 211032 8 7112 0 7111022 3 3 11 2 311012 5 1 3 1 71 0 1 0 1 1 2 g 2 1 0 3 0 1 211133 2711230 201 31 1101131710210100 11 211103 2711200 201 01 12 0 35 4 71 1 3 0 5 An important thing to remember about matrix multiplication is that it is not commutative in general AB BA For example lt11gtlt19gt lt 1gt 1 0 1 1 7 1 1 1 1 0 1 7 1 2 Other operations are 0 conjugation o transpose 0 adj oint Conjugate 1f Z is a matrix then the matrix conjugate is formed by taking the complex conjugate of each entry ExampleLet 71i 2 71 2 1 0 12H ng 114 212 ABi Where 1 2 1 1 3 and 1 0 311 1 The matrix A is the real part of Z and the matrix B is the imaginary part of Z The conjugate of Z is 7 1 7 i 2 Z l3 i 7 ml or using the real and imaginary parts of the matrix7 ZA7Bi Properties of the matrix congugate BX ABXE Transpose The transpose of a matrix A7 At7 is obtained by changing rows to columns or equivalently7 changing columns to rows1 Sometimes the transpose is denoted A rather than Ah 2 1 71 2 0 71 A 0 1 2 AL 1 1 71 0 1 71 2 1 Properties of transpose AB B A AB A B 4 Adjoint The Hermitz39zm transpose or odjoz39nt is the conjugate transpose given by 7t A Ai Example 1 127i3 2i 3 Example For the matrix Z given above 7 it 7 1 7 239 3 239 Z Z l 2 7 221 Properties of Adjoiut ABY BA AB A B a z Dot product Let 1 b and w y be two vectors then the dot product is c 2 given by z vwv w a b c y azbyc2 2 1 z 2 Let 1 1 and w 1 7i then the Hermz39tz39zm dot product is given by i 3 ltvwgt vw iw 1 7 i21 1 7 7i3 3 7 6 it The length of a vector lvl is given by 2 l1 lt1 vgtl Two vectors 1 and w are orthogonal or perpendicular if ltvwgt 0 In general for real vectors ltvwgt cost Where 9 is the angle between the vectors Matrix multiplication and dot product An m X n matrix can be considered as a list of n X 1 column vectors Alv17m7vnl and the transpose as a list of row vectors At B i wt is a k X in matrix given as row vectors7 then wlvl wlvn BA wkvl wkvn is a matrix of dot products Symmetric Matrices A matrix B is symmetric if B Bt Example 3 1 i 2 1 i 0 75 2 75 2 is symmetric A matrix B is self adjoint or Hermitian symmetric if Bquot B Example 2 1i 2i l7i 3 5 72i 5 is self adjoint Determinant The determinant of a 2 X 2 matrix is given by det Z Z Z ad7bc Fora3gtlt3matrix7 a b c detd e f a 7b d cdz g h k g g This is called expansion by minors Likewise the determinant is de ned for any square matrix Properties of determinant detAB detA detB detA 0 implies A71 exists Vector Cross Product a z 122 7 cy b X y 7a2 01 c 2 ay 7 121 b2 7 cyi 7 a2 7 czj ay 7 bzk Where 1 0 0 10 j1 k0 0 0 1 The formula for cross product is often remembered by pretending that i7 j and k are numbers and Writing a z i j k b X y det a b c c z z y 2 Identity Matrix The matrix 1 denotes a square matrix Whose entries are aij were 1ifz39j av Oifiy ji The matrix 1 is called the unit or identity matrix Identity matrices come in different sizes 1 1 0 13 0 l0 1 0 Matrix inverse Let A be a square matrix H detA 0 there is an inverse A71 such that AilA AA 1 I The inverse of a 2 X 2 matrix is easy to nd If A an an 0421 0422 I2 OHO 0 0 1 then A71 l a22 a12 a 7amp21 all Where a anagg 7 algagl is the determinant of A1 Eigenvalue and Eigenvectors The scalar A a real or complex number7 is an eigenvalue of a matrix A corresponding to an eigenvector v f 0 if Av Av 11 11113111 The eigenvalue is 37 and an eigenvector is 11 1 Note that 22t is also an 1 311 1321 111 21321 xample 2 The eigenvalue is i and an eigenvector is 17 filti Example 3 0 71 1 7 7239 7 1 1 0 239 1 1 239 The eigenvalue is 7i 7 and an eigenvector is Lil i Note that the equation in example 3 is the conjugate on the one in example 21 Also note that a matrix With real entries can have complex eigenvalues and eigenvectorsi Eigenvalues of self adjoint matrices are real This fact is essential in many areas of mathematics and is also a key fact in the mathematical formulation of quantum mechanics Here is a proof If A is self adjoint and A1 Av then taking the adjoint of both sides gives v A Xvi Multiplying the first equation on the left by 1 and the second on the right by 1 gives 7 v Av A U Q AM Since v f 0 we have X A and so A is real A consequence is that the eigenvalues of a symmetric real matrix are real Rotation matrices in dimension 2 rotation of a column vector by an angle 9 counterclockwise is given by multiplying on the left by the matrix cos 9 7 sin 9 sin 9 cos 9 in dimension 3 rotations about the three axes are given by 0 Rotation an angle 9 about the z axis 1 0 0 R109 0 3086 7sint9 0 sin 9 cos 9 o Rotation an angle 9 about the y axis cos 9 0 sint9 Ry9 0 l 0 7 sin 9 0 cos 9 o Rotation an angle 9 about the 2 axis cos 9 7 sin 9 0 R209 sint9 3086 0 0 0 l Matrices can be found for rotation of any angle about axis7 Where an axis is given by a direction vector of length onei A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2006 Introduction Linear Algebra also known as matrix theory is an important element of all branches of physics and mathemat ics Very often in this course we study of the shapes and the symmetries of molecules Motion of 3D space which leave molecules rigid can be described by matrices Brie y mentioned in these notes will be quantum mechanics where matrices and their eigen values have an essential rolelab These tools make mul tiplication of matrices very easy and they 2 work with complex numbers The main dif ference between Maple and Matlab is that Maple can work symbolically that is you can use letter as well as numbers for en tries When using numbers M atlab is often faster Below we give a review of a few basic ideas that will be used in the course Matrices Example 2 1 A l 1 l A is a matrix with 2 rows and 2 columns ie a 2 X 2 matrix A matrix with m rows and 71 columns is called an m X 71 matrix A matrix with the same number of rows and columns is called a square matrix 3 X 3 square matrix 317 B 120 015 3 X 2 matrix 2 0 C 9 1O 1 14 A 1 X 1 matrix is the same as a number or scalar 3 3 Vectors Matrices with 1 row are called row vectors and matrices with 1 column are called column vectors are column VGCtOI S C21 D321 are row vectors Usually we will assume vec tors are column vectors A row vector can be converted into a column vector or vice versa by the transpose operation which changes rows to columns Example 1 1 1 of 1 0 Complex Numbers Complex numbers can be used in matrices A number 2 a bi Where a and b are real numbers and Where Z2 1 is called a complex number The number a is called the real part a 3 and b the imaginary part b 2 If 2 a bi Where a and b are real num bers then the complex conjugate of z is Z a bi The number 2 is real if b 0 or equivalently z 2 Basic operations With matrices are 0 addition 0 scalar multiplication o multiplication Matrix Addition Add matrices by adding corresponding entries iiHiSlW l 5 Scalar Multiplication Every matrix en try is multiplied by the scalar 221 42 22 1 3 2i 62 Matrix Multiplication Multiplication AB can be done only if the number of columns of A is the same as the number of rows of B Each entry of the product is the dot product of a row of the rst matrix with a column of the second 31 3111193322 21 1 2J F gt J a 3ll0l2 gt l3gtd O 211133 2711230 2 01 31 110113 1710210 10011 211103 2711200 201 01 12 0 35 4 l 1 3 0 5 6 An important thing to remember about matrix multiplication is that it is not com mutative in general AB y BA For exam mum lt12gtltsigtlt Other operations are c conjugation o transpose o adjoint Conjugate If Z is a matrix then the ma trix conjugate is formed by taking the com plex conjugate of each entry Example Let 71z 2 712 10 Z3 i2i3 12Z ABZ 12 Alwl Where 1 0 B l 1 2l The matrix A is the real part of Z and the matrix B is the imaginary part of Z The conjugate of Z is 1 z 2 Z3r m or using the real and imaginary parts of the matrix ZA m Properties of the matrix eongugate KE ABKE I ranspose The transpose of a matrix A At is obtained by Changing rows to columns or equivalently Changing columns to rows Sometimes the transpose is denoted A rather than At 1 1 12 At 1 gt D 2 A 0 1 Di O Fol Properttes of transpose ABY 13W ABt AtBt Adjoint The Hermtttcm transpose 01 ad jomt is the conjugate transpose given by Aquot At Example 1 12 z 3 2t 3 Example For the matrix Z given above 7771 t3t Z Z 2 22 Properttes of Adjotnt AB BA AB AB Dot product Let 1 b and w x y be two vectors then the dot product 2 is given by vwvtw a b C axbycz z 1Z 2 Let U 1 and w 1 Z then Z 3 the Hermitian dot product is given by ltvwgt vw Ww 1 i211 03 3 ii The length of a vector M is given by M2 lt1 vgt Two vectors 1 and w are orthogonal or perpendicular if U wgt 0 In general for real vectors ltvwgt cos6 Where 8 is the angle between the vectors Matrix multiplication and dot prod uct An m X 71 matrix can be considered as a list of n X 1 column vectors A v1vn and the transpose as a list of row vectors it W lvzl If B wt is a k X m matrix given as row vectors then w1v1 w1vn BA s s wkU1 when is a matrix of dot products Symmetric Matrices A matrix Bis sym metric if B Bt Example 3 1i 2 1i 0 5 l2 5 2 is symmetric A matrix B is self adjoint 0r Hermitian symmetric if 13 13 Example 2 1i 2i 1 i 3 5 2i 5 4 is self adjoint Determinant The determinant of a 2 X 2 matrix is given by ab ab detL d Cdad bc Fora3gtlt3matrix abc detldefla2 bd cdz gym 9 9 This is called expansion by minors Like wise the determinant is de ned for any square | 677.169 | 1 |
A First Course in Mathematical Analysis (2nd Revised edition)
Description
Mathematical Analysis (often called Advanced Calculus) is generally found by students to be one of their hardest courses in Mathematics. This text uses the so-called sequential approach to continuity, differentiability and integration to make it easier to understand the subject.Topics that are generally glossed over in the standard Calculus courses are given careful study here. For example, what exactly is a 'continuous' function? And how exactly can one give a careful definition of 'integral'? The latter question is often one of the mysterious points in a Calculus course - and it is quite difficult to give a rigorous treatment of integration! The text has a large number of diagrams and helpful margin notes; and uses many graded examples and exercises, often with complete solutions, to guide students through the tricky points. It is suitable for self-study or use in parallel with a standard university course on the subject. | 677.169 | 1 |
Introduction to Chaos: Analysis and Mathematics of the Phenomenon
Description
Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena focuses on explaining the fundamentals of the subject by studying examples from one-dimensional maps and simple differential equations. The book includes numerous line diagrams and computer graphics as well as problems and solutions to test readers' understanding. The book is written primarily for advanced undergraduate students in science yet postgraduate students and researchers in mathematics, physics, and other areas of science will also find the book useful. | 677.169 | 1 |
Math Word Problems For Dummies by Mary Jane Sterling
Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or level of problem.
From learning math lingo and performing operations to calculating formulas and writing equations, you'll get all the skills you need to succeed! | 677.169 | 1 |
MATH 2203 Flashcards
MATH 2203 Advice
MATH 2203 Documents
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MIDTERM 3 SOLUTIONS
(CHAPTER 4) MATH 141 FALL 2011 KUNIYUKI 150 POINTS TOTAL: 27 FOR PART 1, AND 123 FOR PART 2 Show all work, simplify as appropriate, and use "good form and procedure" (as in class). Box in your final answers! No notes or books allowed.
(Exercises for Chapter 4: Introduction to Trigonometry) E.4.1
CHAPTER 4: Introduction to Trigonometry
(A) means "refer to Part A," (B) means "refer to Part B," etc. (Calculator) means "use a calculator." Otherwise, do not use a calculator. Write units in
S. F. Ellermeyer
MATH 2203 Exam 1 (Version 2) Solutions September 15, 2010 Name
Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In other words, you must know what you are
S. F. Ellermeyer
MATH 2203 Exam 2 (Version 1) Solutions October 15, 2010 Name
Instructions. Your work on this exam will be graded according to two criteria: mathematical correctness and clarity of presentation. In other words, you must know what you are d
MATH 141 HW INSTRUCTIONS:
SPRING 2012 KUNIYUKI You do not have to do any problems indicated as "ADDITIONAL PROBLEMS." Write your name and Math 141 in the upper right-hand corner of your first sheet (or on a cover sheet). Use your own paper. If you turn in
Math 141
Name: _
FINALMIDTERM 4 SOLUTIONS
(CHAPTERS 5 AND 6) MATH 141 FALL 2011 KUNIYUKI 150 POINTS TOTAL: 44 FOR PART 1, AND 106 FOR PART 2 Show all work, simplify as appropriate, and use "good form and procedure" (as in class). Box in your final answers! No notes or books al
SOLUTIONS TO THE FINAL(Section 6.1: The Law of Sines) 6.01
CHAPTER 6: ADDITIONAL TOPICS IN TRIG
SECTION 6.1: THE LAW OF SINES
PART A: THE SETUP AND THE LAW
The Law of Sines and the Law of Cosines will allow us to analyze and solve oblique
(i.e., non-right) triangles, as well a | 677.169 | 1 |
TracMath is a freely downloadable tutorial (MS Windows) for the basic system of Trachtenberg Multiplication. This system teaches a set of algorithms that perform the same function as the times tables. When students ask "How does it work?" you can use it as an introduction to Algebra. Details at the site. | 677.169 | 1 |
Written for beginners and scholars, for students and teachers, for philosophers and engineers, What is Mathematics?, Second Edition is a sparkling collection of mathematical gems that offers an entertaining and accessible portrait of the mathematical world. Covering everything from natural numbers and the number system to geometrical constructions... more... | 677.169 | 1 |
Factoring & Solving Quadratic Equations
A collection of videos, activities and worksheets that are suitable for A Level Maths. This is part of an online GCSE to A Level Bridging Course. The topics covered are: Factoring and Solving of Quadratic Equations, Completing the Square, Solving Quadratics with Algebraic Fractions, Solve Simultaneous Equations that are linear with non linear | 677.169 | 1 |
Edexcel Level & Subject: AS Maths First teaching: September 2017 First exams: June 2018
Covering all the material needed for AS and A-level Year 1 for the 2017 Edexcel specification, this Student Book combines comprehensive and supportive explanations with plenty of practice to prepare students for the new linear examinations.
Written by expert authors, this Student Book will: • prepare you for linear assessment with chapter links and exam-style practice questions that draw on different areas of mathematics to help build synoptic understanding • help you take control of your learning with prior knowledge checks to assess readiness and end-of-chapter summaries that test understanding • support you through the course with detailed explanations, clear worked examples and plenty of practice on each topic with full workings shown for each answer • build your confidence in the key A-level skills of problem-solving, modelling, communicating mathematically and working with proofs • set maths in the real-world contexts that emphasise practical applications and develop your skills using technology in maths. | 677.169 | 1 |
Purplemath contains lessons, links, and homework tips, all designed to help the high school or college algebra student find...
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Purplemath contains lessons, links, and homework tips, all designed to help the high school or college algebra student find success. The "how to" lessons include tips and hints, point out common errors, and contain cross-links to related materials. The tone of the lessons is informal, and is directed toward students rather than Purplemath - Your Algebra Resource to your Bookmark Collection or Course ePortfolio
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'College Algebra is an introductory text for a college algebra survey course. The material is presented at a level intended...
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'College Algebra is an introductory text for a college algebra survey course. The material is presented at a level intended to prepare students for Calculus while also giving them relevant mathematical skills that can be used in other classes. The authors describe their approach as "Functions First," believing introducing functions first will help students understand new concepts more completely.Each section includes homework exercises, and the answers to most computational questions are included in the text (discussion questions are open-ended). Graphing calculators are used sparingly and only as a tool to enhance the Mathematics, not to replace it Algebra to your Bookmark Collection or Course ePortfolio
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'In writing Elementary Algebra, John Redden had simple but important aims:Lay a solid foundation in mathematics through...
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'In writing Elementary Algebra, John Redden had simple but important aims:Lay a solid foundation in mathematics through algebra, the basis of all mathematical modeling used in a variety of disciplinesGuide students from the basics to more advanced techniques in mathematics PEDAGOGICAL FEATURES:REAL WORLD APPLICATIONS:With diverse exercise sets, students have the opportunity to solve plenty of practice problems. Elementary Algebra has applications incorporated into each and every exercise set. John makes use of the classic "translating English sentences into mathematical statements" subsections in chapter 1 and as the text introduces new key terms.VIDEO EXAMPLES AND ACTIVITIES:Embedded video examples are present, while the importance of practice with pencil and paper is still stressed. This text respects the traditional approaches to algebra instruction while enhancing it with today's technology.OPEN AND MODULAR FORMAT:While algebra is one of the most diversely applied subjects, students find it to be a difficult hurdle in their education. With this in mind, John Redden wrote Elementary Algebra in a format that allows instructors to modify it and leverage their individual expertise and maximize student experience and success.MORE VALUE AT AN AFFORDABLE PRICE:Using the online text in conjunction with a printed version of the text could promote greater understanding (at a lower cost than most algebra texts Algebra to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Elementary Algebra
Select this link to open drop down to add material Elementary and Subtracting Complex Numbers to your Bookmark Collection or Course ePortfolio
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Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Algebra
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"This virtual manipulative allows you to solve simple linear equations through the use of a balance beam. Unit blocks...
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"This virtual manipulative allows you to solve simple linear equations through the use of a balance beam. Unit blocks (representing 1s) and X-boxes (for the unknown, X), are placed on the pans of a balance beam. Once the beam balances to represent the given linear equation, you can choose to perform any arithmetic operation, as long as you DO THE SAME THING TO BOTH SIDES, thus keeping the beam balanced. The goal, of course, is to get a single X-box on one side, with however many unit blocks needed for balance, thus giving the value of X contains Balancing Scales to your Bookmark Collection or Course ePortfolio
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This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more...
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This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and algebraic properties. New concepts include solving two-step equations and inequalities, graphing linear equations, simplifying algebraic expressions with exponents, i.e. monomials and polynomials, factoring, solving systems of equations, and using matrices to organize and interpret data I Online to your Bookmark Collection or Course ePortfolio
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A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video...
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A video instructional series on algebra for college and high school classrooms and adult learners; 26 half-hour video programs and coordinated books In this series, host Sol Garfunkel explains how algebra is used for solving real-world problems and clearly explains concepts that may baffle many students. Graphic illustrations and on-location examples help students connect mathematics to daily life. The series also has applications in geometry and calculus instruction In Simplest Terms to your Bookmark Collection or Course ePortfolio
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With this Bottomless Worksheet you can get endless practice on adding complex numbers (numbers including the imaginary number...
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With this Bottomless Worksheet you can get endless practice on adding complex numbers (numbers including the imaginary number i). At the click of a button, there are ten more problems for you to solve. A printed copy and answer sheet is Adding Complex Numbers to your Bookmark Collection or Course ePortfolio
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Get endless practice right here in dividing complex numbers! At the click of a button, this Bottomless Worksheet generates...
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Get endless practice right here in dividing complex numbers! At the click of a button, this Bottomless Worksheet generates ten more problems for you to solve. Plus, a printed copy or answer sheet is only a click away Dividing Complex Numbers to your Bookmark Collection or Course ePortfolio
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MATH 322 and vectors Linear independence Vector space Rank Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices and vectors Linear in 39 39 1 Matrices and vectors 0 An m X n matrix is an array with m rows and n columns It is typically written in the form all a12 a1n a21 a22 a2n A Z I 7 am1 am2 amn where i is the row index andj is the column index a A column vector is an m x 1 matrix Similarly a row vector is a 1 X n matrix 0 The entries aij of a matrix A may be real or complex Chapters 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Matrices a n ln Transposition Matrices and vectors continued 0 Examples 0 A 1 i J is a 2 X 2 square matrix with real entries 9 u 16 J is a column vector of A 0 0 3 7i complex entries 1 0 0 1 a BLO i 0 Jisa3gtlt3diagonal matrix with a An n X n diagonal matrix whose entries are all ones is called the n X n identity matrix 1 6 8 0 l 39S a 2 X 4 matrIX Wlth real entries Chapters 7 8 Linear Algebra C1 2 3 10 Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices and vectors Linear indepe Vectc Matrix addition and scalar multiplication Let A aij and B be two m x n matrices and let c be a scalar o The matrices A and B are equal if and only if they have the same entries ABltgtgtaijb forallij1 i m1 j n U7 0 The sum of A and B is the m x n matrix obtained by adding the entries of A to those of B Al B aij l bij o The product of A with the scalar c is the m x n matrix obtained by multiplying the entries of A by c CA caj Chapters 7 8 Linear Algebra Matrices and vectors Linear independence Vector Ra n k Mia n 0 Let A aij be an m x n matrix and B be an n x p matrix The product C A8 of A and B is an m X p matrix whose entries are obtained by multiplying each row of A with each column of B as follows n CU E aik bkj k1 1 2 1 2 3 10 ltgtExampesLetA 3 4andC1 6 8 0 o Is the product AC defined If so evaluate it 0 Same question with the product CA 0 What is the product of A with the third column vector of C Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrix multiplication continued o More examples 9 Consider the system of equations 3X1l 2X2 X3 4 X2 7X30 X1 l 4X2 6X3 10 Write this system in the form AX Y where A is a matrix and X and Y are two column vectors 12 56 A3 4 and B7 8 Calculate the products AB and BA 0 Let Chapters 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication 0 Matrix multiplication Rani Rules forimatrix addition and multiplication TranspQSItIon Matrices and vectors Linear indepe Vectc 3 Rules for matrix addition and multiplication o The rules for matrix addition and multiplication by a scalar are the same as the rules for addition and multiplication of real or complex numbers 0 In particular if A and B are matrices and C1 and C2 are scalars then ABBA A l B l CA l B l C C1ABC1A l C1B C1C2A C1Al C2A C1 C2 A C1 C2A whenever the above quantities make sense Chapters 7 8 Linear Algebra Definitions Matrix addition and scalar multiplication Matrix multiplication Rules for matrix addition and multiplication Transposition Matrices a nd Linear indep Vect Rank Rules for matrix addition and multiplication continued 0 The product of two matrices is associative and distributive ie ABC ABC 2 ABC ABCABAC A l BCAC l BC 0 However the product of two matrices is not commutative If A and B are two square matrices we typically have ABy BA o For two square matrices A and B the commutator of A and B is defined as A B 2 AB BA In general A7 8 32E 0 If A7 8 0 one says that the matrices A and B commute Chapters 7 8 Linear Algebra o The transpose of an m x n matrix A is the n x m matrix AT obtained from A by switching its rows and columns ie if A aij then AT aji 12310 0 Example Find the transpose of C 1 6 8 0 0 Some properties of transposition If A and B are matrices and c is a scalar then ABTATBT CATCAT T A BT BTAT AT 2 A whenever the above quantities make sense Matrices and vectors Linear independence De nitions Vector E a i i R3 71 k in ience o A linear combination of the n vectors 31 32 3 is an expression of the form c131 l C232 l l CH3 where the C s are scalars 0 A set of vectors 31 32 3n is linearly independent if the only way of having a linear combination of these vectors equal to zero is by choosing all of the coefficients equal to zero In other words 31 32 3n is linearly independent if and only if C131C232Cn3nOgtc1ZC2cn0 Matrices and w 39s Linear indepen Vector Evfmmioilag i Ra n k Limear i39h J i continued 0 Examples 0 Are the columns of the matrix A 16 i linearly independent 0 Same question with the columns of the matrix C 1 2 3 10 1 6 8 0 39 0 Same question with the rows of the matrix C defined above 0 A set that is not linearly independent is called linearly dependent 0 Can you find a condition on a set of n vectors which would guarantee that these vectors are linearly dependent Matrices and in Linear indepc Vector o A real or complex vector space is a non empty set V whose elements are called vectors and which is equipped with two operations called vector addition and multiplication by a scalar o The vector addition satisfies the following properties The sum of two vectors a E V and b E V is denoted by a b and is an element of V It is commutative a b b a for all a b E V It is associative a b c a b c for all a b c E V There exists a unique zero vector denoted by 0 such that for every vector 3 E V a 0 a 9 For each a E V there exists a unique vector a E V such that a a 0 Matrices and in Linear indepc Vector Vector f o The multiplication by a scalar satisfies the following properties 0 The multiplication of a vector 3 E V by a scalar or E R or or E C is denoted by aa and is an element of V 9 Multiplication by a scalar is distributive aabaaab oz aoza a for all ab Vand 0566R or C It is associative a a or 6 a for all a E V and 056 6 R or C GD Multiplying a vector by 1 gives back that vector ie 1 a a for all a E V Matrices an Linear indepel Vector o The span of set of vectors U 31 32 an is the set of all linear combinations of vectors in M It is denoted by Spanal7 327 7an or and is a subspace of V o A basis 8 of a subspace S of V is a set of vectors of S such that Q SpanB S B is a linearly independent set 0 Theorem If a basis 8 of a subspace S of V has n vectors then all other bases of 5 have exactly n vectors 0 The dimension of a vector space V or of a subspace S of V spanned by a finite number of vectors is the number of vectors in any of its bases T The row space of an m x n matrix A is the span of the row vectors of A If A has real entries the row space of A is a subspace of IR Similarly the column space of A is the span of the column vectors of A and is a subspace of Rm The rank of a matrix A is the dimension of its column space Theorem The dimensions of the row and column spaces of a matrix A are the same They are equal to the rank of A Example Check that the row and column spaces of C 1 2 3 10 l 1 6 8 0 dimension are vector subspaces and find their h atnces and vectors Linearind c i Vector The wank tlhero em Rank o The null space of an m x n matrix A NA is the set of vectors u such that Au 2 0 If A has real entries then NA is a subspace of IR 0 The rank theorem states that if A is an m x n matrix then rankA l dim n 0 Example Find the rank and the null space of the matrix 12 3 10 Cl16 8 ol39 Check that the rank theorem applies | 677.169 | 1 |
This textbook is intended to provide the requisite mathematical background for an introductory college course in general physics. The topics covered are: Numbers and Units; Experimental Error; Basic Algebra, Geometry and Trigonometry; Functions and Graphs; Vectors; The Slide Rule; and Physical Units and Conversions. Where possible, mathematical ideas are introduced via physical examples (mostly from mechanics), and the emphasis is on operational techniques rather than mathematical rigor. There are numerous worked examples as well as exercises with answers provided. (MM) | 677.169 | 1 |
Logarithmic Differentiation
Twelfth graders investigate logarithm differentiation. For this calculus lesson, 12th graders explore situations in which one would use logarithmic differentiation as an appropriate method of solution. Students should have already studied the chain rule.
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Literary usage of Graphs
Below you will find example usage of this term as found in modern and/or classical literature:
1.Learning about Our World and Our Past: Using the Tools & Resources of by Evelyn Hawkins (1999) "For example, students regularly encounter charts, graphs, and tables in daily
newspapers, on television programs, in school textbooks, and in many other ..."
2.Main Economic Indicators: Sources and Definitions (1996) "Explanatory notes on country graphs Part Two of Main Economic Indicators (MEI),
contains for each of the 27 OECD Member countries a page of graphs and a ..."
3.Handbook of Building Construction: Data for Architects, Designing and by George Albert Hool, Nathan Clarke Johnson (1920) "Column graphs. — Diagram 1 is a graph of the proposed column design stresses ...
Plotting Column graphs. — The best paper to use for plotting column graphs ..."
4.Proceedings of the Cambridge Philosophical Society by Cambridge Philosophical Society (1843) "A FAMILY OF CUBICAL graphs BY WT TUTTE Received 23 August 1946 We begin with some
... (But most of the theorems apply to other s-regular cubical graphs ..."
5.Central Stations by Terrell Croft (1917) "SECTION 6 LOAD graphs AND THEIR SIGNIFICANCE 118. A Load Graph, or as it is
sometimes called, a load curve, is merely a graphic record of the power loads ..."
6.Transactions of the Cambridge Philosophical Society by Cambridge Philosophical Society (1899) "V. Partitions of Numbers whose graphs -possess ... The enumeration of the
three-dimensional graphs of given weight (number of nodes), the numbers of nodes ..."
7.The Place of the Elementary Calculus in the Senior High-school Mathematics by Noah Bryan Rosenberger (1921) "THE MATERIAL The Use of graphs. The simplest vehicle for the presentation of the
theory to beginners is that of graphs. This simplicity is increased in the ..." | 677.169 | 1 |
Mathematics Quiz Book
Product Code : 9788122303636
Quick Overview
If you like to challenge your own mental capabilities or are hooked by the quiz bug, this book is for you. Just browse a few pages of this book and you will never want to leave it. This book is a must for anybody interested in mathematics and its brain teasers. The abstract and mysterious world of mathematics has great attraction for those who are interested in numbers and in exploring or decoding the language in which nature writes its laws. If you are inclined to play with brain teasers, or want to understand the riddles of numbers and here is an easy short cut to develop familiarity about the subject. Mathematics has its own world and with the help of quizzes you can gain entry into this world from backdoor, provided you show enough familiarity with its whereabouts. For some student mathematics is a constant headache and they develop fobias about it and for others it is a fascinating subject full of interesting facts. This book is helpful for both. Nature communicates in the language of mathematics and mathematicians try to decode that language using their great knowledge and insights. You'll never have a dull moment with this extraordinary compendium of fascinating facts, interesting information, and tantalizing trivia. If you're even remotely interested in quiz shows, this book will transport you to exhilarating heights. But you don't have to be a quiz buff to take pleasure from it. Filled with fascinating information on various topics of Mathematics, this slim book has answers to all your general queries about mathematics. This exciting trivia book is packed with enough quizzes, lists, and definitions to please even the most ardent trivia buff | 677.169 | 1 |
Handbook ofMathematical Functions
The Handbook ofMathematical Functions is a standard reference for those in physics, mathematics, engineering, and economics. We chose to carry this book because we use it ourselves. When ... as high as 20 places.
Look the book over! Click for the back cover, Table of Contents, or a pdf with sample pages.
- Paperback: 1046 pages
- Publisher: Dover Publications (1965)
- Language ...
Building Blocks ofMathematics Paperbacks- Addition
... boring? Not the Building Blocks ofMathematics! With Building Blocks ofMathematics: Addition, your kids will be submersed in a whimsical, illustrated world that will teach them about the principles of addition. Kids love comics ...
Building Blocks ofMathematics Paperbacks- Division
... much more fun than World Book's Building Blocks ofMathematics. With Build Blocks ofMathematics: Division, your kids will learn about division through the use of comics, the same medium they use to read ... Building Blocks are available in paperback, you can take Math wherever you want!
Building Blocks of Math, About the Illustrator:
Building Blocks ofMathematics Paperbacks- Fractions
Get the Full Story of Fractions with Building Blocks: Fractions!
With Building Blocks ofMathematics: Fractions, World Book gives fractions the full attention they deserve! World Book developed the Building ...
Building Blocks ofMathematics Paperbacks- Multiplication
Children and visual learners of all ages will just love World Book's Building Blocks ofMathematics: Multiplication! This great kids' math book uses the same visual elements found in comic books and graphic novels to teach your children the basic principles of multiplication! Have ...
Building Blocks ofMathematics Paperbacks- Numbers
... Blocks ofMathematics Series uses colorful characters and comic book style storytelling to teach your children basic mathematic principles. With this fun math book, Building Blocks: Numbers, your kids will learn about all of ...
Building Blocks ofMathematics Paperbacks- Subtraction
... develop students' conceptual understanding of the mathematical operations and teach them techniques for solving real-life math problems. Your children will have so much fun learning about subtraction with Building Blocks ofMathematics: Subtraction.
Building Blocks ofMathematics Paperback
World Book Building Blocks of Math now available in paperback! The fun and inventive series was designed in collaboration with elementary mathematics education experts.
Titles in this series:
Addition
Division
... | 677.169 | 1 |
ISBN 13: 9780007194940
Exam Practice - GCSE Maths
High GCSE grades are gained through a combination of good knowledge, good understanding and good exam technique. Exam Practice is all about exam technique and because it's written by the people who mark the exams, it really will help students improve their performance.
The content of Exam Practice GCSE Maths has been thoroughly updated to match the very latest GCSE exams and provides examples of exam questions across all the exam boards. With key skills highlighted, lots of questions to try and tips from top examiners, this book really will give students the confidence to do brilliantly…
This new edition includes Tony Buzan's revolutionary Mind Maps which will help to make study and revision easier.
"synopsis" may belong to another edition of this title.
About the Author:
Paul Metcalf if a freelance Consultant in Mathematics and is Principal Moderator for a major examining group. He was formerly a Head of Mathematics and a Deputy Headteacher. | 677.169 | 1 |
Know It All! Grades 9-12 Math
4.11 - 1251 ratings - Source
We Get Results We know what it takes to succeed in the classroom and on tests. This book includes strategies that are proven to improve student performance. We provide a€c content review, detailed lessons, and practice exercises modeled on the skills tested by standardized tests a€c proven test-taking skills and techniques such as how to solve word problems and answer open-ended questionsWe Get Results We know what it takes to succeed in the classroom and on tests. This book includes strategies that are proven to improve student performance.
Title
:
Know It All! Grades 9-12 Math
Author
:
James Flynn, Princeton Review (Firm)
Publisher
:
The Princeton Review - 2004
ISBN-13
:
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How to Write Mathematics
4.11 - 1251 ratings - Source
This classic guide contains four essays on writing mathematical books and papers at the research level and at the level of graduate texts. The authors are all well known for their writing skills, as well as their mathematical accomplishments. The first essay, by Steenrod, discusses writing books, either monographs or textbooks. He gives both general and specific advice, getting into such details as the need for a good introduction. The longest essay is by Halmos, and contains many of the pieces of his advice that are repeated even today: In order to say something well you must have something to say; write for someone; think about the alphabet. Halmos's advice is systematic and practical. Schiffer addresses the issue by examining four types of mathematical writing: research paper, monograph, survey, and textbook, and gives advice for each form of exposition. Dieudonne's contribution is mostly a commentary on the earlier essays, with clear statements of where he disagrees with his coauthors. The advice in this small book will be useful to mathematicians at all levels.Dieudonnea#39;s contribution is mostly a commentary on the earlier essays, with clear statements of where he disagrees with his coauthors. The advice in this small book will be useful to mathematicians at all levels.
Title
:
How to Write Mathematics
Author
:
Norman Earl Steenrod
Publisher
:
American Mathematical Soc. - 1973-12-31
ISBN-13
:
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Maths the Basic Skills Curriculum Edition - Student Book (E3-L2) (Levels 1 and 2 and 3) (Paperback). These resources have been designed specifically for the Adult Numeracy Curriculum, covering Entry L...
World of Books was founded in 2005, recycling books sold to us through charities either directly or indirectly. Practice in the Basic Skills (9) - Maths Book 4: Maths Bk.4. While we do our best to pro...
Title: Practice in the Basic Skills (9) - Maths Book 4: Mathematics Bk. World of Books was founded in 2005, recycling books sold to us through charities either directly or indirectly. Practice in the ...
World of Books was founded in 2005, recycling books sold to us through charities either directly or indirectly. Practice in the Basic Skills (10) - Maths Book 5: Maths Bk.5. While we do our best to prMaths the Basic Skills Handling Data Workbook E1/E2 (Paperback). You can choose what method you would like your order sent via when going through the order process. EAN: 9780748783328. Binding: Paperb...
Maths the Basic Skills Measures, Shape & Space Workbook E1/E2: Measures, Shape and Space (Paperback). You can choose what method you would like your order sent via when going through the order process...
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World of Books was founded in 2005, recycling books sold to us through charities either directly or indirectly. Practice in the Basic Skills: Maths. While we do our best to provide good quality books ...
World of Books was founded in 2005, recycling books sold to us through charities either directly or indirectly. Practice in the Basic Skills (7) - Maths Book 2: Maths Bk.2. While we do our best to pro...
The product supplied may vary slightly from the image shown. e.g. cover image may be updated to a new edition. This item is obtained direct from the respective publishers/suppliers. Publisher : Collin... | 677.169 | 1 |
Project #38933 - Week 9 Homework Discussion and Quiz
24 Mathematics Tutors Online
Discussion Item
Discuss characteristics of integer programming problems
Select one (1) of the following topics for your primary discussion posting:
·Explain how the applications of Integer programming differ from those of linear programming. Give specific instances in which you would use an integer programming model rather than an LP model. Provide real-world examples.
Identify any challenges you have in setting up an integer programming problem in Excel, and solving it with Solver. Explain exactly what the challenges are and why they are challenging. Identify resources that can help you with that.
Homework
Click the link above to submit your homework.
Complete the following problems from Chapter 5:
Problems 6, 10, 14, 20
Chapter problems will be attached upon acceptance.
Quiz #2
Scholar needs to contact me to discuss completion of the quiz assignment. | 677.169 | 1 |
Institutional Access
Description
This 4e includes new exercises in each chapter that provide practice in a technique immediately after discussion or example and encourage self-study. The early chapters are constructed around a sequence of mathematical topics, with a gradual progression into more advanced material. A final chapter discusses mathematical topics needed in the analysis of experimental data.
Key Features
Numerous examples and problems interspersed throughout the presentations
Each extensive chapter contains a preview and objectives
Includes topics not found in similar books, such as a review of general algebra and an introduction to group theory
Provides chemistry-specific instruction without the distraction of abstract concepts or theoretical issues in pure mathematics
Readership
New chemistry researchers; freshmen through juniors, seniors and graduates students enrolled in general through physical chemistry courses; especially students in lower- and upper-division honors chemistry courses
Table of Contents
Dedication
Preface
Chapter 1. Problem Solving and Numerical Mathematics
1.1 Problem Solving
1.2 Numbers and Measurements
1.3 Numerical Mathematical Operations
1.4 Units of Measurement
1.5 The Factor-Label Method
1.6 Measurements, Accuracy, and Significant Digits
Problems
Chapter 2. Mathematical Functions
2.1 Mathematical Functions in Physical Chemistry
2.2 Important Families of Functions
2.3 Generating Approximate Graphs
Problems
Chapter 3. Problem Solving and Symbolic Mathematics: Algebra
3.1 The Algebra of Real Scalar Variables
3.2 Coordinate Systems In Two Dimensions
3.3 Coordinate Systems in Three Dimensions
3.4 Imaginary and Complex Numbers
3.5 Problem Solving and Symbolic Mathematics
Problems
Chapter 4. Vectors and Vector Algebra
4.1 Vectors in Two Dimensions
4.2 Vectors in Three Dimensions
4.3 Physical Examples of Vector Products
Problems
Chapter 5. Problem Solving and the Solution of Algebraic Equations
5.1 Algebraic Methods for Solving One Equation with One Unknown
5.2 Numerical Solution of Algebraic Equations
5.3 A Brief Introduction to Mathematica
5.4 Simultaneous Equations: Two Equations with Two Unknowns
Problems
Chapter 6. Differential Calculus
6.1 The Tangent Line and the Derivative of a Function
6.2 Differentials
6.3 Some Useful Derivative Identities
6.4 Newton's Method
6.5 Higher-Order Derivatives
6.6 Maximum–Minimum Problems
6.7 Limiting Values of Functions
6.8 l'Hôpital's Rule
Chapter 7. Integral Calculus
7.1 The Antiderivative of a Function
7.1.1 Position, Velocity, and Acceleration
7.2 The Process of Integration
7.2.1 The Definite Integral as an Area</p
Details
Reviews
"The text is extremely clear and concise delivering exactly what the student needs to know in a pinch – nothing more, nothing less. It is an indispensable resource for any student of physical chemistry." --Gregory S. Engel, Harvard University
"Mathematics for Physical Chemistry is a comprehensive review of many useful mathematical topics...The book would be useful for anyone studying physical chemistry." --Daniel B. Lawson, University of Michigan-Dearborn
"The student will derive benefit from the clarity, and the professional from a concise compilation of techniques stressing application rather than theory.… Recommended." --John A. Wass, Scientific Computing and Instrumentation
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Quantitative Methods For Banking & Finance
Overview of Quantitative Methods For Banking & Finance
The study of quantitative methods has become essential for effective functioning of today's bankers and financial analysts; however, many of them have had no formal introduction to the subject. quantitative methods for banking & finance attempts to bridge the gap by providing useful concepts in mathematics and statistics like time value of money, differential calculus, statistical measures and probability theory. it also demonstrates the application of these concepts in the area of banking and finance with the help of illustrations. in addition, several box items are included in the book to provide a better insight into some of the interesting concepts. simplicity and lucidity are the key words throughout the book. | 677.169 | 1 |
This paper presents a teaching proposal for the introduction of elementary algebra in the middle school which aims the students' understanding of the necessity of algebraic language using, initially, the native language. We developed this proposal after a theoretical study about the existent approaches between the native language and mathematics, about the most frequent difficulties among students and about the historical evolution of algebraic notation. For a better understanding of the issue, we also defined what we'll call as teaching, communication, speech, language, algebra, algebraic thinking and algebraic language. The motivation to write about this subject arose because we believe that algebraic language, which is used by the teacher in elementary and high school, may be incomprehensible to the students, and algebra may be like a useless content to them. Assuming that these conditions may be coming from a lack of understanding of algebra practical applications and from the reason of using letters in mathematics, the teaching proposal presented here seeks to use the students' native language in solutions for problem-situations to show them the necessity of algebra and of algebraic | 677.169 | 1 |
Discovering / Dimensions Mathematics (Gr. 7-10)
In general, the newer Singapore material is visually more appealing and this series is no exception. Perhaps it's the full-color texts, or the graphic icons, or the occasional photograph or illustration, but there's really no doubt. Compared to New Elementary Math (the other Singapore series at the secondary level), visually, it's a no-brainer. However, there are other considerations and I suspect that choices will be made in both directions.
A comparison of the scope and sequence of the two courses shows surprisingly few differences, although they are there if you look closely. Both include integrated coverage of beginning and advanced algebra topics, geometry, and some trigonometry. DM lacks an in-depth review of arithmetic topics but includes matrices, radian measures, some additional data presentation forms (box & whisker plots; stem & leaf plots), and standard deviation. DM does not include as many advanced trig topics (see description for Discovering Additional Mathematics). There is also a difference in the degree of difficulty in terms of problems worked. DM has fewer challenging problems and exercises are slightly shorter, although they are divided as to problem difficulty. As in NEM, lessons in the text consist of explanations and worked examples, but in DM each worked example is followed by a Try It! challenge that allows students to see if they have understood. Each DM chapter concludes with review, compared to every 3-4 chapters in NEM. Additionally, there are some open-ended problems and journal writing questions. Sidebar information includes brief biographical information on notable mathematicians, discussion questions, recall points, and information tidbits. Textbooks have answers to the reviews and most exercises in the back, but the Textbook Teacher's Guide includes fully worked solutions to all problems along with weekly lesson plans that include helpful website resources. Textbooks vary from 150 to 215 pgs with most running in the 200 pg neighborhood. Teacher's Guides also vary in terms of page length but tend to be 150 pgs on average, both paperback.
Workbooks (with an accent color) for each level are designed to give students more practice. The publisher suggests that the student refer to the summary of important concepts in the text before tackling the workbook problems. The workbook questions for each chapter divided into basic, further, and challenging practice categories as well as enrichment. As with the textbook there is a complete answer key in the back of the workbook, but the Workbook Teacher's Edition shows worked solutions for every problem. Avg. 160 pgs, pb
Question Banks provide an unusual tool. Included are problems for each textbook chapter (number per chapter varies) labeled by difficulty level (low, medium, high). Each problem is shown with a worked solution. These problems allow the teacher to make selections for tests and assessments. The drawback is that each problem must be hand copied/typed as the solutions are "right there" and can take up a good part of each page. Avg. 390 pgs, pb ~ Janice | 677.169 | 1 |
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Venn Diagrams
Author(s):
Alfredo Jiminez
Alfredo Jiminez Penn State - Hazleton
This application offers a rich set of tutorials in a review/self-test format implemented for browsers with the free Flash player plugin.
Students are given exercises in which the number of elements in each region of a three set Venn diagram is given, and a question is asked for which the student must successfully interpret a set expression in terms of the Venn diagram to find an answer. | 677.169 | 1 |
A graphing calculator (GC), or graphic calculator, is a handheld mathematical tool that could plot graphs, solve simultaneous equations, and perform other arithmetic operations on variables in real-time. The tool is specifically designed for enhancing the mathematics/science-related learning as well as teaching experience at the school or college level.
The report analyzes and presents an overview of Graphing Calculator market worldwide. The report also provides a review of market trends, growth drivers, and strategic industry activities of major companies worldwide. The report further discusses about various types of Graphing Calculators including Numerical/non-CAS, and Symbolic/CAS calculators. In addition, companies operating in the Graphing Calculators arena worldwide including Casio Computer Co. Ltd., Desmos Inc., Hewlett-Packard Development Company LP, Liquidware, Orbit Research, Sharp Electronics Corporation, and Texas Instruments, Inc. are profiled.
Summary Strategic Defense Intelligence in its report "Global Defense Spends on Homeland Security, 2016 to 2025" has considered Homeland security (HLS) to include protection of a Global's civilians and ... | 677.169 | 1 |
Math WizardsClick to expand...
Do yourself a HUGE favor and get a TI-89 Platinum if you're allowed to use it in class. | 677.169 | 1 |
Proof and Proving in Mathematics Education
4.11 - 1251 ratings - Source
One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.Representational systems, learning, and problem solving in mathematics. ...
Studenta#39;s proof schemes: Results from exploratory studies. ... What can pre-
service teachers learn from interviewing high school students on proof and
proving? (Vol.
Title
:
Proof and Proving in Mathematics Education
Author
:
Gila Hanna, Michael de Villiers
Publisher
:
Springer Science & Business Media - 2012-06-14
ISBN-13
:
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TOPICSApplications of Quadratic EquationsInequalitiesInequalities in Two VariablesMatrices and DeterminantsSystems of Linear EquationsSystems of InequalitiesApp of Sys of Linear Equations and Inequalities LONG QUIZ 3Partial FractionsFunctions and Relations 1 (Defns and Operns)Functions and Relations 2 (Inverse Fxn and Graphs)Ratio, Proportion, and VariationCombinatorial Mathematics: PermutationCombinatorial Mathematics: Combination LONG QUIZ 4 FINAL EXAM TOPICSSystems of Equations and Inequalities 1Systems of Equations and Inequalities 2Systems of Equations and Inequalities 3Systems of Equations and Inequalities 4Systems of Equations and Inequalities 5 LONG QUIZ 3Relations Functions and Graphs 1Relations Functions and Graphs 2Polynomial Functions 1Polynomial Functions 2 LONG QUIZ 4Sequences and Series, Arithmetic SequencesHarmonic sequence, Geometric SequenceFactorial of a Number, Binomial Theorem LONG QUIZ 5 FINAL EXAM | 677.169 | 1 |
Elements of Set Theory
4.11 - 1251 ratings - Source
This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.
Title
:
Elements of Set Theory
Author
:
Herbert B. Enderton
Publisher
:
Gulf Professional Publishing - 1977
ISBN-13
:
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Mathematics for Queensland Yr 11 B Author: Kiddy Bolger
ISBN: 9780195508529 Format: Book with Other Items Number Of Pages: 488 Published: 15 September 2001 Country of Publication: AU Dimensions (cm): 25.5 x 19.000 Description: • Syllabus subject matter at the start of each chapter • Text written at appropriate reading level for senior students • Life-related applications • Historical notes • Clearly worked examples which allow students to study independently • Definitions and rules are highlighted • Numerous, well graded exercises • Class discussion topics • Appropriate use of technology using graphics calculator activities which are integrated throughout the text • Problem solving strategies are developed in all topics • Mathematical modeling is integrated throughout the text • Investigations to develop concepts • Statistics using real-life datasets • 'Words you need to know' section in each chapter • End of chapter review • End of chapter tests • End of chapter extension activities • Cumulative review exercises • An appendix to maintain knowledge and procedures • Glossary | 677.169 | 1 |
Arithmetic With Polynomials And Rational Expressions
Circles
HSG.C: Understand And Apply Theorems About Circles
HSG.C.1
Prove that all circles are similar.HSG.C.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
HSG.C.4
(+) Construct a tangent line from a point outside a given circle to the circle.
Find Arc Lengths And Areas Of Sectors Of Circles
HSG.C.5
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Conditional Probability And The Rules Of Probability
HSS.CP: Understand Independence And Conditional Probability And Use Them To Interpret Data
HSS.CP.1
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
HSS.CP.2
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
HSS.CP.3
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.
HSS.CP.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
HSS.CP.5
Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
Use The Rules Of Probability To Compute Probabilities Of Compound Events In A Uniform Probability Model
HSS.CP.6
Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
HSS.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
HSS.CP.8
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
HSS.CP.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Congruence
HSG.CO: Understand Congruence In Terms Of Rigid Motions
HSG.CO.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
HSG.CO.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
HSG.CO.8
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Experiment With Transformations In The Plane
HSG.CO.1
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
HSG.CO.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
HSG.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
HSG.CO.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Prove Geometric Theorems
HSG.CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
HSG.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
HSG.CO.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Make Geometric Constructions
HSG.CO.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
HSG.CO.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.HSA.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
HSA.CED.3
Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non- viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
HSA.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
Functions
8.F: Define, Evaluate, And Compare Functions.
8.F.1Geometric Measurement And Dimension
HSG.GMD: Explain Volume Formulas And Use Them To Solve Problems
HSG.GMD.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
HSG.GMD.2
(+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.
HSG.GMD.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ and
Class
Verify experimentally the properties of rotations, reflections, and translations:
8.G.1.a
Lines are taken to lines, and line segments to line segments of the same length.
8.G.1.b
Angles are taken to angles of the same measure.
8.G.1.c
Parallel lines are taken to parallel lines.
8.G.2
Understand
8.G.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.4
Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them.
8.G.5
Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
Understand And Apply The Pythagorean Theorem.
8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
8.G.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Making Inferences And Justifying Conclusions
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
HSS.IC.2
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
Make Inferences And Justify Conclusions From Sample Surveys, Experiments, And Observational Studies
HSS.IC.3
Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
HSS.IC.4
Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.
HSS.IC.5
Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
HSS.IC.6
Evaluate reports based on data.
Measurement And DataClassify Objects And Count The Number Of Objects In Each Category.
K.MD.3
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.3Tell And Write Time.
1.MD.3
Tell and write time in hours and half-hours using analog and digital clocksMeasureGeometric Measurement: Understand Concepts Of Area And Relate Area To Multiplication And To Addition.
3.MD.5
Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.5.a
A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
3.MD.5.b
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
3.MD.7.b
Multiply side lengths to find areas of rectangles with whole- number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
3.MD.7.c
Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
3.MD.7.d
Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
Ge
4.MD: Solve Problems Involving Measurement And Conversion Of Measurements From A Larger Unit To A Smaller Unit.
4.MD.1
Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two- column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
4.MD.2
4.MD.3
Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.
RepresentRecognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
4.MD.5.a
An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
4.MD.5.b
An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.7
Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.
5.MD: Convert Like Measurement Units Within A Given Measurement System.
5.MD.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.
Represent And Interpret Data.
5.MD.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.
Geometric Measurement: Understand Concepts Of Volume And Relate Volume To Multiplication And To Addition.
5.MD.3
Recognize volume as an attribute of solid figures and understand concepts of volume measurement.
5.MD.3.a
A cube with side length 1 unit, called a "unit cube," is said to have "one cubic unit" of volume, and can be used to measure volume.
5.MD.3.b
A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units.
Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.
5.MD.5.a
Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.
5.MD.5.b
Apply the formulas V=l×w×handV=b×h for rectangular prisms to find volumes of right rectangular prisms with whole- number edge lengths in the context of solving real world and mathematical problems.
5.MD.5.c
Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.
Modeling With Geometry
HSG.MG: Apply Geometric Concepts In Modeling Situations
HSG.MG.1
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).★
HSG.MG.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).★
HSG.MG.3
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).★
Number And Operations In Base Ten
K.NBT: Work With Numbers 11–19 To Gain Foundations For Place Value.
K.NBT.1
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Understand Place Value.
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <5
Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used2.NBT: Understand Place Value.
2.NBT.1
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.4
Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
Use written method. Understand that in adding or subtracting three- digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.8
Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties of operations.3
3.NBT: Use Place Value Understanding And Properties Of Operations To Perform Multi-Digit Arithmetic.4
3.NBT.1
Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
4.NBT.2
Read4.NBT.3
Use place value understanding to round multi-digit whole numbers to any place.
Use Place Value Understanding And Properties Of Operations To Perform Multi-Digit Arithmetic.: Understand The Place Value System.
5.NBT.1
Recogn
ExplainQuantities
HSN.Q: Reason Quantitatively And Use Units To Solve Problems.
HSN.Q .1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
HSN.Q .2
Define appropriate quantities for the purpose of descriptive modeling.
HSN.Q .3
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."1
6.RP.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
6.RP.3.a
Make tables of equivalent ratios relating quantities with whole- number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
6.RP.3.b
Solve6.RP.3.c
Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
7.RP.2.d
Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Reasoning With Equations And Inequalitiesa
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this formUnderstand
HSA.REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
HSA.REI.8
(+) Represent a system of linear equations as a single matrix equation in a vector variable.
HSA.REI.9
(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
RepresentThe Complex Number System
HSN.CN: Perform Arithmetic Operations With Complex Numbers.
HSN.CN.1
Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
HSN.CN.2
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
HSN.CN.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
Represent Complex Numbers And Their Operations On The Complex Plane.
HSN.CN.4
(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
The Number System
6.NS: CompApply div
Trigonometric Functions
HSF.TF: Extend The Domain Of Trigonometric Functions Using The Unit Circle
HSF.TF.1
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
HSF.TF.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
HSF.TF.3
(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
HSF.TF.4
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
HSF.TF.7
(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.★
Pro Perform Operations On Vectors.
HSN.VM.4
(+) Add and subtract vectors.
HSN.VM.4.a
Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
HSN.VM.4.b
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
HSN.VM.4.c
Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).
RepresentPerform Operations On Matrices And Use Matrices In Applications.
HSN.VM.6
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
HSN.VM.7
(+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
HSN.VM.8
(+) Add, subtract, and multiply matrices of appropriate dimensions.
HSN.VM.9
(+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
HSN.VM.10
(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
HSN.VM.11
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
HSN.VM.12
(+) Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. | 677.169 | 1 |
Solutions manual to accompany Elementary statistics
4.11 - 1251 ratings - Source
Succeed in statistics with ELEMENTARY STATISTICS! Including relevant examples, exercises, and applications, this textbook gives you the tools you need to get a good grade in your statistics course. Struggling with a specific concept? Log onto Personal Tutor with SMARTHINKING to get live, one-on-one online tutoring from a statistician who has a copy of the textbook. Video Skillbuilders and StatisticsNow (an online learning tool built around your individual progress that gives you a simple pre-test, and then focuses your learning experience on your studying needs) provide additional online support. Learning to use MINITAB, Excel, and the TI-83/84 graphing calculator is made easy with instructions included in relevant sections throughout the text.Learning to use MINITAB, Excel, and the TI-83/84 graphing calculator is made easy with instructions included in relevant sections throughout the text.
Title
:
Solutions manual to accompany Elementary statistics
Author
:
Robert Russell Johnson
Publisher
:
Brooks/Cole - 1984
ISBN-13
:
Continue
You Must CONTINUE and create a free account to access unlimited downloads & streaming | 677.169 | 1 |
Short Cuts in Math
4.11 - 1251 ratings - Source
qSHORT CUTS IN MATHq is a fast paced way to review or learn the basics of math in a non- academic method using a lot of short cuts to many problem solutions. It also shows the difference between the regular procedure and the short cut.This book makes it interesting to learn math.75 X 4 = 300 2. A studenta#39;s average of September and Octobera#39;s English test is
72%. If Septembera#39;s grade was 70% what was Octobera#39;s grade? Solution: Step 1
Total of Sept. and Oct. grades: 2 X 72 = 144 Step 2 Octobera#39;s grade = 144 a€" 70anbsp;...
Title
:
Short Cuts in Math
Author
:
Mounir Samaan
Publisher
:
Lulu.com - 2011-08
ISBN-13
:
Continue
You Must CONTINUE and create a free account to access unlimited downloads & streaming | 677.169 | 1 |
Easy Learning - GCSE Maths Revision Guide for Edexcel A: Higher
Easy Learning GCSE Maths Revision Guide for Edexcel A includes revision content with highlighted grade levels so that students know exactly which grade they are working at, making revising for GCSE Maths easy.
•Easy to use – clear and comprehensive structure and design
•Easy to revise – colour-coded to show grade level of content
•Easy to remember – concise information organised in memorable chunks
Together with the accompanying Easy Learning GCSE Maths Exam Practice Workbook, the two books provide complete revision coverage of the new GCSE Maths Edexcel A specification.
"synopsis" may belong to another edition of this title.
Review:
" They are the best we have seen for the new Edexcel specification, particularly as theya re fully graded." Paula Wright, Stage 4 Co-ordinator, City of London Academy
About the Author:
Keith Gordon is a highly experienced examiner and renowned author of various secondary Maths publications. | 677.169 | 1 |
Description
Mary Everest Boole (1832-1916) was born Mary Everest in England and spent her early years in France. She married mathematician George Boole. She was the author of several works on teaching and teaching mathematics in particular. This short book,…..
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A humorous book of how the author overcame depression with his admiration of nature. The author discusses the two principal methods of treating disease; the combative and the preventive, and goes on… [more..]
These 19 lessons provide initial instruction or intervention on linear equations and inequalities of two variables and functions. The first 4 lessons define those equations and their solutions,Great for children of all ages!Over 13 Unique Characters to play with more Goonsters on their way!!!Fantastic interactive puzzle game, because of its colorful atmosphere, fun characters and exciting… [more..]
*** Amazing 11 musical instruments inside *** PLAY MUSIC LIKE NEVER BEFORE The largest, most robust collection of fun & playful musical instruments on the App-Store. A lasting visual and audio… [more..] | 677.169 | 1 |
Showing 1 to 3 of 3
This was a great starter class for me, the classroom environment was one of a learning environment. Mrs.Ceithlm cares greatly for the students and wants us to excel. Excel is exactly what I did, I grasped a better understanding of math problems that I could have not finished at the beginning of the semester.
Course highlights:
Mrs.Ceithlm teaches MA 107 online and in class, I took the in class course. I felt like I was always learning and grasping new ideas. The homework and the cool downs helped me greatly. At the beginning of each class period we would go over any homework questions. That made me feel more comfortable in asking questions and needing help.
Hours per week:
9-11 hours
Advice for students:
Do your homework! Sometimes people don't do their homework, which only hurts them in the long run. The home work is about 5-10 points, so if you end up not doing your homework it is greatly going to effect your grade. Also, a graphing calculator would be very useful in this course, all that is required is scientific, but if you have the resources, buy a graphing calculator.
Course Term:Winter 2016
Professor:Cynthia Ceithlm
Course Required?Yes
Course Tags:Math-heavyGreat Intro to the SubjectMany Small Assignments
Apr 14, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
It's a great class to get a background in math, especially if you are continuing a career in math.
Course highlights:
I learned how to solve word problems, and get a better understanding. I learned quadratic formula and graphing them. | 677.169 | 1 |
Math 2412 Final Exam Review
Name_
Calculator allowed. One small note card is allowed.
Show your work for each item. Turn in your work on the final exam review items in order to earn up to 5 points extra credit.
Find all solutions of the equation.
Solve th
Section 7.3
Law of Sines and Law of Cosines
We use these laws to find angles and side lengths for triangles of any type (not just right
triangles).
Law of Sines
C
a,b,c are lengths of sides
a
A,B,C are angles (A is opposite a, etc.)
b
A
B
c
Law of Sines:
Showing 1 to 3 of 4
Pre-calculus is a great "vetting-ground" to help you decide if you're ready to dip your toes into higher math. However, unless you are extremely dedicated or naturally gifted in mathematics, I advise against taking Pre-calculus during an abbreviated summer semester (especially not online).
As with every math class, if you enjoy solving puzzles and building to your knowledge of the language of mathematics, it is fun to take.
Course highlights:
In this course you spend a good amount of time observing how trigonometry is translated into other concepts. You will learn how to convert a trigonometric expression from one type of function to another and learn how to graph and observe said function on different planes. My favorite part of this is rotating functions by an angle.
Hours per week:
12+ hours
Advice for students:
Take advantage of the practice problems after every chapter in your text book.
Course Term:Summer 2016
Professor:Muhammad Afane
Course Required?Yes
Course Tags:Math-heavyAlways Do the ReadingMany Small Assignments
Aug 03, 2016
| Would highly recommend.
This class was tough.
Course Overview:
An outstanding professor with an organized regimen that completely keeps the student on his toes, waiting to see how far his mind can exceed to gather up the information needed. I came in taking Math 1316 the semester before, which truly prepared me with the necessary skill set to excel in Precalculus.
Course highlights:
I learned beyond the math from the textbook. Professor gave us real life scenarios and engineering problems that we would similarly do in the future as engineers in various industries. Beyond the problems in arithmetic, Professor Moebes gave us amazing tips and strategies that we could utilize in the near future to succeed.
Hours per week:
12+ hours
Advice for students:
Make sure you know your trigonometric identities real well, because they will haunt and taunt if you do not. it is wise to consider knowing the trigonometric functions of angles within the circle and to also read ahead in the precal text book if you have got it already. Study whenever you can because this will help you in the long run. | 677.169 | 1 |
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Scientific computing, also known as computational science, uses computational methods to solve science and engineering problems. The modeling of natural systems using numerical simulation is an important area of focus within scientific computing. Learn how to use MATLAB for scientific computing applications. Resources include videos, examples, and documentation for numeric linear algebra, numerical integration, and other topics | 677.169 | 1 |
...
Show More with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It | 677.169 | 1 |
Swain Chemistry & Chemical Engineering Library
Polymath
Polymath educational version
POLYMATH is a computational system, which has been specifically created for educational or professional use. This software executes on XP, Vista, Windows 7, 8, 8.1 and 10. ( It operates on 32-bit or 64-bit systems.) The various POLYMATH programs allow the user to apply effective numerical analysis techniques during interactive problem solving on personal computers. Results are presented graphically for easy understanding and for incorporation into papers and reports. Engineers, mathematicians, scientists, students or anyone with a need to solve problems will appreciate the efficiency and speed of problem solution. | 677.169 | 1 |
Su questo libro:
This is a practical workbook on calculating the prices and returns on complex financial instruments. Designed for working professionals on Wall Street, this workbook is sprinkled with "fast facts" about the inner workings of the capital markets. Its lively and accessible style communicates math concepts accurately on the first reading, with plenty of exercises integrated into the text so you can test your understanding as you go. Depending on your personal knowledge base, you can work through each bite-sized chapter or skip a topic, knowing you have a quick reference.
Who needs this workbook? Professionals working on trading desks and students of money and capital markets, investments, financial institutions, and corporate finance. Excellent math supplement to traditional finance text books.
Appendices: Conventions used in the capital markets for settlement, day counts and bad dates; Newton-Raphson Search; Excel functions useful for capital markets math; regression analysis; the top 100+ formulas; a glossary; and solutions to all the exercises.
L'autore:
Dr. Norman Toy is one of Wall Street's best-known and most beloved instructors. Dr. Toy has taught financial math, capital markets, corporate finance and financial analysis at investment banks for over 30 years. His students range from beginning analysts to executives and directors of major financial institutions.
A member of the faculty of Columbia Graduate School of Business since 1969, he teaches advanced corporate finance, debt markets and investment strategies in the school's Executive MBA Program. His experience includes management posts at Columbia University's Health Sciences Division and its College of Physicians and Surgeons, and he was a director and principal of the financial counseling firm of Brownson, Rehmus & Foxworth, Inc.
Dr. Norman Toy was educated at the University of Florida, where he earned a BA in mathematics, and the Harvard Business School, where he earned an MBA and Doctorate in managerial economics (management science and operations research. Dr. Toy's research interests include capital markets, valuation and corporate finance and he is the author of articles on finance, marketing and statistics.
His books include Introduction to Financial Math and Capital Markets Math, both published by Adkins Matchett & Toy, which delivers practical, hands-on training to investment banks, financial institutions, corporations, asset managers and government agencies | 677.169 | 1 |
Algebra and Trigonometry, 3rd Edition
Using outside resources may lead to improper placement and ultimately course failure. We studied Inverses of Functions here; we remember that getting the inverse of a function is basically switching the x and y values, and the inverse of a function is symmetrical (a mirror image) around the line The same principles apply for the inverses of six trigonometric functions; the notation does look a little funny at first, but you'll get used to it.
The owner of this blog makes no representations as to the accuracy or completeness of any information on this site or found by following any link on this site. The Dictionary of Algebra, Arithmetic, and Trigonometry (Comprehensive Dictionary of Mathematics). Bhaskara II developed spherical trigonometry, and discovered many trigonometric results. Bhaskara II was the first to discover The Indian works were later translated and expanded in the classical Islamic world by Islamic mathematicians of mostly Persian and Arab descent, who enunciated a large number of theorems which freed the subject of trigonometry from dependence upon the complete quadrilateral, as was the case in Hellenistic mathematics due to the application of Menelaus' theorem Functions and Change: A Modeling Approach to College Algebra: 3rd (Third) edition. Using only a pencil, compass, and straightedge, students begin by drawing lines, bisecting angles, and reproducing segments. Later they do sophisticated constructions involving over a dozen steps-and are prompted to form their own generalizations read Algebra and Trigonometry, 3rd Edition online. Analytic trigonometry combines the use of a coordinate system, such as the Cartesian coordinate system used in analytic geometry, with algebraic manipulation of the various trigonometry functions to obtain formulas useful for scientific and engineering applications Elements of geometry and plane trigonometry. With an appendix, and copious notes and illustrations. Their work includes commentaries on the proofs of the arctan series in Yuktibhāṣā given in two papers, [76] [77] a commentary on the Yuktibhāṣā's proof of the sine and cosine series [78] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). [79] [80] The Kerala mathematicians included Narayana Pandit [ dubious – discuss ] (c. 1340–1400), who composed two works, an arithmetical treatise, Ganita Kaumudi, and an algebraic treatise, Bijganita Vatamsa An Elementary Treatise on Plane Trigonometry; With Numerous Examples and Applications.
Atkins is one of the best science writers alive. The book by the Goldsteins does a thorough job of discussing the history and concepts of thermodynamics. The Refrigerator and the Universe: Understanding the Laws of Energy. Harvard. 1993. 0674753240 This book assumes no knowledge of probability The Theory of Strains. a Compendium for the Calculation and Construction of Bridges, Roofs and Cranes, With the Application of Trigonometrical Notes. Mcdougal littell math"algebra 2 answers, factoring polynomials for dummies, math cheats linear and nonlinear equations, plotting coordinates worksheet, difficult, Answers to Evens Algebra: Structure and Method, "prentice hall mathematics texas algebra 2", solving roots polynomials in excel Zuckerman Algebra & Trigonometry Workbook. This is a simple application that will factor quadratic equations, if possible. If you need to *solve* quadratics as well as factor them, you should check out iFactor and Solve Quadratics in the app store. It can handle any quadratic, including ones with complex roots. Jumbo Calculator is great how needs a calculator with large buttons. Math Sheet is the calculator that mimics sheet paper that you can use to write down your calculations Nociones de álgebra y trigonometría.. 3 MB Logical consequence is the relation that obtains between premises and conclusion(s) in a valid argument. Orthodoxy has it that valid arguments are n... 11 MB A Complete Introduction to probability AND its computer Science Applications USING R Probability with R serves as a comprehensive and introd... 2016 3 MB A Concise Introduction to Geometric Numerical Integration presents the main themes, techniques, and applications of geometric integrators for resear.. College Trigonometry Fifth Edition. It is suggested that you study for this test. If you do not satisfy requirements 1 or 3 then you MUST take this test (or else enroll in MATH 1316: Trigonometry prior to enrolling in MATH 2413: Calculus) When you come to campus to meet with an advisor in Academic Advising and you have indicated any of the above majors, they will automatically schedule a one-hour block for you to take the test Plane Geometry.
Euclid's Elements of Geometry: From the Latin Translation of Commandine. to Which Is Added, a Treatise of the Nature of Arithmetic of Logarithms ; ... and Spherical Trigonometry ; with a Preface
Plane and spherical trigonometry
Four place logarithmic tables: containing the logarithms of numbers and of the trigonometric functions, arranged for use in the entrance examinations ... scientific school of Yale university | 677.169 | 1 |
Maths in Focus HSC Student Book Plus Access Card for 4 Years
Supplementary material such as digital access codes, CDs or DVDs are only available with new copies of this book.
Overview
The bestselling Maths in Focus series has been extensively revised and features a refreshed design that aims to promote visual and textual clarity and accessibility. All chapters contain comprehensive fully-worked examples and explanations, as well as ample sets of graded exercises for continual revision. NelsonNetBook versions of some of the student books are available on the NelsonNet student website to booklisting schools to complement the printed text. | 677.169 | 1 |
Quick Overview
The B.Com Entrance Examination is conducted by the Aligarh Muslim University for admitting students to Bachelor of Commerce (B.Com) programme of the university. The marks obtained by the candidates in the entrance exam are used for selecting the students for admission into B.Com programme. This book has been designed for the students preparing for AMU B.Com Entrance Examination. The present study guide for AMU B.Com Entrance Examination has been divided into four sections namely Commerce, Mathematics, General English and General Intelligence & Reasoning, each sub-divided into number of chapters as per the syllabi of the entrance examination for admission to B.Com at AMU. Each chapter in the book contains ample number of solved problems which have been designed on the lines of questions asked in previous years' AMU B.Com. Entrance Examination. Also practice exercises have been covered at the end of each chapter to help aspirants revise the concepts discussed in the chapters. The Commerce section covers Accounting, Partnership, National Income & Consumer Producer Equilibrium, Money Market & Indian Banking System, Financial Management, Marketing Management & Consumer Protection Act, National Income & Consumer Producer Equilibrium, etc whereas the Mathematics section covers Probability, Differentiation, Matrices, Indefinite Integrals, Linear Programming, Vector Algebra, etc. The General Intelligence & Reasoning section covers both Verbal and Non-Verbal Reasoning ability. The book also contains 2015 AMU B.Com. Entrance Examination which will help in knowing the pattern of examination and the types of questions asked in previous years' AMU B.Com. Entrance Examination. As the book contains ample number of solved problems as well as practice material, it for sure will help aspirants score high in the upcoming AMU B.Com. Entrance Examination 2015. | 677.169 | 1 |
Polynomials
In this College algebra activity, students determine the degree of polynomials. Students add, subtract, multiply, divided and factor polynomials. The two page activity contains thirty-eight problems. Answers are no provided. | 677.169 | 1 |
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This book, first published in 2002,
"This book will fill an interesting niche in a library collection...it should be used by browsing students interested in making sure that they are prepared for success in their graduate programs." Choice
"All the Mathematics You Missed...is a help for students going on to graduate school..Since many students beginning graduate school do not have the mathematical knowledge needed, All the Mathematics You Missed aims to fill in the gaps." Berkshire Eagle, Pittsfield, MA
"From the preface: 'The goal of this book is to give people at least a rough idea of many topics that beginning graduate students at the best graduate schools are assumed to know." Mathematical Reviews
"The writing is lucid mathematical exposition, at a level quite appropriate to beginning graduate students." The American Statistician
"Before classes began, I jump started my graduate career with the help of this book. Even though I didn't believe that I could have missed much math, it became clear that my belief was wrong during the first week of class. While proving a theorem, my professor asked if anyone remembered a previous result from calculus. While I did not remember it from my days as an undergraduate, I had read about the theorem and had even seen a sketch of the proof in Garrity's book...This will be one of the books that I keep with me as I continue as a graduate student. It has certainly helped me understand concepts that I have missed." Elizabeth D. Russell, Math Horizons
"Point set topology, complex analysis, differential forms, the curvature of surfaces, the axiom of choice, Lebesgue integration, Fourier analysis, algorithms, and differential equations.... I found these sections to be the high points of the book. They were a sound introduction to material that some but not all graduate students will need." Charles Ashbacher, School Science and Mathematics
Book Description:
This book, first published in 2002, will help students in mathematics and other quantitative subjects to fill in the gaps in their preparation for graduate school. It presents the basic points, a few key results, and an annotated reading list of the most important undergraduate topics in mathematics: linear algebra, vector calculus, geometry, real analysis, point-set topology, and more.
Descrizione libro77753 Cambridge University Press. Paperback. Condizione libro: new. BRAND NEW, All the Mathematics You Missed: But Need to Know for Graduate School, Thomas A. Garrity, Lori Pedersen, B9780521797078 | 677.169 | 1 |
Popular in Mathematics (M)
Reviews for MODERN GEOMET 5327361 LECTURE NOTES 10 Notes on Translations and Rotations at We associate with each point Ly the column y which we will sometimes write as x y 1T A translatton of the Euclidean plane is a function f which maps each point z y to z a y b for some real numbers a and b To make matters more precise we shall refer to f as a translation by a b We may view such a translation as mapping z y 1T into z a y b 1T Since xa 1 0 a z yb 01 b y 1 0 01 1 we may therefore think of f as simply being multiplication by the matrix above We shall refer to the above matrix as T0117 If P represents the point a b we will sometimes write Tp Thus 1 0 a Tp 0 1 b 0 0 1 represents a translation of the Euclidean plane by P If P 0 0 observe that Tp maps each point to itself In this case we will call Tp the identity transformation Now consider a point A 1111 and a real number 15 A rotatton of the Euclidean plane about A by an angle t is a function f which maps each point B Ly to O z y where O is the same distance as B from A and where the angle measured counterclockwise from the vector B to the vector E is t It will be convenient to also nd a matrix representation of such a rotation Suppose for the moment that A 0 0 We can write B in polar coordinates as r 6 Then 0 has the polar coordinate representation r 6 15 Hence x r cos0 b 71030 cos 7 rsin6 sin acos 7 y sin and 1 rsin0 b 71030 sin rsin6 cos xsin y cos In matrix notation we may combine these as a 7 cos isin y sin COS y 39 In general with A 1111 we may obtain z y by translating the Euclidean plane rst by izl 711 and then performing the above rotation about the origin and then translating the Euclidean plane by 1111 Thus 1 zit 2222 132 1 ma 7 ysma no 7 coslt gtgt 111 ma z s1n y COS 7 961 81H y11 COS We may rewrite this as x 003 7 3in 1l 7 003 yl 3in z y 3in 003 7z1 3in y1l 7 003 y 1 0 0 1 1 Thus a rotation f can also be viewed in terms of matrix multiplication We call the above 3 gtlt 3 matrix R A With the above information we may now view a combination of translations and rotations in terms of matrix multiplication For example if we wish to translate the Euclidean plane by A 2 3 and then rotate about the point B 11 by 7r 6 and then translate by O 75 7 each point z y in the Euclidean plane will be moved to z y where z z y TORw6BTA y 1 1 This is a good place to do some examples and to make up some related homework Our main goal here is to establish and apply the following result Theorem Let Di and 6 be real numbers not necessarily distinct and let A and B be points not necessarily distinct If Oi 6 is not an integer multiple of 27139 then there is point C such that RagRa Range fa 6 is an integer multiple 0f27r then R BRaA is a translation Before demonstrating the theorem it would be a good idea to discuss the analogous result for a composition of 2 translations the rst by a b and the second by c d Geometrically it should be clear that the result of such a composition is a translation by a c b d Alternatively one can show by taking the product of matrices that TabTcd Tltacbd To see why the theorem holds write A zhyl and B 2112 Then 0035 Sir15 9621 COSWD 12 9115 R BRaA 3in6 0036 7z2 3in6 y2l 7 0036 0 0 1 003Oi 7 3ina 1l 7 003a yl 3ina gtlt 3ina 003Oi 7z1 3inOi y11 7 003a 0 0 1 003 3in a6 73ina6 u am mmm v 0 0 1 where u 1 0036l 7 003a yl 3ina 0036 1 3ina 3in6 7mamml7cmo mu7cmw mamm 1l 7 003Oi 6 yl 3in04 6 m7muwmmw7wmm and 1 1 sin l 7 cosa yl sina sin 7 1 cosa sin yl cos l 7 cosa 7 2 sin y2l 7 0036 7z1 sina B y1l 7 cosa 6 i 952 i 951 31115 12 7 1100 0035 Observe that if a B is an integer multiple of 27139 then the above matrix represents a translation by u 1 so that the second part of the theorem follows Suppose now that a B is not an integer multiple of 27139 We will have that there is a C such that R BRaA is a rotation at O by the angle a B if we can nd a pair 3113 such that 9631 7 00w 6 ya sina 6 7 962 7 9601 7 0086 12 7 in 31MB and 7 smlta B y3lt17 coslta 6 7 7ltz2 7 z1gtsinlt gtlty2 7 y1gtlt17 cow We have two equations in the 2 unknowns 3 and y3 There is a solution provided that l 7 cosa B 311104 5 detlt 7sinoz B 17 00304 6 7g 039 Observe that one does not need to use anything fancy here simply solve for 3 and y3 above and the equivalent of the determinant being non zero above follows We get that 0 exists provided that 272cosa 7amp0 Since we are now only considering a B which are not integer multiples of 27139 the theorem is established | 677.169 | 1 |
ISBN 13: 9780123910974
Elements of the Theory of Numbers
Elements of the Theory of Numbers teaches students how to develop, implement, and test numerical methods for standard mathematical problems. The authors have created a two-pronged pedagogical approach that integrates analysis and algebra with classical number theory. Making greater use of the language and concepts in algebra and analysis than is traditionally encountered in introductory courses, this pedagogical approach helps to instill in the minds of the students the idea of the unity of mathematics. Elements of the Theory of Numbers is a superb summary of classical material as well as allowing the reader to take a look at the exciting role of analysis and algebra in number theory.
"synopsis" may belong to another edition of this title.
From the Back Cover:
Elements of the Theory of Numbers is a comprehensive and contemporary introduction for a first course in classical number theory. The authors offer an integrated approach to the subject, making greater use than usual of the language and concepts of algebra, mathematical proof, and analysis. The book offers a wealth of topics in two parts. Part I consists of fundamental or core material. It includes primes, congruences, primitive roots, residues, and multiplicative functions. Part II is a collection of more specialized topics, such as a brief look at number fields, recurrence relations, and additive number theory. Throughout the text, the authors offer historical references and introduce topics in their historical context. Over 900 exercises are included. "I definitely appreciate the unified approach. I think it is important that the students realize that mathematics does not consist of separate entities. --Maureen Fenrick, Mankato State University "The authors communicate successfully the joy they find in number theory. Students will be excited by learning from this (text)." --Frank DeMeyer, Colorado State University "The book's biggest advantage is its thorough integration of the relevant algebra into the development. It's about time!" --Thomas McLaughlin, Texas Tech University | 677.169 | 1 |
Enhancing Learning Environments Through Curriculum Integration
4.11 - 1251 ratings - Source
The first 18 units make up Applied Math I and are designed to be covered in the
ninth grade. The remaining 18 units constitute Applied Math II and are to be
completed at the tenth grade. The 36 units ... Using Graphs, Charts, and Tables 5.
Dealing ... Using Formulas to Solve Problems 16. ... Geometry in the Workplace 1
29.
Title
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Enhancing Learning Environments Through Curriculum Integration
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DIANE Publishing - 1993-02-01
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Buy Mathematics Books Online on Junglee
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It is not possible for a person to become a good mathematician simply by attending a good school. It is important that you get hold of mathematical books that help you grapple with mathematical ideas and make them your own. We at Junglee stock a large number of geometry books, algebra books, calculus books, statistics books, engineering books among others from some of the leading Indian and foreign authors like Aditi Singhal, Michael J Schramm, Richard Ku, Serge Lang and Shakuntala Devi among others. You can find at Junglee mathematics books that can act as your trusty friend and guide in understanding and learning the ideas of mathematics. | 677.169 | 1 |
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CentralBusiness Mathematics amp Statistics MTH 302 Lesson 22 Lesson 23 Lesson 24 Lesson 25 Lesson 26 Lesson 27 Lesson 28 Lesson 29 Lesson 30 Lesson 31 Lesson 32 Lesson 33 Lesson 34 Lesson 35 Lesson 36 Business Mathematics amp Statistics MTH 302 VU TABLE OF CONTENTS Lesson 1 COURSE OVERVIEW 3 Lesson 2 APPLICATION OF BASIC MATHEMATICS 12 Lesson 3 APPLICATION OF BASIC MATHEMATICS 22 Lesson 4 APPLICATION OF BASIC MATHEMATICS 29 Lesson 5 APPLICATION OF BASIC MATHEMATICS 399 Lesson 6 APPLICATION OF BASIC MATHEMATICS Error Bookmark not defined8 Lesson 7 APPLICATION OF BASIC MATHEMATICS Error Bookmark not defined9 Lesson 8 COMPOUND INTEREST 709 Lesson 9 COMPOUND INTEREST 776 Lesson 10MATRCES 809 Lesson 11 MATRICES 854 Lesson 12 RATIO AND PROPORTION 94 Lesson 13 MATHEMATICS OF MERCHANDISING 1009 Lesson 14 MATHEMATICS OF MERCHANDISING 105 Lesson 15 MATHEMATICS OF MERCHANDISING 11211 Lesson 16 MATHEMATICS OF MERCHANDISING 12120 Lesson 17 MATHEMATICS FINANCIAL MATHEMATICS 12524 Lesson 18 MATHEMATICS FINANCIAL MATHEMATICS 13029 Lesson 19 PERFORM BREAKEVEN ANALYSIS 13433 Lesson 20 PERFORM BREAKEVEN ANALYSIS 14241 Lesson 21 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS 14746 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS 15049 STATISTICAL DATA REPRESENTATION 1587 STATISTICAL REPRESENTATION 16362 STATISTICAL REPRESENTATION 17170 STATISTICAL REPRESENTATION 18079 STATISTICAL REPRESENTATION 18988 MEASURES OF DISPERSION 20099 MEASURES OF DISPERSION 207 MEASURE OF DISPERASION 217 LINE FITTING 22524 TIME SERIES AND 24039 TIME SERIES AND EXPONENTIAL SMOOTHING 25352 FACTORIALS 26059 COMBINATIONS 269 ELEMENTARY PROBABILITY 27675 Lesson 37PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 27978 Lesson 38PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS Lesson 41 Lesson 42 Lesson 43 Lesson 44 Lesson 45 28483 Lesson 39PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 29796 Lesson 40PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS 302 ESTIMATING FROM SAMPLES INFERENCE 314 ESTIMATING FROM SAMPLE INFERENCE 320 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION 325 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION 328 PLANNING PRODUCTION LEVELS LINEAR PROGRAMMING 335 2 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MTH 302 LECTURE 1 COURSE OVERVIEW COURSE TITLE The title of this course is BUSINESS MATHEMATICS AND STATISTICSquot Instructor s Resume The instructor of the course is Dr Zahir Fikri who holds a PhD in Electric Power Systems Engineering from the Royal Institute of Technology Stockholm Sweden The title of Dr Fikri s thesis was Statistical Load Forecasting for Distribution Network Planningquot Obiective The purpose of the course is to provide the student with a mathematical basis for personal and business financial decisions through eight instructional modules The course stresses business applications using arithmetic algebra and ratioproportion and graphing Applications include payroll costvolumeprofit analysis and merchandising mathematics The course also includes Statistical Representation of Data Correlation Time Series and Exponential Smoothing Elementary Probability and Probability Distributions This course stresses logical reasoning and problem solving skills Access to Microsoft Excel software is required for the course Course Outcomes Successful completion of this course will enable the student to 1 Apply arithmetic and algebraic skills to everyday business problems 2 Use ratio proportion and percent in the solution of business problems 3 Solve business problems involving commercial discount markup and markdown 4 Solve systems of linear equations graphically and algebraically and apply to cost volume profit analysis 5 Apply Statistical Representation of Data Correlation Time Series and Exponential Smoothing methods in business decision making 6 Use elementary probability theory and knowledge about probability distributions in developing profitable business strategies Unit Outcomes ResourcesTestsAssiqnments Successful completion of the following units will enable the student to apply mathematical methods to business problems solving Required Student Resources lncludinq textbook nd workbooks Text Selected books on Business Mathematics and Statistics Optional Resources Handouts supplied by the professor Instructor s Slides Online or CD based learning materials Prereguisites The students are not required to have any mathematical skills Basic knowledge of Microsoft Excel will be an advantage but not a requirement Evaluation In order to successfully complete this course the student is required to meet the following evaluation criteria Full participation is expected for this course All assignments must be completed by the closing date Overall grade will be based on VU existing Grading Rules All requirements must be met in order to pass the course Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COURSE MODULES The following are the main modules of this course Module 1 0 Overview Lecture 1 0 Perform arithmetic operations in their proper order Lecture 2 0 Convert fractions their percent and decimal equivalents Lecture 2 o Solve for any one of percent portion or base given the other two quantities Lecture 2 0 Using Microsoft Excel Lecture 2 Calculate the gross earnings of employees paid a salary an hourly wage or commissions Lecture 3 0 Calculate the simple average or weighted average given a set of values Lecture 4 Perform basic calculations of the percentages averages commission brokerage and discount Lecture 5 0 Simple and compound interest Lecture 6 0 Average due date interest on drawings and calendar Lecture 6 Module 2 o Exponents and radicals Lecture 7 o Solve linear equations in one variable Lecture 7 o Rearrange formulas to solve for any of its contained variables Lecture 7 o Solve problems involving a series of compounding percent changes Lecture 8 0 Calculate returns from investments Lecture 8 0 Calculate a single percent change equivalent to a series of percent changes Lecture 8 o Matrices Lecture 9 o Ratios and Proportions Lecture10 0 Set up and manipulate ratios Lecture11 o Allocate an amount on a prorata basis using proportions Lecture11 0 Assignment Module 12 Module 3 0 Discounts Lectures 12 0 Mathematics of Merchandising Lectures 1316 Module 4 0 Applications of Linear Equations Lecture 1718 0 Breakeven Analysis Lecture 1922 0 Assignment Module 34 0 MidTerm Examination Module 5 0 Statistical data Lectures 23 0 Measures of central tendency Lectures 2425 0 Measures of dispersion and skewness Lectures 2627 Module 6 0 Correlation Lectures 2829 0 Line Fitting Lectures 3031 0 Time Series and Exponential Smoothing Lectures 3133 0 Assignment Module 56 Module 7 o Factorials Lecture 34 o Permutations and Combinations Lecture 34 0 Elementary Probability Lectures 3536 0 Patterns of probability Binomial Poisson and Normal Distributions Lecture 3740 Module 8 0 Estimating from Samples Inference Lectures 4142 0 Hypothesis testing ChiSquare Distribution Lectures 4344 0 Planning Production Levels Linear Programming Lecture 45 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0 Assignment Module 78 0 EndTerm Examination Note The course modules are subject to change MARKING SCHEME As per VU Rules ESCRIPTION OF TOPICS RECOMMENDED NO MAIN TOPIC TOPICS READING 10 Module Applications of o Overviewew Lecture 1 Reference 1 1 Basic Mathematics Lectures 16 Reference 2 MOdUIe Earnineefi c gvzgtions amp Lecwre 2 1 U M p It E I Tool Microsoft 0 sung Icroso xce Excel Reference 2 Module 0 Calculate Gross Earnings Lecture 3 1 0 Using Microsoft Excel Tool Microsoft Excel Reference 2 Module 0 Calculating simple or Lecture 4 1 weighted averages Tool Microsoft 0 Using Microsoft Excel Excel Reference 6 0 Basic calculations of 5353 2 percentages averages commission nOdUIe brokerage and discount using g rvi gs c r 3 0 Microsoft Excel Excel Reference 2 0 Simple and compound Lecture 6 Module interest Reference 3 Ch 3 1 o Average due date interest on drawings and calendar Tool Microsoft Excel 0 Exponents and radicals Reference 2 20 x rquotsscrr111llfy algebraic Lecture 7 Module Applications of o p S I I t Reference 3 Ch 2 2 Basic Algebra variable W 39quotear equa mm m one Tool Microsoft Lectures 79 o Rearrange formulas to solve for any of its contained variables Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 8 0 Calculate returns from investments Reference 2 0 Problems involving a Series Lecture 8 of Reference 3 Ch 3 compounding percent changes 0 I SIngle percent change Tool Microsoft equwalent Excel to a series of percent changes 9 Reference 2 Lecture 9 Matrices Reference 3 Ch 4 Tool Microsoft Excel 10 0 Set up and manipulate ratios Reference 2 30 0 Set up and solve proportIons Lecture 10 Applications o Express percent differences Reference 3 Ch 3 Module of Ratio and using proportions 2 Proportion o Allocate an amount on a gill ectures 10 prorata basls usmg Tool Microsoft proportlons Excel 11 Reference 2 Module 0 Set up and manipulate ratios Eig g gg 3 Ch 3 2 AHOFate an amount 0quot a Tool Microsoft prorata baSlS usmg proportIons Excel 12 40 0 Calculate the net price of an Reference 2 Merchandising item after single or multiple trade Lecture 12 Module and Financial discounts Reference 3 Ch 3 3 Mathematics 0 Calculate an equivalent single Lectures 12 discount rate given a series of Tool Microsoft 16 discounts Excel 13 Reference 2 o Solve merchandising pricing Lecwre 13 Module Reference 3 Ch 3 3 problems Involvmg markup and Tool Microsoft markdown Excel 14 Reference 2 Lecture 14 Module Reference 3 Ch 3 3 0 Financial Mathematics Part 1 Reference 5 Ch 16 Tool Microsoft Excel Module 15 o Financial Mathematics Part 2 Reference 2 3 Lecture 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 3 50 BreakEven Module Analysis 4 Lectures 17 22 Module 4 Module 4 Module 4 Module 4 Module 4 6 Statistical Module Representation 5 of Data Lectures 23 16 17 18 19 20 21 22 23 o Financial Mathematics Part 3 0 Graph a linear equation in two variables 0 Solve two linear equations with two unknowns 0 Perform linear costvolume profit and breakeven analysis 0 Using a breakeven chart 0 Perform linear costvolume profit and breakeven analysis 0 Using the algebraic approach of solving the cost and revenue func ons 0 Perform linear costvolume profit and breakeven analysis 0 Using the contribution margin approach 0 Perform linear costvolume profit and breakeven analysis 0 Using Microsoft Excel o Assignment Module 34 0 MidTerm Examination 0 Statistical Data Reference 3 Ch 3 Reference 5 Ch 16 Tool Microsoft Excel Reference 2 Lecture 16 Reference 3 Ch 3 Reference 5 Ch 16 Tool Microsoft Excel Reference 2 Lecture 17 Reference 3 Ch 3 Reference 5 Ch 16 amp 18 Tool Microsoft Excel Reference 2 Lecture 18 Reference 3 Ch 2 Reference 5 Ch 1 Tool Microsoft Excel Reference 2 Lecture 19 Tool Microsoft Excel Reference 2 Lecture 20 Tool Microsoft Excel Reference 2 Lecture 21 Tool Microsoft Excel Reference 2 Lecture 22 Tool Microsoft Excel Reference 2 Lecture 23 Reference 5 Ch 5 Tool Microsoft Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 5 Module 5 Module 5 Module 5 Module 6 27 24 0 Statistical Representation Measures of Central Tendency Part 1 25 o StatIstIcal RepresentatIon 0 Measures of Central Tendency Part 2 26 0 Measures of Dispersion and Skewness Part 1 27 0 Measures of Dispersion and Skewness Part 2 7 Correlation 28 Time Series and Exponential 0 Correlation Smoothing Part 1 Lectures 28 33 29 0 Correlation Part 2 30 0 Line Fitting Part 1 Excel Reference 2 Lecture 24 Reference 4 Ch 3 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 25 Reference 4 Ch 3 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 26 Reference 4 Ch 4 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 27 Reference 4 Ch 4 Reference 5 Ch 6 Tool Microsoft Excel Reference 2 Lecture 28 Reference 5 Ch 13 Tool Microsoft Excel Reference 2 Lecture 29 Reference 5 Ch 13 Tool Microsoft Excel Reference 2 Lecture 30 Reference 5 Ch 14 Tool Microsoft Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 7 Elementary Probability Lectures 34 38 Module 7 Module 7 Module 7 Module 7 Module 7 31 32 33 34 35 36 37 38 0 Line Fitting Part 2 Time Series and Exponential Smoothing Part 1 Time Series and Exponential Smoothing Part 2 o Assignment Module 56 o Factorials o Permutations and Combinations 0 Elementary Probability Part 1 0 Elementary Probability Part 2 o Patterns of probability Binomial Poisson and Normal Distributions Part 1 0 Patterns of probability Binomial Poisson and Normal Distributions Part 2 Reference 2 Lecture 31 Tool Microsoft Excel Reference 2 Lecture 32 Reference 5 Ch 15 Tool Microsoft Excel Reference 2 Lecture 33 Reference 5 Ch 15 Tool Microsoft Excel Reference 2 Lecture 34 Reference 3 Ch 2 Tool Microsoft Excel Reference 2 Lecture 35 Reference 5 Ch 8 Tool Microsoft Excel Reference 2 Lecture 36 Reference 5 Ch 8 Tool Microsoft Excel Reference 2 Lecture 39 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 40 Reference 5 Ch 9 Tool Microsoft Excel Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Module 7 Module 7 8 Probability Distributions Lectures 39 Module 44 8 9 Linear Programming Lecture 45 Module 8 Module Module Module Methodology 39 40 41 42 43 44 45 0 Patterns of probability Binomial Poisson and Normal Distributions Part 3 0 Patterns of probability Binomial Poisson and Normal Distributions Part 4 o Estimating from Samples Inference Part 1 o Estimating from Samples Inference Part 2 o Hypothesis testing Chi Square Distribution Part 1 o Hypothesis testing Chi Square Distribution Part 2 0 Production Planning Linear Programming o Assignment Module 78 0 End Term Examination Reference 2 Lecture 41 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 41 Reference 5 Ch 9 Tool Microsoft Excel Reference 2 Lecture 42 Reference 5 Ch 10 Tool Microsoft Excel Reference 2 Lecture 43 Reference 5 Ch 10 Tool Microsoft Excel Reference 2 Lecture 44 Reference 5 Ch 11 Tool Microsoft Excel Reference 2 Lecture 45 Reference 5 Ch 1 1 Tool Microsoft Excel Reference 2 Lecture 45 Reference 5 Ch 18 Tool Microsoft Excel There will be 45 lectures each of 50 minutes duration as indicated above The lectures will be delivered in a mixture of Urdu and English The lectures will be heavily supported by slide presentations The slides for a lecture will be made available on the VU website for the course a few days before the actual lecture is televised This will allow students to carry out preparatory reading before the lecture The course will be provided its own page on the VU s web site This will be used to 10 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU provide lecture and other supporting material from the course to the students The page will have a link to a webbased discussion and bulletin board for the students Teaching assistants will be assigned by VU to provide various forms of assistance such as grading answering questions posted by students and preparation of slides Gradin There will be a term exam and one final examination There will also be 4 assignments each covering two modules The final exam will be comprehensive These will contribute the following percentages to the final grade Mid Term Exam 35 Final 50 4 Assignments 15 Text and Reference Material The course is based on material from different sources Topics for reading will be indicated on course web site and in professor s handouts also to be posted on the course web site A list of reference books will also be posted and updated on the course web site The following material will be used by the students as reference Reference 1 Course Outline Instructor s Power Point Slides Business Mathematics amp Statistics by Prof Miraj Din Mirza Elements of statistics amp Probability by Shahid Jamal Quantitative Approaches in Business studies by Clare Morris Microsoft Excel Help File Schedule of Lectures Given above is the tentative schedule of topics to be covered Minor changes may occur but these will be announced well in advance 11 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 2 Applications of Basic Mathematics Part 1 OBJECTIVES The objectives of the lecture are to learn about 0 Different course modules 0 Basic Arithmetic Operations 0 Starting Microsoft MS Excel 0 Using MS Excel to carry out arithmetic operations COURSE MODULES This course comprises 8 modules as under 0 Modules 14 Mathematics 0 Modules 58 Statistics Details of modules are given in handout for lecture 01 BA SIC ARIT HME TIC OPERA T IONS Five arithmetic operations provide the foundation for all mathematical operations These are 0 Addition 0 Subtraction o Multiplication 0 Division 0 Exponents Example Addition 12 5 17 mple Subtraction 12 5 7 mple Multiplication 12 x 5 60 Example Exponent 4quot2 16 4quot12 2 4quot12 14quot12 12 05 MICROSOFT EXCEL IN BUSINESS MATHEMATICS amp STATISTICS Microsoft Corporation s Spreadsheet software Excel is widely used in business mathematics and statistical applications The latest version of this software is EXCEL 2002 XP This course is based on wide applications of EXCEL 2002 It is recommended that you install EXCEL 2002 XP software on your computer If your computer has Windows 2000 and EXCEL 2000 even that version of EXCEL can be used as the applications we intend to learn can be done using the earlier version of EXCEL Those of you who are still working with Windows 98 and have EXCEL 97 installed are encouraged to migrate to newer version of EXCEL software 12 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Startinq EXCEL 2000 XP EXCEL 2000 XP can be started by going through the following steps Click Start on your computer Click All Programs Click Microsoft Excel The following slides show the operations i EIi39l39laiil r391III39iII li Horosol t Wire Horosol t Excel r Horosol t Frooieoe 7h wif Re note existence M E Adobe Fieecler en Horosol t PamePoin L l allquot Humor E New ISIFFiEI Eizcurl39u39lt ti opanorricaoommt Elf Set Program notes and Defaults lEia Window Cider Win nite Ltdate Geniee h Acceeezries b lr39icroiofl IIIFFico Toois Iquot F39rlrtlle Ji39itemel F39rntlng Stortup If Adzhe Fieezler EI Internet EEcplz er ll39icroiofl Fliozoii rillroeor39t Ezozel lr39itroeoft Dutlozit lr39lcroeofl Word rr39icrosoft FrotF39ege ll39icroioil PowerPoint I39I39EHExpluzirer i3 MoohEIp39cee Remote Meetmce window li39ieizlle Flafer 39iilinzlziio lihsionuer Snain I 3945 IZIIZIIII39 i The EXCEL window opens and a blank worksheet becomes available as shown below Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU M Icmsnf l Esme Harald Elle gdt Elanu mart Fgrrnat cola gate nderu Ijelp v 39 r E If g l g il r g v gt gd ma E i itsta m 1 Big E 12 Egg 335123 3931 r 1 A E E D E F G H IT Hm Wartthunk 1r n i penamthnIt Miami3 Lecture jiEEUUIUntereal Arlthmetlt paratlnre LectureJlIi More hunch19kt lilil HEW El Blank Workbook HEW Iimnt tstin Hmikhllt E Choose workbook HEW Iimnt template General Tan rplatae Tarnplatas on my Wet Sites Tan platae on I39quotIlru IftIm Add ll tmrltFlace Hmtt Emil Help quot If Ericw at startup H i r ill llieetl shaa l39ee f M i 39ii FlIIIII A 2 3 4 E E T E 9 10 11 12 13 11 15 13 1 13 1E 33 21 22 23 9 u i il39i39l39HElL39EiIEIZEI39JE hand I3939Iiurur39uufl Excel Banal1 The slide shows a Workbook by the name book1 with three sheets Sheet1 Sheet2 and Sheet3 The Excel Window has Column numbers starting from A and row numbers starting from 1 the intersection of a row and column is called a Cell The first cell is A1 which is the intersection of column A and row 1 All cells in a Sheet are referenced by a combination of Column name and row number Example 1 B15 means cell in column B and row 15 Example 2 A cell in row 12 and column C has reference C12 A Range defines all cells starting from the leftmost corner where the range starts to the rightmost corner in the last row The Range is specified by the starting cell a colon and the ending cell Example 3 A Range which starts from A1 and ends at D15 is referenced by A1 D15 and has all the cells in columns A to D up to and including row 15 A value can be entered into a cell by clicking that cell The mouse pointer which is a rectangle moves to the selected cell Simply enter the value followed by the Enter key The mouse pointer moves to the cell below If you make a mistake while entering the value select the cell again by clicking it Enter the new value The old value is replaced by the new value 14 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If only one or more digits are to be changed then select the cell Then double click the mouse The blinking cursor appears Either move the arrow key to move to the digit to be changed or move the cursor to the desired position Enter the new value and delete the undesired value by using the Del key suggest that you learn the basic operations of entering deleting and changing data in a worksheet About calculation operators in Excel In Excel there are four different types of operators Arithmetic operators Comparison operators Text concatenation operator Reference operators The following descriptions are reproduced from Excel s Help file for your ready reference In the present lecture you are directly concerned with arithmetic operators However it is important to learn that the comparison operators are used where calculations are made on the basis of comparisons The text concatenation operator is used to combine two text strings The reference operators include and or as the case maybe We shall learn the use of these operators in different worksheets You should look through the Excel Help file to see examples of these functions Selected material from Excel Help File relating to arithmetic operations is given in in a separate file The Excel arithmetic operators are as follows Addition Symbol Example 54 Result 9 Subtraction Symbol Example 54 Result 1 Multiplication Symbol Example 54 Result 20 Division Symbol Example 124 Result 3 Percent Symbol Example 20 Result 02 Exponentiation quot Example 5quot2 Result 25 Excel Formulas for Addition All calculations in Excel are made through formulas which are written in cells where result is required Let us do addition of two numbers 5 and 10 We wish to calculate the addition of two numbers 10 and 5 Let us see how we can add these two numbers in Excel 1 Open a blank worksheet 2 Click on a cell where you would like to enter the number 10 Say cell A15 3 Enter 10 in cell A15 4 Click cell where you would like to enter the number 5 Say cell B15 5 Click cell where you would like to get the sum of 10 and 5 Say cell C15 6 Start the formula Write equal sign in cell C15 7 After write left bracket in cell C15 8 Move mouse and left click on value 10 which is in cell A15 ln cell C15 the cell reference A15 is written 9 Write quot after A15 in cell C15 10 Move mouse and left click on value 5 which is in cell B15 ln cell C15 the cell reference B15 is written 11 Write right bracket in cell C15 12 Press Enter key The answer 15 is shown in cell C15 If you click on cell C15 the formula A15B15quot is displayed the formula bar to the right of fX in the Toolbar 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The main steps along with the entries are shown in the slide below The worksheet H302 lec 02 contains the actual entries 7 T Micreseft Excel Arithmetic pere ene F Elle Edit Eiew insert Fgrmat Innis Data window elp Adobe PDF TWZHEEDUESGQlDDlDi iTEiD v 51 x neee weem Ev ferial vznve ettev vf 1 E El 99 ii 5 v eeleele El D E F A r e H I J 2 ADDING TWO NUMBERS 5 and 10 eleTEPe e l 1 Entereln eell A15 e l e Enterie in cell 315 e l 3 Write eln in Cell ele i 4 Piece meuee en eell A115 3 i E type ein e l 39r eete eell Bie iEI E Preee Enter key 11 Reeult Shewn in Cell C15 12 Q Cli l39f en Cell 315 13 Reeult FermuleA15BiEie el lewn in fermuleleer 15 5 1U 15 HI l h H Sheetl etE A Sheet3 f i r r I l ean e eeeeee x a a e 41 e e v iv e E e e e l ll g iii 2 l 5 g l Tg E l3 V lllrerr Ele iceit l l3939iilrIIItt F39IZI393939El39F39IIirlt IZIIZIE39E 43 The next slide shows addition of 6 numbers 5 10 15 20 30 and 40 The entries were made in row 34 The values were entered as follows Cell A34 5 Cell B34 10 Cell C34 15 Cell D34 20 Cell E34 30 Cell F34 40 The formula was written in cell G34 The formula was 51015203040 The answer was 120 You can use an Excel function SUM along with the cell range A34F34 to calculate the sum of the above numbers The formula in such a case will be SUMA34F34 You enter quot followed by SUM followed by Click on the cell with value 5reference A34 Drag the mouse to cell with value 40reference F34 and drop the mouse Enter quot and then press the Enter key 16 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel hrithmetic peratinns Elle Edit ew lnsert Fgrmat Iools Qatar indow elp Atler PDF TWEEEUESHUW FEW 0310 v 539 X a ggg rm Ev jmial vlov viv E i3 99 Tl 7 J35 v 5 4 E c o E F c H I J H L T 19 ADDING SIX N U M BERS 51015203040 2 STEPS 21 1 Enter 5 in cell 11 Type in cell C34 or formula Iquot 22 2 Enter 10in cell 334 12 Click on cell C34 23 Enter 15 in cell C34 13 Type in cell C34 or formula ll l39 24 4 Enter 20 in cell D34 14 Click on cell D34 25 5 Enter 30 in cell E34 15 Type in cell C34 or formula ll f 39 25 3913 Enter 40 in cell F34 113 Click on cell E34 2 T Enter in cell C34 1quot Type in cell C34 or formula llquot 28 13 Click on cell 113 Click on cell F34 23 9 Type in cell C34 or formula liar 13l Fll39E Enter 3 1 Click on cell 334 Ftesult In cell C34 31 32 E312 tltl 51 15203040 in a row 51 1520304012 33 34 5 10 15 20 30 40 120 y a 7quot H 1 r H heetl 6 SheetE Sheet3 I i l Draw 3 Agto hapesquot K E D Q 3 L d quot i quot E E i 1 Ready 7 Pelicans3ft EIIEl Firit 7 El l3939liIrIIIIft F39III39IerF39IIint 393939 239E In the above two examples you learnt how formulas for addition are written in Excel Excel Formula for Subtraction Excel formulas for subtraction are similar to those of addition but with the minus sign Let us go through the steps for subtracting 15 from 25 Enter values in row 50 as follows Cell A50 25 Cell B50 15 Write the formula in cell C50 as follows A50B50 To write this formula click cell 050 where you want the result Enter Click on cell with value 25 referenceA50 Enter quot minus sign Click on cell with value 15 reference B50 Press enter key If you enter 15 first and 25 later then the question will be to find result of subtraction 15 25 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Arithmetic pere uns Elle Edit iew lnsert Fgrmat Innis gate lndew elp Fidelge F39DF TWDEEEIUESUDWFDM HEHD v 539 X D gci39n v Evl fnrial vznvg g s 5s E Tin esn v s 4seEise1 4391 E C Formula Elarl E F G H I J K L 44 SUBTRACTING 110i NUMBERS 25 and 15 4n n STEPS 42 1 Enter 25 in eell A55 5 Type in eell C55 er f39ermula bar 43 2 Enter 15 in sell 350 Cli l t en sell 350 44 3 Write in eell C50 F Press Enter key 45 41 Clielt en eell A50 Result In eell C50 45 39 4 4s 4s a 25 15 ml 51 Yr H heetl Sheet2 4 Sheet3 f r Drawr 33 eute hapesr I 3 Ready l3939liIrIIsIIFt EIe Firit El l3939liurIIsIIFt F39III39IerF39IIint 54 k Ijlj35 Excel Formula for Multiplication Excel formula for multiplication is also similar to the formula for addition Only the sig of multiplication will be used The Excel multiplication operator is Micrusuft Excel Arithmetic pera nns Elle Edit Eiew lnsert Fgrmat Tools gate window elp Fidelge PDF WDEEEUESUDW FDD D HEHD 539 X D 39 g r Evlferial vzsvggg g f e4 ns El 99 T113 fist it sen v 8 Aenese I 43934 E C Formula Ear E F G H I J K 44 MULTIPLYING TW NUMBERS 25 and 15 52 53 54 STEPS 55 1 Enter 25 in eell A5 5 Type in eell C5D er fermula bar 55 2 Enter 15 in cell BEE 5 Click en cell 3513 5 3 1Ilulii39rite in cell GEE T Press Enter keyr 58 4 Click en eell A Result In eell GEE 59 44 25 1 5 375 E 52 ES 541 as quoti H 4 r H Sheetl SheetESheet f DLBW39 ls setu hapesv x III 4411 3 Ready l39391ir5ft EEI Arit 7 El l39391irsFt F39IZI393939Er39F39IIiFIt 55439 quotl Let us look at the multiplication of two numbers 25 and 15 The entries will be made in row 60 Enter values as under Cell A50 25 Cell 850 15 The formula for multiplication is A50BSO 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Click on cell C50 to write the formula in that cell Enter Click on cell with number 25 reference A50 Enter Click on cell with number 15 reference B50 Press Enter key The answer is 375 in cell C50 Excel Formula for Division The formula for division is similar to that of multiplication with the difference that the division sign I will be used Micrueuft Excel hrithmetic pera une TEX Elle Edit Eiew insert Fgrmet IDDIS gate window Help Adelge PDF lecfiiti39mei e e e v e v Dagger v Ev ferial Tzsvg e e E 99 Q c cre v e AFEIB75 A E 3 Formula Ber E F 3 H J K ee DIVIDING A NUMBER 240 BY 15 E STEPS ES 1 Enter 24D in cell A75 5 Type l in cell 75 cr fermule her ES 2 Enter 15 in cell B75 5 Cliclt cn cell BTE 7 3 Write in cell 375 7 Freee Enter keyr r1 4 Cliclt en cell A75 Reeult In cell 375 r2 r3 r4 r5 240 15 16 H 4 r n Sheetl rf SheetE rf Sheet3 r Dtaw39 it eetn hapesr e Al 3 H Reedy I Eta l39391iurIIIft Ee Firit i l39391iIrII5IIFt F39III39IerF39IIint Let us divide 240 by 15using Excel formula for division Let us enter numbers in row 75 as follows Cell A75 240 Cell B75 15 The formula for division will be written in cell C75 as under A75B75 The steps are as follows Click the cell A75 Enter 240 in cell A75 Click cell B75 Enter 15 Click cell C75 Enter Click on cell with value 240 reference A75 Enter Click cell with number 15 reference B75 Press enter key The answer 16 will be displayed in cell C75 Excel Formula for Percent The formula for converting percent to fraction uses the symbol To convert 20 to fraction the formula is as under 20 If you enter 20 in cell A99 you can write formula for conversion to fraction by doing the following Enter 20 in cell A99 ln cell B99 enter Click on cell A99 Enterquotquot Press Enter key The answer 02 is given in cell B99 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrsft Excel Arithmetic pera ns Elle Edit ew insert Fgrmat Tools gate window Help Typeequeetinnforhe D ig ft refn EEr ltl aWnv Arial 110v HIE EEEE T63493 E E1IIIIII v I A El 3 D l E F e H an ERTING PERCENT 91 92 93 STEPS Em 1 Enter in cell t E15 Write in cell 399 BE IEliclt en cell A99 9 El Write 93 Frees Enter key Result cell BEE 99 2D 02 Excel Formula for Exponentigtion The symbol for exponentiation is quot The formula for calculating exponents is similar to multiplication with the difference that the carat symbol quot will be used Let us calculate 16 raised to the power 2 by Excel formula for exponentiation The values will be entered in row 85 The steps are Select Cell A85 Enter 16 in this cell Select cell B85 Enter 2 in this cell Select cell C85 Enter quot Select cell with value 16 referenceA85 Enter Select number 2 reference B85 Press Enter key The result 256 is displayed in cell C85 20 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Arithmetic pera nne Eile Edit iew insert Fgrmet leels gate indew elp adage F39DF DWEQJE v Ev jerial C85 v f3 a9HEEILElEE e e r FermuleEer E F G H I J K I r CALCULATING 15 iii FE re STEPS El 1 Enter l in eellAEEn Bi 2 Enter in eell BEE 82 Write in eell C35 83 1 Cliek en eell A35 35 1E E Type A in eell C35 erferrnule leer E Cliek en eell BEE T Preee Enter key Reeult In eell C35 H 4 r gtISheet15heet2j3heet3f lil Il Drew 3 AgteShepESTK 3 I Reedy l39391iIrII5IIFt EIE Ftr39it i IE l39391iIrII5IIt39t F39III393939Er39F39IIirIl Sill KIDNE Recommended Homework Download worksheet MTH302 Iec 02xls from the course web site Change values to see change in results Set up new worksheets for each Excel operator with different values Set up worksheets with combinations of operations 21 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 3 Applications of Basic Mathematics Part 2 OBJECTIVES The objectives of the lecture are to learn about o Evaluations 0 Calculate Gross Earnings 0 Using Microsoft Excel Evaluation In order to successfully complete this course the student is required to meet the evaluation criteria 0 Evaluation Criterion 1 0 Full participation is expected for this course 0 Evaluation Criterion 2 o All assignments must be completed by the closing date o Evaluation Criterion 3 0 Overall grade will be based on VU existing Grading Rules o Evaluation Criterion 4 o All requirements must be met in order to pass the course My There will be a term exam and one final exam there will also be 4 assignments The final exam will be comprehensive These will contribute the following percentages to the final grade Mid Term Exam 35 Final 50 4Assignments 15 Collaboration The students are encouraged to develop collaboration in studying this course You are advised to carry out discussions with other students on different topics It will be in your own interest to prepare your own solutions to Assignments You are advised to make your original original submissions as copying other students assignments will have negative impact on your studies ETHICS Be advised that as good students your motto should be 0 No copying o No cheating o No short cuts 22 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Methodology There will be 45 lectures each of 50 minutes duration The lectures will be delivered in a mixture of Urdu and Englis The lectures will be heavily supported by slide presentations The slides available on the VU website before the actual lecture is televised Students are encouraged to carry out preparatory reading before the lecture This course has its own page on the VU s web site There are lecture slides as well as other supporting material available on the web site Links to a webbased discussion and bulletin board will also been provided Teaching assistants will be assigned by VU to provide various forms of assistance such as grading answering questions posted by students and preparation of slides Text and Reference Material This course is based on material from different sources Topics for reading will be indicated on course web site and in professor s handouts A list of reference books to be posted and updated on course web site You are encouraged to regularly visit the course web site for latest guidelines for text and reference material PROBLEMS If you have any problems with understanding of the course please contact mth302vuedupk Types of Employees There may be three types of employees in a company 0 Regular employees drawing a monthly salary 0 Part time employees paid on hourly basis 0 Payments on per piece basis To be able to understand how calculations of gross earnings are done it is important to understand what gross earnings include GROSS EARNLNGSSALARY Gross salary includes the following 0 Basic salary o Allowances Gross salary may include 0 Basic salary 0 House Rent 0 Conveyance allowance 0 Utilities allowance Accordance to the taxation rules if allowances are 50 of basic salary the amount is treated as tax free Any allowances that exceed this amount are considered taxable both for the employee as well as the company Example 1 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs 23 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU What is the taxable income of employee Is any add back to the income of the company Allowances 500010000 x 100 50 Hence allowances are not taxable Total taxable income 10000 Rs Add back to the income of the company 0 Example 2 The salary of an employee is as follows Basic salary 10000 Rs Allowances 7000 Rs What is the taxable income of employee Is any add back to the income of the company Allowances 700010000 x 100 70 Allowed nontaxable allowances 50 05 x 10000 5000 Rs Taxable allowances 70 50 7000 5000 2000 Rs Hence 2000 Rs of allowances are taxable Total taxable income 10000 2000 12000 Rs Add back to the income of the company 20 allowances 2000 Rs Structure of Allowances The common structure of allowances is as under 0 House Rent 45 o Conveyance allowance 25 0 Utilities allowance 25 Example 3 The salary of an employee is as follows Basic salary 10000 Rs What is the amount of allowances if House Rent 45 Conveyance allowance 25 and Utilities allowance 25 House rent allowances 045 x 10000 4500 Rs Conveyance allowance 0025 x 10000 250 Rs Utilities allowance 0025 x 10000 250 Rs Thus total allowances are 4500250250 5000Rs Provident Fund According to local laws a company can establish a Provident Trust Fund for the benefit of the employees By law 111th of Basic Salary per month is deducted by the company from the gross earnings of the employee An equal amount ie 111th of basic salary per month is contributed by the company to the Provident Fund to the account of the employee The company can invest the savings in Provident Fund in Government Approved securities such as defense saving Certificates Interest earned on investments in Provident Fund is credited to the account of the employees in proportion to their share in the Provident Fund Example 4 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs 24 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU What is the amount of deduction on account of contribution to the Provident Trust Fund What is the contribution of the company What is the total saving of the employee per month on account of Provident Trust Fund Employee contribution to Provident Fund 111 x 10000 9091 Rs Company contribution to Provident Fund 111 x 10000 909 1 Rs Total savings of employee in Provident Fund 909 1 909 1 18182 Rs Gratuity Fund According to local laws 3 company can establish a Gratuity Trust Fund for the benefit of the employees By law 111th of Basic Salary per month is contributed by the company to the Gratuity Fund to the account of the employee Thus there is a saving of 111 h of basic salary on behalf of the employee in Gratuity Fund The company can invest the savings in Gratuity Fund in Government Approved securities such as defence saving Certificates Interest earned on investments in Gratuity Fund is credited to the account of the employees in proportion to their share in the Gratuity Fund Example 5 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the contribution of the company on account of gratuity to the Gratuity Trust Fund Company contribution to Gratuity Fund Total savings of employee in Gratuity Fund 111 x 10000 909 1 Rs Leaves All companies have a clear leaves policy The number of leaves allowed varies from company to company Typical leaves allowed may be as under 0 Casual Leave 18 Days per year o Earned Leave 18 Days per year 0 Sick Leave 12 Days per year Example 6 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the cost on account of casual earned and sick lea ves per year if normal working days per month is 22 What is the total cost of lea ves as percent of gross salary Gross salary 10000 5000 15000 Rs Cost of casual leaves per year 18 22 x 12 x 15000 x 12 122727 Rs Cost of earned leaves per year 18 22 x 12 x 15000 x 12 122727 Rs Cost of Sick leaves per year 12 22 x 12 x 15000 x 12 81818 Rs Total cost of leaves per year 122727 122727 81818 327273 Rs Total cost of leaves as percent of gross salary 32727312 x 15000x 100 182 Social Charqes 25 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Social charges comprise leaves group insurance and medical Typical medicalgroup insurance is about 5 of gross salary Other social benefits may include contribution to employee s children s education club membership leave fare assistance etc Such benefits may be about 58 Leaves are 182 of gross salary as calculated in above example Total social charges therefore may be 182 5 58 29 of gross salary Other companies may have more social benefits The 29 social charges are quite common Example 7 The salary of an employee is as follows Basic salary 10000 Rs Allowances 5000 Rs What is the cost of the company on account of leaves 182 group insurancemedical 5 and other social benefits 58 Leaves cost 0 182 x 15000 2730 Rs Group insurancemedical 005 x 15000 750 Rs Other social benefits 0058 x 15000 870 Rs Total social charges 2730 750 870 4350 Rs W Summary of different components of salary is as follows Basic salary Allowances 50 of basic salary Gratuity 909 of basic salary Provident Fund 909 of basic salary Social Charges 29 of gross salary Gross remuneration is pay or salary typically monetary payment for services rendered as in an employment It includes Group insurance medical etc Miscellaneous social charges 1 Basic Salary 2 House rent allowance 3 Conveyance allowance 4 Utilities 5 Provident fund 6 Gratuity fund 7 Leaves 8 9 Bene ts can also include more factors and are not limited to the above list The purpose of the benefits is to increase the economic security of employees Example 8 The salary of an employee is as follows Basic salary 6000 Rs 26 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The calculations are shown in the slide below There is mistake in calculating gross remuneration in example given below Total amount of leaves are 19636 which is the amount for 1 year not 1 month So divide the amount of leaves by 12 and then calculate gross remuneration Microsoft Excel liliHIi 2lec Eile Edit Eiew insert Fgrn39iat Innis gate indow elp D n j g v 11 559611 gzv l l l mnvo Ariel no 32g E E 2341 J15 v 1 3 A E G D E F G H 2 EK rMPLE GRSS ShLARY 3 Eaeicealaly 1311111 1 House Rent Allowance 1145 22111 5 Gouveyauce allowance 2511 151 E Utlitiee tllowauce 2511 151 7 Total llowaucee 15 311111 E F rovitleut Fuutl 555 555 El wu contribution 555 55151 1 Gratuity fuutl 555 5515 11 Earuetl Leave 15 tlaye 22134 12 Gaeual leave 11 tlaye 22134 13 Sick leave 12 tlaye 45115 14 Group luequotlrletlical 511 3111 l ieceocial Gltargee 53 522 15 Total llowaucee 511111 1 Gl ealaly 5111111 15 F rovitleut Fuutl I3 5515 19 Gratuityr Fuutl 5515 2 Leavee 21 Iifiitltereocial Gltargee 13122 22 Total Social Gltargee 211455 23 Grow remuneration 211551quot Convertinq frgction to percent Calculate percent by multiplying fraction by 100 and put the percent sign Percent Fraction X 100 Example 9 Convert 0 1 to percent 01 X 100 10 Common Frgction Common fraction is a fraction having an integer as a numerator and an integer as a denominator For example 12 10100 are common fractions Converting percent into Common Fraction Example 11 20 20 100 02 Decimal fraction Any number written in the form an integer followed by a decimal point followed by a possibly in nite string of digits For example 25 39 etc Converting percent into decimal fraction 27 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 20 02 Percent 20 or 2010002 M Percentage is formed by multiplying a number called the base by a percent called the rate Thus Percentage Base x Rate Example 13 What percentage is 20 x of 120 Here rate 20 20 100 02 Base 120 Percentage 20100 x 120 0r 02 X 120 24 Example 14 What Percentage is 6 of 40 Percentage Rate X Base 006 X 40 24 Base Base PercentageRate Example 15 Find base if Rate 240 024 Percentage 96 Base 96024400 28 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 4 Applications of Basic Mathematics Part 3 OBJECTIVES The objectives of the lecture are to learn about Review Lecture 3 Calculating simple or weighted averages Using Microsoft Excel Gross Remuneration The following slide shows worksheet calculation of Gross remuneration on the basis of 6000 Rs basic salary As explained earlier house rent is 45 of basic salary Conveyance and Utilities Allowance are both 25 of basic salary Both Gratuity and Provident fund are 111th of basic salary The arithmetic formulas are as follows Excel formulas are within brackets Basic salary 6000 Rs House rent 045 x 6000 2700 Rs Excel formula B93045 Conveyance Allowance 0025 x 6000 150 Rs Excel formula B930025 Utilities allowance 0025 x 6000 150 Rs Excel formula B930025 Gross salary 6000 2700 150 150 9000 Rs Excel formula SUM893BQ6 Gratuity 111 x 6000 545 Excel formula ROUND111B930 In the Excel formulas the sign is used before the row and column reference to fix the location of the cell Since house rent CA utilities gratuity and provident fund are calculated with respect to basic salary so by using B93 we fixes the location of cell 893 This feature can be used for quick and correct calculation of all allowances and benefits 39 Microsoft Excel Gross ren39iuneration Elle Edit Eiew insert Fgrmet Tools gate window elp s e ii 3i it s as few M e v vi 2 viii ilit s 125 v Enrial 71o THBIQHEEE i a I i gj39g iiazi quotquotquot 3944quotth H Hse a e e c D E F G EGROSS REMUNERATION Rs 91 3 Amount in Rs Percentage Basic Salary 6000 House Rent 200 045 E13131 150 0025 EUtilities 150 0025 E Gratuity 5454545455 009090909 ER Fund 5454545455 009090909 E E Follow these steps 1 In cell 394 write B93DQ4 2 Press Enter 3 Click on the cell 894 E 4 Move your cursor to the bottom righthand cornor of cell 394 E till the cursor symbol become a plus sign 5 Hold the left mouse button and drag downward till 398 E 6 This lls all the cells by multiplying 393 cell yalue with the m respective percentage 109 29 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Now let us see cell by cell calculation EmitStart39Eastl7tiiiii i ii i iai mats Eiew insert Fgrmal nulls Eata induw elp Adulge PDF ViEWFDrl39I39IUlEIE stav r E39i SUM 139 X J 52 El93 45 a E c D 91 GROSS REMUNERSTIGN Rs 92 a Basic salary l EUUUISEharges 1740 gHouse Rent B93m45l 118130 a CA 15039 as Utilities 150 a Total salary SUD as Gratuity 545 sgPFund 545 IIIIEI satiaastaa its iiiii iiii ii itjittii jiietjtiti Elsiii insert Fgrmat Inuls Eata induw elp Adulge F DF ViEWFEIrlTIUIEIE SET Ei SUM 1quot X J f3 El93l25 a E C D a GROSS REMUNERATIGN Rs 92 93 Basic salary SUUU SCl1arges 1MB 94 House Rent 27011 G Remun 1181310 ea B93ra025 as Utilities 150 a Total salary 9000 as Gratuity 545 as P Fund 545 IIIlEl Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 engage ear quoti a ai g a seagemeJed E Eile Edit yew insert Fgrmet eels gate llIn ow elp ACIDQE F39DF yiewfermulee a 3 e v 2 v are 39 SUM v 4 9quot f3 SLJMlElE13zElEIEj a e e u 91 GROSS REMUNERETIDN Rs 92 as Basic salary 5000 SCharges 1MB 94 Huse Rent 2WD G Remun 11830 as CE 150 as Utilities 15l Tetal salary SUMBQ3BQE ee Gratuity 5amp5 as P Fund 545 lIIIIII In Gratuity and provident calculations the function ROUND is used to round off values to desired number of decimals In our case we used the value after the semicolon to indicate that no decimal is required If you want 1 decimal use the value 1 for 2 decimals use 2 as the second parameter to the ROUND function The first parameter is the expression for calculation 111B93 ailar aaetaa I1mmtastelessea Elle Edit ierv insert Fgrrnet eels gate window Help adage F39DF yiewforrnulee a re e v 2 v i sum r X J a eeurlurnnlreeauj a e e u a REMUNERATIDN Rs 92 as Basic salary 5000 SEharges 1740 a Heuse Rent 2700 G Remun 11830 as CA 150 as Utilities 150 a Total salary 9000 Gratuity eeunnu1r11lrea30l as P Fund 545 1IIIII Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In the calculation for social charges the formula is B9329100 Here 29100 means 29 social charges The sign was not used here to explain another feature of excel If the formula in cell D93 is copied to cell E93 say the cell reference B93 in formula changes to C93 B93 would be needed to fix the value of basic salary in cell E93 Microsoft Excel AddinLNurnherLEaernplesEel Elle Edit Eiew insert Format eels gate window Help Fidelge F39DF ViEWFUrmulaE 139 5 K ya in 21 an r x u e eeeteenee 77777777777 N e e C e E T a REMUNERATIGN as Basic salary SGhares 393201100 a House Rent G Remun 11830 as 150 as Utilities 150 a Total salary ee Gratuity 545 as P Fund 545 Microsoft Excel Adding umherijarnplesEx1 Eile Edit iew insert Format eels gate window Help Ade e F39DF yiewfermulee 1r 5 x DE Elt Ev f Ariel v1e g f g gamp f E a 1quot use v e s e e e E j 91 GROSS REMUNER iTIGN Rs as as Basic salary 5000 SCharges 1740 at House Rent 2700 G Remun 11830 as CA 150 as Utilities 150 a Total salary 9000 as Gratuity 545 as P Fund 545 me 139 32 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU AVERAGE Average Arithmetic Mean Sum N Sum Sum of all data values N number of data values EXAMPLE 1 Data 10 7 9 27 2 Sum 1079272 55 There are 5 data values Average 555 11 ADDING NUMBERS USING MICROSOFT EXCEL Add numbers in a cell Add all contiguous numbers in a row or column Add noncontiguous numbers Add numbers based on one condition Add numbers based on multiple conditions Add numbers based on criteria stored in a separate range Add numbers based on multiple conditions with the Conditional Sum Wizard NQWEWNT A I Add numbers in a cell Type 510 in a cell Result 15 See Example 2 Microsoft Excel Hool Elle Edit ew Insert Fgrmat Iools Qatar window elp Ader PDF 1 to m Eve 1 o SLIM r X oquot r 51E A a C n E F 3 1 2 NUM1NUM2 3 4 5 51I 2 Add all contiguous numbers in a row or column using AutoSum If data values are in contiguous cells of a column click a cell next to last data value in the same column If data values are in contiguous cells of a row then click a cell at right side of last data value Click AutoSum symbol 2 in tool bar 33 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Press ENTER This will add all the data values See Example sasisaiariquotssi Lhasa ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff quot Eile Edit ew insert Fgrmat Inuls Qata induw Help Adulge F39DF Iris Evil 3 Arial vIIII e as ENE 1r 5 A E I D E F 3 H r s ADDING AutSum 11 12 10 13 3915 14 E 3 Add noncontiguous numbers Use the SUM function See Example Micrnsnft Excel Baum Eile Edit ew insert Fgrmat Inuls ats induw Help Adobe PDF it PM Ev Elf SLIM v E J 5 Sumt iiz t123 A a c D E F G H s ADDING USING AutSum Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 4 Add numbers based on one condition Use the SUMIF function to create a total value for one range based on a value in another range Micruauft Excel Addini umher5ExamplesEx1 Elle Edit ew insert Fgrrnat Innls Eata indcw Help indulge F39DF E galI T E T I SLIM v x u a SUMIFia3FM2quotBuchananquot5334234I A E c n I 35 Salcancracn lnucicc 3 Buchanan 38 Buchanan 39 Suuama an Suuama 41 Buchanan 42 Dcdawcrth 22500 43 a quotBuchananquotB3 4 B42 Microst Excel iiiuninitialleheragExampllEsE11 E Ede Edit yaw naert F grmat incl esta Diet ndalgae REF an Ariai urn i a 1 a Halal 139 e Surn Elf inunices fur Buchanan 29mm I A Fnrmula Ear 35 SEIES E II SOM Invoice 3 Buchanan MENU 3 Buchanan 39 Suvama BUM a Suuama 20000 41 Buchanan lm 42 odlawcrth 43 I15 Sum aiinveicecicr Buchanan 39 quot 45 5 Add numbers based on multiple conditions Use the IF and SUM functions to do this task 6 Add numbers based on criteria stored in a separate range Copyright Virtual University of Pakistan VU 35 Business Mathematics amp Statistics MTH 302 VU Use the DSUM function to do this task DSUM Adds the numbers in a column of a list or database that match conditions you specify Syntax DSUMdatabasefieldcriteria Database is the range of cells that makes up the list or database Field indicates which column is used in the function Criteria is the range of cells that contains the conditions you specify DSUM EXAMPLE DSUMA4E10quotProfit A1F2 The total profit from apple trees with a height between 10 and 16 75 AVERAGE USING MICROSOFT EXCEL AVERAGE Returns the average arithmetic mean of the arguments Syntax AVERAGEnumber1number2 Number1 number2 are 1 to 30 numeric arguments for which you want the average Calculate the average of numbers in a contiguous row or column Elle Edit Elevll insert Fgrrnat IDEIIS Eata indnw elp FadInge F39DF DEEJE tag u Ev l 3 Arial 43 39E El 9amp1 T 1 A4 139 E A El 3 D 55 HER tGE EE 5 Data 53 1 ES F39III m 27 2 a 1 1 31 I I 5 TH Calculate the average of numbers not in contiguous row or column 36 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hiicrnsnft Excel AddingNun1her5Exan1plesE151 Eile Edit ew Lnsert Fgrmat luuls Qata induw Help adage F39DF using autnsum cit E v I r E r 3 SLIM v H v a A ERAGE AFEz EEI BEj a a r U E F as VERAGE OF NUMBERS NOT CONTIGUOUS a Data a 3910 F9 El a 2 82 33 a AUERAGEM73MBUmBS BE Micrnsn Excel hddini umherijampleaE31 Eile Edit ew insert Fgrmat Iciclls gaita window Help Adnge F39DF using autDSUITI r Ev l 1fj iiirial 1DTBEEEEE quot El ti EI V r a E t D E F as AVERAGE OF NUMBERS NOT CONTIGUOUS a Data a 3910 Ei El a 2 82 EB an L5 EEI I HI i 37 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU WEIGHTED AVERAGE Weighted average is one type of arithmetic mean of a set of data in which some elements of the set carry more importance weight than others If X1 X2 X3 Xn is a set of n number of data and w1 wz W3 wn are corresponding weights of the data then Weighted average X1W1 X2W2 X3W3 XnXWn Be careful about one thing that the weights should be in fraction Grades are often computed using a weighted average Suppose the weightage of homework is 10 quizzes 20 and tests 70 Here weights of homework quizzes tests are already in fraction ie10 01 20 02 70 07 respectively If Ahmad has a homework grade of 92 a quiz grade of 68 and a test grade of 81then Ahmad39s overall grade 01092 02068 07081 795 Let us see another example Labor hours per Grade of Labor unit of labor Hourly wages Rs Skilled 6 300 Semiskilled 3 200 Unskilled l 100 Here weights Labor hours per unit of labor are not in fraction So rst we convert them to fraction Total labor hours 6 3 1 10 Grade of Labor hours per unit of labor labor in fraction Skilled 6 10 06 Semiskilled 3 10 03 Unskilled l 10 01 Weighted average 06300 03200 01100 38 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 250 Rs per hour LECTURE 5 Applications of Basic Mathematics Part 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review of Lecture 4 0 Basic calculations of percentages salaries and investments using Microsoft Excel PERCENTAGE CHANGE Monday s Sales were Rs 1000 and grew to Rs 2500 the next day Find the percent change METHOD Change Final value initial value Percentage change Change initial value x 100 CALCULATION Initial value 1000 Final value 2500 Change 1500 Change 15001000 x 100 150 The calculations using Excel are given below First the entries of data were made as follows Cell C4 1000 Cell C5 2500 In cell C6 the formula for increase was C5 C4 The result was 1500 In cell C7 the formula for percentage change was C6C4100 39 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The result 150 is shown in the next slide Micruen Excel Perce ntageji hange Eile Edit Eiew insert Fgrmat Inels gate indew Help adage F39DF 3957quot Evif Furial 20E e as T lg ml L B 79 l 1 i163 39III39 E 139 39 A El 393 I D E F 393 2 PERCENTAGE CHANGE R5 3 4 Men ay 5 Next Day 5 Increase r Increase 150 a 1 5 j l 40 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EXAMPLE 1 How many Percent is Next Day s sale with reference to Monday s Sale Monday s sale 1000 Next day s sale 2500 Next day s sale as 25001000 x 100 250 Two and a half times Nicrnxnft Excel MTHEElec handnut E Eile Edit ew Lnserl Fgrmal Innls gate window elp El if x v r v E v 43 1 I SLIM v E J 5 D13fD121DD x a E E F e H many Percent is Next Day s sale m 1 ith refernee t Mndey s Sale 11 12 Mndey39e sale 13 Next day39s Sale 14 Next day39s sale as 15 f Mnday s sale D13jD12t1UU IE 41 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hicrnsuft Excel tumIBElec handout Eile Edit ew insert Fgrmat Innls Qatar indnw elp nemaaaenses iiirial 3qu e Di 1 A E r E F 9 Hw many Percent is Next Day s sale mith refernce t Mnday s Sale 5 F E Inc El El 43 H mm M iii 11 12 Mnay s sale 13 Next ay39s sale H Next day39s sale as f Mnay39s sale 250 15 15 EXAMPLE 2 In the making of dried fruit 15kg of fresh fruit shrinks to 3 kg of dried fruit Find the percentage change Calculation Original fruit 15 kg Final fruit 3 kg Change 315 12 change 1215x 100 80 Size was reduced by 80 42 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Percentagejihange Eile Edit iew insert Fgrmat Innis Qatar indow Help Fido e F39DF El gt 21 up SLIP391 139 X all t D2HD19 UU A E C D E F G 19 riinal fruit 15 20 Final fruit 21 Chane in meiht 12 Chane DZ1jD191UU 23 21 Microsoft Excel Percentagejihange Eile Edit ew insert Fgrmat Iools Eata window Help Fider F DF D l i a r 3 3 E T 5 TIDTB at v r A E I13 1 lN WEIGHT IE 19 riinal fruit 15 2n Final fruit 21 Chane in xElht 12 thehange 22 23 a 25 Calculations in Excel were done as follows Data entry Cell D19 15 Cell D20 3 Formulas 43 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Formula for change in Cell D21 D20 D19 Formula for change in Cell D22 D21D19100 Resuhs Cell D21 12 kg Cell D22 80 EXAMPLE 3 After mixing with water the weight of cotton increased from 3 kg to 15 kg Find the percentage change CALCULATION Original weight 3 kg Final weight 15 kg Change 153 12 change 123 X 100 400 Weight increased by 400 Micrnenft Excel ercentageEhange E Eile Edit ew insert Fgrn39iat Innls Qatar indnw Help Filing3 F DF a asquot I iv 2 if Firial 10 w a a t in a JED 139 r A E C D I E F 23 a CHANGE IN WEIGHT 25 25 riginal weight f cttn kg 2 Final weight f cttn 15 kg 23 Change in weight 12 kg 29 all change 400 la y l 31 Calculations in Excel were done as follows Data entry Cell D26 3 Cell D27 15 Formulas Formula for change in Cell D28 D27 D26 Formula for change in Cell D29 D28D26100 Resuhs 44 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cell D28 12 kg Cell D29 400 EXAMPLE 4 A union signed a three year collective agreement that provided for wage increases of 3 2 and l in successive years An employee is currently earning 5000 rupees per month What will be the salary per month at the end of the term of the contract Calculation 50001 31 21 1 5000x 103x 102x 101 5306 Rs Calculations using Excel are shown in the following slides Microsoft Excel Percentage hange Eile Edit ew insert Fgrmalz Iools gate window elp Fider F39DF e v 2 v 39L 39JJZ E 1 SLIM v 1 J r RDUNDEC351C3l3l lIIIIIIIIIjI a El e D E F 3392 39 33 SALARY IN YEAR 1 2 AND 31 35 Salary year 1 Rs 35 Increase year 1 3 In SaIary year 2 RUNDC351C36l ea 1 D 39 45 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Percentage l1ange Eile Edit ew insert Fgrmet leels gete indew Help Adelge F39DF El as 2 L e 7e TE it v M v x J r eeumeressrne nnnnmnj I e E F I A El 33 SALARY IN YEAR 1 2 AND 34 as Salary year 1 Re 35 Increase year 1 3 Salary year 2 5150 Re 33 Increase year 2 2 39 Salary year Re 4 Increase year 1 Salary end f year RUNDCSQ1C4UI 42 1 I RDUHDI numter numdigitsl I LI39I 1 43 Ufa micrusuft Excel ercentage hange Eile Edit Eiew insert Fgrmet eels gete indew elp Adege PDF ll E a El E 1quot f PurlEll 139 1D 1 B e 1 i L quotIquot 2 1quot 39 i hr Itquot 4 e D E F A El 33 SALARY IN YEAR 1 2 AND 34 as Salary year 1 35 Increase year 1 3 Salary year 2 33 Increase year 2 39 Salary year 4 Increase year 1 41 Salary end 1quot year Re E I I 43913 46 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Calculations in Excel were done as follows Data entry Cell C35 5000 Cell C36 3 Cell C38 2 Cell C40 1 Formulas Formula for salary in year 2 in Cell C37 ROUNDC351C361000 Formula for salary year 3 in Cell C39 ROUND C371C381000 Formula for salary at the end of year 3 in Cell C41 ROUNDC391C391000 Resu s Cell C37 5150 Rs Cell C39 5253 Rs Cell C41 5306 Rs EXAMPLE 5 An investment has been made for a period of 4 years Rates of return for each year are 4 8 10 and 9 respectively If you invested Rs 100000 at the beginning of the term how much will you have at the end of the last year Micrnsuft Excel ercentage l1ange Eile Edit Eiew insert Fgrmat Innls Qatar indnw elp Fadage F39DF E E v E 3 1 3939 sum v x J a RDUNDECdE1C4F I1IIIIIIJHIIJ A El 3 n E F 45 INVESTMENT HT THE END IF 4 YEHRS 15 Investment year 1 100000 Rs 4 Increase year 1 4 3911 VaIue in year 2 RUNIZIDCLE IEii11 Cir171i 49 Increase year 2 1000i a Value in year 112320 Rs 51 Increase year 10 In 52 Value in year 101088 Rs 53 Increase year 4 36 54 Ualyeendyear 110130 Rs 47 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel percentage hange E Eile Edit View insert Fgrmat Incls Qatar indcw Help Rulings P39DF D g v v Evf Arial 20 s a is 1L 391 a 54 v s RDUNDEC521C53I1IIIIIIJJII a E c D E F 45 INVESTMENT HT THE END NF 4 YEHRS 45 Investment year 1 100000 Rs 4 Increase year 1 4 fe a Value in year 2 104000 Rs 49 Increase year 2 Va 5 Value in year 112320 Rs 51 Increase year 3910 Va 52 Value in year 101088 Rs 53 Increase year 4 In 54lVaIue end year 4 110185le Calculations in Excel were done as follows Data entry Cell C46 100000 Cell C47 4 Cell C49 8 Cell C51 10 Cell C53 9 Formulas Formula for value in year 2 in Cell C48 ROUNDC461C471000 Formula for value in year 3 in Cell C50 ROUNDC481C491000 Formula for value in year 4 in Cell C52 ROUNDC501C511000 Formula for salary end of year 4 in Cell 054 ROUNDC521CS31000 Resu s Cell C48 104000 Rs Cell C50 112320 Rs Cell C52 101088 Rs Cell C54 110186Rs 48 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 6 Applications of Basic Mathematics Part 5 OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 5 0 Discount 0 Simple and compound interest o Average due date interest on drawings and calendar REVISION LECTURE 5 A chartered bank is lowering the interest rate on its loans m 9 to 7 What will be the percent decrease in the interest rate on a given balance A chartered bank is increasing the interest rate on its loans m 7 to 9 What will be the percent increase in the interest rate on a given balance As we learnt in lecture 5 the calculation will be as follows Decrease in interest rate 79 2 decrease 29 x 100 222 Increase in interest rate 97 2 decrease 27 x 100 286 The calculations in Excel are shown in the following slides DECREASE IN RATE Data entry Cell F4 9 Cell F5 7 Formulas Formula for decrease in Cell F6 F5F4 Formula for decrease in Cell F7 F6F4100 Resu s Cell F6 2 Cell F7 222 INCREASE IN RATE Data entry Cell F14 7 Cell F15 9 Formulas Formula for increase in Cell F16 F15F14 Formula for increase in Cell F17 F16F14100 Resu s Cell F16 2 Cell F17 286 49 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU IIIIicrsft Excel IHITHIIELecE handut Eile Edit Eiew Insert Fgrmat Innls gate indnw Help nenseees rarer IEreiI as vwv aueeeaeessaees Ii Big 2 D E F G H 2 DEC IN INTEEST 4 rim inal Interest ate 5 rise Interest te a Decrease D wa s E39E Atii r nrial 10 BI E eg i iii i i BEE quotF F3 a E r n E F e H 12 IN INTEEST 13 14 rizj inal Interest Rate 15 Revise Interest Rate re Increase 1 Increase 235 re 50 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The Definition of a Stock Plain and simple stock is a share in the ownership of a company Stock represents a claim on the company39s assets and earnings As you acquire more stock your ownership stake in the company becomes greater Whether you say shares equity or stock it all means the same thing Stock yield With stocks yield can refer to the rate of income generated from a stock in the form of regular dividends This is often represented in percentage form calculated as the annual dividend payments divided by the stock39s current share price Earnings per share EPS The EPS is the total profits of a company divided by the number of shares A company with 1 billion in earnings and 200 million shares would have earnings of 5 per share Priceearnings ratio A valuation ratio of a company39s current share price compared to its pershare earnings Calculated as Market Value per Share Earnings per Share EPS For example if a company is currently trading at 43 a share and earnings over the last 12 months were 195 per share the PE ratio for the stock would be 2205 43195 Outstanding shares Stock currently held by investors including restricted shares owned by the company39s officers and insiders as well as those held by the public Shares that have been repurchased by the company are not considered outstanding stock Net current asset value per shareNCAVPS NCAVPS is calculated by taking a company39s current assets and subtracting the total liabilities and then dividing the result by the total number of shares outstanding Currentquot laaeta Tatal Lialiilitiea H iul39PE a t Eltarea uta tan ling Current Assets The value of all assets that are reasonably expected to be converted into cash within one year in the normal course of business Current assets include cash accounts receivable inventory marketable securities prepaid expenses and other liquid assets that can be readily converted to cash 51 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Liabilities A company39s legal debts or obligations that arise during the course of business operations Market value The price at which investors buy or sell a share of stock at a given time Face value Original cost of a share of stock which is shown on the certificate Also referred to as quotpar valuequot Face value is usually a very small amount that bears no relationship to its market price Dividend Usually a company distributes a part of the profit it earns as dividend For example A company may have earned a profit of Rs 1 crore in 200304 It keeps half that amount within the company This will be utilised on buying new machinery or more raw materials or even to reduce its borrowing from the bank It distributes the other half as dividend Assume that the capital of this company is divided into 10000 shares That would mean half the profit ie Rs 50 lakh Rs 5 million would be divided by 10000 shares each share would earn Rs 500 The dividend would then be Rs 500 per share If you own 100 shares of the company you will get a cheque of Rs 50000 100 shares x Rs 500 from the company Sometimes the dividend is given as a percentage i e the company says it has declared a dividend of 50 percent It39s important to remember that this dividend is a percentage of the share39s face value This means if the face value of your share is Rs 10 a 50 percent dividend will mean a dividend of Rs 5 per share BUYING SHARES If you buy 100 shares at Rs 6250 per share with a 2 commission calculate your total cost m 100 Rs 6250 Rs 6250 002 Rs 6250 125 Total Rs 6375 RETURN ON INVESTMENT Suppose you bought 100 shares at Rs 5225 and sold them after 1 year at Rs 68 With a 1 commission rate of buying selling the stock and 10 dividend per share is due on these shares Face value of each share is 10Rs What is your return on investment Bought 52 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 100 shares at Rs 5225 522500 Commission at 1 5225 Total Costs 5225 5225 527725 Sold 100 shares at Rs 68 680000 Commission at 1 6800 Total Sale 6800 68 673200 Gain Net receipts 673200 Total cost 527725 Net Gain 6732 527725 1 45475 Dividends 1001010 10000 Total Gain 145475 100 155475 Return on investment 155475527725100 2946 The calculations using Excel were made as follows BOUGHT Data entry Cell B21 100 Cell B22 5225 Formulas Formula for Cost of 100 shares at Rs 5225 in Cell B23 B21B22 Formula for Commission at 1 in Cell B24 B23001 Formula for Total Costs in Cell B25 B23B24 Resu s Cell B23 5225 Cell B24 5225 Cell B25 527725 53 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel lullllH ELec handout Elle Edit Eiew insert Format Innis gate window elp neusssos s E E39ilil al i39ic i nrial Tan T g EH 1 r EEEIEEE1DD A E i an B uiht 21 Shares 10 22 Rate 5225 23 Cost of1 shares at Rs 5225 5225 24 Commission 5225 25 Total Costs 527725 25 s a 3393 so m Data entry Cell B28 68 Formulas Formula for sale of 100 shares at Rs 68 in Cell B29 B21B28 Formula for Commission at 1 in Cell B30 B29001 Formula for Total Sale in Cell 831 829BBO Resu s Cell B29 6800 Cell B30 68 Cell B31 6732 lulicrnsnft Excel iiilTl39lE ELecll handout Eile Edit glow insert Format Innis Qatar indnw elp D o El Hist E if it is Arial 39239 ratr E E as v or Bssrsssnnn A E C quot quot quot ss so 2 ss t 2v 23 29 100 shares atRs68 6800 an Commission at1i 68 31 Total Sale 54 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GAIN Formulas Formulator Net receipts in Cell B34 B31 Formula for Total cost in Cell B35 B25 Formula for Net Gain in Cell 836 BB1825 Formula for Gain in Cell BB7 B36B35100 Resu s Cell B34 6732 Cell B35 527725 Cell B36 145475 Cell B37 2757 DISCOUNT Discount is Rebate or reduction in price Discount is expressed as of list price Example List price 2200 Discount Rate 15 Discount 2200 x 0 15 330 Calculation using Excel along with formula is given in the following slide Microsoft Excel lil39l39IIEDELec handout Eile Edit iew insert Fgrmal Innis gate window elp iVquotVii W Dena am it w azvmiii newera v Arial m HIE EEE EEEE E v v n E51 139 i3 A El 3 D E I F G 33 QJSCUNT 39 List price 4 Discunt Rate 15 a 11 Discunt Frmula BBtB lUi1UU 42 43 NET COST PRICE Net Cost Price List price Discount Example List price 4500 Rs Discount 20 Net cost price 55 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Net cost price 4500 20 of 4500 4500 02 x4500 4500 900 3600 Rs Calculation using Excel along with formula is given in the following slide Microsoft Excel MTllii ELoo handout Eile Edit Eiew insert Formal Tools gate window elp gene eee it e m eaveee new iirial TIDTBIHEEE E133 3 39 D54 1 ii A E C D E F Iiii 45 NET CST PRICE 45 d 4 List price Rs 43 in Diseunt est 1 3934 19 Net Get Price Re El 51 Fr mula Cell 19 Bd IB4Bi1 52 SIMPLE INTEREST P Principal R Rate of interest pereent per annum T Time in years I Simple interest then I P R T 100 Thus total amount A to be paid at the end of T years P 1 Example P Rs 500 T 4 years R 11 Find simple interest I P x Tx R 100 500x4x 11100 RS 220 Calculation using Excel along with formula is given in the following slide 56 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lulicruenft Excel MTHE ELec handuut Eile Edit ew insert Fgrmal IIZIIZIIS Eata induw Help D tttevr v Elii Furial vl vBI 39 gf g g FEE T A E I D E 55 SIMPLE INTEREST 5 5E Principal P Rs 59 Time Jeritl T 1 Year El R 1 1 51 Interest Rs 52 53 Fr mula in Cell 61 53i39559i39i t1 Ed COMPOUND INTEREST Compound Interest also attracts interest Example P 800 Interest year 1 0 1 x 800 80 New P 800 80 880 Interest on 880 0 1 X 880 88 New P 880 88 968 Calculation using Excel along with formula is given in the following slide 57 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel MTHIi ELec handuut Eile Edit Eiew insert Fgrmat eels Qatar window Help D r 5E1 iiirial 1 1 r E I H HFE 1quot 33 A 3 EE CMPUND INTEREST Ei39 res Principal P Rs EB Interest 10 quotA re Interest year1 80 Rs Frmule BEB tBBB100 r1 New P 880 Rs Frmule BEBBTU 2 Interest n 880 88 Rs Frmula B1BEBI100 3 New P 368 Frmule B1B2 r4 Co Compound Interest Formula S Money accrued after n years also called compound amount P Principal r Rate of interest n Number of periods S P1 r100 n Compound interest S P Example Calculate compound interest earned on Rs 750 invested at 12 per annum for 8 years S P1r1008 750112100A8 185 7 Rs Compound interest 1857 750 1107 Rs esvesttl memen v ee 3 use ee 39i v 3 D E F 3 H 58 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Calculation using Excel along with formula is given in the following lide lulicrueuft Excel MWB ZLec handuut Eile Edit Eiew insert Fgrmat eels gate window elp D 5ev E El r mne E 3395 BEES iii atrial v 1 1 B I H em 1 r e e D E F rr CMPUND INTEREST USING FE re Princiel P 5 Re El Interest 12 e1 Perio 8 Years 32 Meney accrue 1857 Rs 33 34 Formula RUNDB7Q1BBUI1UU BB1U 35 III 59 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 7 Applications of Basic Mathematics OBJECTIVES The objectives of the lecture are to learn about 0 Scope of Module 2 Review of lecture 6 Annuity Accumulated value Accumulation Factor Discount Factor Discounted value Algebraic operations Exponents Solving Linear equations Module 2 Module 2 covers the following lectures Linear Equations Lectures 7 Investments Lectures 8 Matrices Lecture 9 Ratios amp Proportions and Index Numbers Lecture 10 Annuity It some point in your life you may have had to make a series of fixed payments over a period of time such as rent or car payments or have received a series of payments over a period of time such as bond coupons These are called annuities Annuities are essentially series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed period of time An annuity is a type of investment that can provide a steady stream of income over a long period of time For this reason annuities are typically used to build retirement income although they can also be a tool to save for a child s education create a trust fund or provide for a surviving spouse or heirs The most common payment frequencies are yearly once a year semiannually twice a year quarterly four times a year and monthly once a month Qlculatinq the Future value or accumulated value of an Annuitv If you know how much you can invest per period for a certain time period the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate If you are making payments on a loan the future value is useful for determining the total cost of the loan Let39s now run through Example 1 Consider the following annuity cash flow schedule 60 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU End f each period ifquot i n 1 1 3 4 5 l l l l l ll l l H H H ll 5mm 51 EDD 51am 51 EDD 51mm in J Payment paid er minivan at end of each patina In order to calculate the future value of the annuity we have to calculate the future value of each cash flow Let39s assume that you are receiving 1000 every year for the next five years and you invested each payment at 5 The following diagram shows how much you would have at the end of the five yearpenod El 1 E 3 4 5 ll lll lll lll lll lll II III III III III III 51993 Ei E l Ei E l H Si litEJE it e sin oilmai39 a Magnet 3 5t tl t 5l stinger 1 5i iliioeilil warm Ei it f gma i Future allu nf an ndinary Ann witty S a d Since we have to add the future value of each payment you may have noticed that if you have an annuity with many cash flows it would take a long time to calculate all the future values and then add them together Fortunately mathematics provides a formula that serves as a short cut for finding the accumulated value of all cash flows received from an annuity anquot 1 Iquotquot39IICII39Iinalr39I IEIII39Inu39rlvI I Hi CPayment per period or amount of annuity i interest rate n number of payments 1 in 1 i is called accumulation factor for n periods Accumulated value of n period payment per period x accumulation factor for n periods If we were to use the above formula for Example 1 above this is the result 1 005315 1 Fri 1000 liltlruzllnar39I F39II39II39I39J39t39r39 El 05 1ooo553 552563 61 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Note that the 001 difference between 552564 and 552563 is due to a rounding error in the first calculation Each of the values of the first calculation must be rounded to the nearest penny the more you have to round numbers in a calculation the more likely rounding errors will occur So the above formula not only provides a shortcut to finding FV of an ordinary annuity but also gives a more accurate result Calculating the Present Value or discounted value of an Annuity If you would like to determine today39s value of a series of future payments you need to use the formula that calculates the present value of an ordinary annuity For Example 2 we39ll use the same annuity cash flow schedule as we did in Example 1 To obtain the total discounted value we need to take the present value of each future payment and as we did in Example 1 add the cash flows together u 1 1 s 4 s lll m lll m lll lll ll ll ll ll ll Ill smuu smuu smuu smuu smuu iltlili z sususs nus sumu Lusii smuu E assumius nusi i39h iii sass t 3 us m mums trust Allis Prusun ltiiutallmu uf an rdiuuw nmuity 54319913 Again calculating and adding all these values will take a considerable amount of time especially if we expect many future payments As such there is a mathematical shortcut we can use for PV of ordinary annuity l 1i39quot F39I39i39lllCtlrzlinstquotI Finnuit39I I I C Cash flow per period i interest rate n number of payments 1 1 in i is called discount factor for n periods Thus Discounted value of n period payment per period x discount factor for n period The formula provides us with the PV in a few easy steps Here is the calculation of the annuity represented in the diagram for Example 2 Fquot39quot 1000 Hi ICirzllnsirlI anurtlI D IDS 1ooo433 432948 1 1 un t 5 62 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NOTATIONS The following notations are used in calculations of Annuity R Amount of annuity N Number of payments Interest rater per conversion period S Accumulated value A Discounted or present worth of an annuity ACCUMULATED VALUE The accumulated value S of an annuity is the total payments made including the interest The formula for Accumulated Value S is as follows S r 1iquotn 1i Accumulation factor for n payments 1 in 1 i It may be seen that Accumulated value Payment per period x Accumulation factor for n payments The discounted or present worth of an annuity is the value in today s rupee value As an example if we deposit 100 rupees and get 110 rupees ie 10 interest on 100 Rs which is 10010100 10 Rs so total amount is 10010 110 or simply 100 110100 100 1 01 10011 110 after one year the Present Worth or of 110 rupees will be 100 Here 110 will be future value of 100 at the end of year 1 The amount 110 if invested again can be Rs 121 after year 2 Le 10 of110 is 11010100 11 so total amount is 11011 121 The present value of Rs 121 at the end of year 2 will also be 100 DISCOUNT FACTOR ANQISCOUNTQ VA When future value is converted into present worth the rate at which the calculations are made is called Discount factor rate In the previous example 10 was used to make the calculations This rate is called Discount Rate The present worth of future payments is called Discounted Value EXAMPLE 1 ACCUMULATION FACTOR AF FOR n PAYMENTS Calculate Accumulation Factor and Accumulated value when rate of interest i 425 Number of periods n 18 Amount of Annuity R 10000 Rs Accumulation Factor AF 1 00425quot18100425 2624 Accumulated Value S 10000x 2624 260240 Rs EXAMPLE 2 DISCOUNT VALUE DV In the above example calculate the value of all payments at the beginning of term of annuity ie present value or discounted value Discount rate 425 Number of periods 18 63 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Amount of annuity 10000 Rs Value of all payments at the beginning of term of Annuity or discounted value Payment per period x Discount Factor DF Formula for Discount Factor 111iquotni 11100425quot1800425 124059 Discounted value 10000 X 124059 124059 Rs EXAMPLE 3 DISCOUNTE VALUE DV How much money deposited now will provide payments of Rs 2000 at the end of each half year for 10 years if interest is 11 compounded sixmonthly Amount of annuity 2000Rs Rate of interest i 11 2 0055 Number of periods n 10 X 2 20 DISCOUNTED VALUE 2000 x 11 10055A20 0055 2000 X1195 2390077 ALGEBRAIC OPERATIONS Algebraic Expression indicates the mathematical operations to be carried out on a combination of NUMBERS and VARIABLES The components of an algebraic expression are separated by Addition and Subtraction In the expression 2x2 3x 1 the components 2x2 3x and 1 are separated by minus sngn Algebraic 3 I H H E aperations J a I H 54 4IIquotquot39T39E quotTI 39 quot quotquot1 Mammal Binnmial TrinnmialiPa numial HUT F iEIHg I I I y y 55 mailE than H f 1Tern1 41 rizl J 39 xi 33 3 xy4 f In algebraic expressions there are four types of terms 0 Monomial ie 1 term Example 3x2 0 Binomial ie 2 terms Example 3x2xy o Trinomial ie 3 terms Example 3x2xy6y2 o Polynomial ie more than 1 term Binomial and trinomial examples are also polynomial Algebraic operations in an expression consist of one or more FACTORs separated by MULTIPLICATION or DIVISION sign Multiplication is assumed when two factors are written beside each other Example xy xy Division is assumed when one factor is written under an other Example 36x2y 50xy2 64 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Algebraic 51233 31 I peratinns39 Hiquot In 51 1 I i 39 i 535 llitlr Term each BE in an Ergmssmr f le 139 n 7 II I I7 AETEEE sqmmte hjr r mural two when on fan hrs arew thn factor I39E 39 i39h39l39l 39I 39I 39I I I ZITE eat1 ENEEIquot u mfe 39 an I rFE rr39i Factors can be further subdivided into NUMERICAL and LITERAL coefficients r Algebraic wilt1 a H I H H AIDE Illii a I 1 lgebraic EHpI EEEi I I n i iiiT in 397 39i 1 TEI ITIEI n 39i uiilliil IHLi a i w nmial Binmi l Trinumial Pa mumial IiEHigur IL 41 fitquot rt FEET HES L 393 39 Numerical Literal Eingf itm E39mgf c m39 There are two steps for Division by a monomial 1 Identify factors in the numerator and denominator 2 Cancel factors in the numerator and denominator Example 36x y 60xy2 36 can be factored as 3 x 12 60 can be factored as 5 x 12 x2y can be factored as xxy xy2 can be factored as xyy Thus the expression is converted to 3 x 12xxy 5 x 12xyy 65 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 12xxy in both numerator and denominator cancel each other The result is 3x5y a 7 E59111 3395 1 39 5E Rim it Fam FACTUES H1 m u FulfilEf i f 11 if 1 39 39 139439539 i39EI39J NEEHIE n4 swimmer r nittracer Factors f in u FEEJFFEEi39tli f 11 I 39 Ill1 i E L3939 iquotiquotE 2 F212 n4 quot Another example of division by a monomial is 48a2 32ab8a Here the steps are 1 Divide each term in the numerator by the denominator 2 Cancel factors in the numerator and denominator 48a2 8a 8x6aa 8a 6a 32ab 8a 4x8ab 8a 4b The answer is 6a 4b War he Ex wh 777i7rir 7 I 7 7 39I39quot39quot39 t J H HitI 39I I II J39 I139I IiiE1 I LI J J I I I 39I 39l m 39E rzzirrze1mtl1lrr a Jr E 39I 39 NEE Ill39E39 i ELT39i i Hi Eili i39 II39 quot mud191 Factors 7 l El 1 mill FEEJf39f39iEh li i and i 13942quot i E I393939 i i E 2 Hill 13quot How to multiply polynomials Look at the example x2x2 3x 1 Here each term in the trinomial22x2 3x 1 is multiplied by x X2X X3X X1 2x3 3x x 66 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Please note that product of two negatives is positive Mll pljting Putin mi elf Emmett5 KE3I it 1139 3116tfxIExltliffllli39u Thepm citiet efttee negeiitie quantities is peeiiiiie 3X6y3 X2232 Exponent of a term means calculating some power of that term In the example we are required to work out exponent of 3x6y3 x223 to the power of 2 The steps in this calculation are 1 Simplify inside the brackets first 2 Square each factor 3 Simplify In the first step the expression 3x6y3 x223 is first simplified to 3x4y323 In the next step we take squares The resulting expression is 32X42y32z32 9X8 y6 z6 67 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Step 2 Step 3 mp image Sg39mzre ench z ctnr mp the brackets mt LINEAR EQUATION If there is an expression A 9 137 how do we calculate the value of A A 137 9 128 As you see the term 9 was shifted to the right of the equality To solve linear equations 1 Collect like terms 2 Divide both sides by numerical coefficient Step 1 X 34125 0025X X 0025X 34125 X10025 34125 0975X 34125 Step 2 X 341 250975 350 68 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU l 25 Divicfe bath itquot SEE 95 by H975 69 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 8 Compound Interest Calculate returns from investments Annuities ExcelFunc ons OBJECTIVES The objectives of the lecture are to learn about 0 Review of lecture 7 0 Compound Interest 0 Calculate returns from investments 0 Annuities 0 Excel Functions m Returns the cumulative interest paid on a loan between startperiod and endperiod If this function is not available and returns the NAME error install and load the Analysis ToolPak addin The syntax is as follows CUMIPMTratenperpvstartperiodendperiodtype Rate interest rate Nper total number of payment periods Pv present value Startperiod first period in the calculation Endperiod last period in the calculation Type timing of the payment Type Timing 0 zero Payment at the end of the period 1 Payment at the beginning of the period CUMIPMTEXAMPLE Following is an example of CUMIPMT function In this example in the first case the objective is to find total interest paid in the second year of payments for periods 13 to 24 Please note there are 12 periods per year The second case is for the first payment penod In the first formula the Annual interest rate 9 is cell A2 not shown here The Years of the loan are given in cell A3 The Present value is in cell A4 For the Start period the value 13 was entered For the End period the value 24 has been specified The value of Type is 0 which means that the payment will be at the end of the period Please note that the annual interest is first divided by 12 to arrive at monthly interest Then the Years of the loan are multiplied by 12 to get total number of months in the Term of the loan The answer is 1113523 In the second formula which gives Interest paid in a single payment in the first month 1 was specified as the Start period For the End period also the value 1 was enteredThis is because only 1 period is under study All other inputs were the same The answer is 93750 70 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Data Description 9 Annual interest rate 30 Years of the loan 125000 Present value CUMIPMTA212A312A413240Total interest paid in the second year of payments periods 13 through 24 1113523 CUMIPMT A212A312A4110lnterest paid in a single payment in the first month 93750 CUMPRINC The CUMPRINC function returns the cumulative principal paid on a loan between two penods The syntax is as under CUMPRINCratenperpvstartperiodendperiodtype Rate interest rate Nper total number of payment periods Pv present value Startperiod period in the calculation Payment Endperiod last period in the calculation Type timing of the payment 0 or 1 as above CUMPRINC EXAMPE Following is an example of CUMPRINC function In this example in the first case the objective is to find the total principal paid in the second year of payments periods 13 through 24 Please note there are 12 periods per year The second case is for the principal paid in a single payment in the first month In the first formula the Interest rate per annum 9 is in cell A2 not shown here The Term in years 30 is given in cell A3 The Present value is in cell A4 For the Start period the value 13 was entered For the End period the value 24 has been specified The value of Type is 0 which means that the payment will be at the end of the period Please note that the interest is first divided by 12 to arrive at monthly interest Then the years of loan are multiplied by 12 to get total number of months in the term of the loan The answer is 9341071 In the second formula which gives the principal paid in a single payment in the first month 1 was specified as the start period For the end period also the value 1 was enteredThis is because only 1 period is under study All other inputs were the same The answer is 6827827 EXAMPLE Data Description 900 Interest rate per annum 30 Term in years 125000 Present value CUMPRNCA212A312A413240The total principal paid in the second year of payments periods 13 through 24 9341071 CUMPRNCA212A312A4110The principal paid in a single payment in the first month 6827827 EFFECT Returns the effective annual interest rate As you see there are only two inputs namely the nominal interest Nominalrate and the number of compounding periods per year Npery 71 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EFFECTnominalratenpery Nominalrate nominal interest rate Npery number of compounding periods per year EFFECTEXAMPLE Here Nominalrate 525 in cell A2 Npery 4 in cell A3 The answer is 0053543 or 53543 You should round off the value to 2 decimals 535 525 Nominal interest rate 4 Number of compounding periods per year EFFECTA2A3 Effective interest rate with the terms above 0053543 or 53543 percent FV Returns the future value of an investment There are 5 inputs namely Rate the interest rate Nper number of periods Pmt payment per period Pv present value and Type FVratenperpmtpvtype Rate interest rate per period Nper total number of payment periods Pmt payment made each period Pv present value orthe lumpsum amount Type number 0 or 1 due FVEXAMPLE 1 In the formula there are 5 inputs namely Rate 6 in cell A2 as the interest rate 10 as Nper number of periods in cell A3 200 notice the minus sign as Pmt payment per period in cell A4 500 notice the minus sign as Pv present value in cell A4 and 1 as Type in cell A6 The answer is 258140 mm Dweri im EMWJJLE l it Aliuml intimatequot rate 391 Iii H un39iJH ef Ili i39l39l i5 Jill An39mnt ef the peymmt 45 P ri ent H39 lll 1 P Hyman ie tlue atthe beguiling if the uselied W quot l A3 A4 A5 e6 Future value ef an inveetm entwi 39i the ebeue term 5 2531 4131 Fiuinretenperpmtputype FVEXAMPLE 2 72 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 In the formula there are 3 inputs namely Rate 12 in cell A2 as the interest rate 12 as Nper number of periods in cell A3 1000 notice the minus sign as Pmt payment per period in cell A4 Pv present value and Type are not specified Both are not required as we are calculating the Future value of the investment The answer is 1268250 EXAMPLE 12 nnuel intereet rate 12 Number ef paymente 1 Ameunt ef the neyment FW lEl1E Ad Future HEIUE ef en ineeetrnent With the eleeve terrne thslrete per enttpmtype FVEXAMPLE 3 In the formula there are 4 inputs namely Rate 11 in cell A2 as the interest rate 35 as Nper number of periods in cell A3 2000 notice the minus sign as Pmt payment per period in cell A4 1as Type in cell A5 The value of Pv was omitted by entering a blank for the value note the double commas The answer is 8284625 EXAM PL E 11 Annual intereet r e 35 Number ef pnymente eeee meunt ef the nnylnent 1 Payment ie tlue at the heginning ef the perietl FWHEP12 A3 A4 A5 Future unlue ef an investment with the nheve terlne 132 Erie 25 FWrntem1erilntiuteen VU 73 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FV SCHEDULE Returns the future value of an initial principal after applying a series of compound interest rates FVSCHEDULEprincipal schedule Principal present value Schedule an array of interest rates to apply FV SCHEDULEEXAMPLE In this example the Principal is 1 The compound rates 009 01101 are given within curly brackets The answer is 133089 FVSCHEDULEprincipalschedule FVSCHEDULE100901101 Future value of 1 with compound interest rates of 00901 101 133089 IPMT Returns the interest payment for an investment for a given period lPMTratepernperpvfvtype Rate interest rate per period Per period to find the interest Nper total number of payment periods Pv present value or the lumpsum amount Fv future value or a cash balance Type number 0 or 1 ISPMT Calculates the interest paid during a specific period of an investment lSPMTratepernperpv Rate interest rate Pen penod Nper total number of payment periods Pv present value For a loan pv is the loan amount NOMINAL Returns the annual nominal interest rate NOMINALeffectratenpery Effectrate effective interest rate Npery number of compounding periods per year NPER Returns the number of periods for an investment NPERrate pmt pv fv type Rate the interest rate per period Pmt payment made each period Pv present value or the lumpsum amount Fv future value or a cash balance Type number 0 or 1 due NPV Returns the net present value of an investment based on a series of periodic cash flows and a discount rate lts syntax is NPVratevalue1value2 Rate is the rate of discount over the length of one period Valu1value2 are 1 to 29 arguments representing the payments and income 74 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU M Returns the periodic payment for an annuity PMTratenperpvfvtype Rate interest rate Nper total number of payments Pv present value Fv future value Type number 0 zero or 1 PPMT Returns the payment on the principal for an investment for a given period PPMTratepernperpvfvtype Rate interest rate per period Per period and must be in the range 1 to nper Nper total number of payment periods Pv the present value Fv future value 0 Type the number 0 or 1 due PV Returns the present value of an investment PVratenperpmtfvtype Rate interest rate per period Nper total number of payment periods in an annuity Pmt payment made each period and cannot change over the life of the annuity Fv future value or a cash balance Type number 0 or 1 and indicates when payments are due RATE Returns the interest rate per period of an annuity RATEnperpmtpvfvtypeguess Nper total number of payment periods Pmt payment made each period Pv present value Fv future value or a cash balance 0 Type number 0 or 1 due Guess 10 RATEEXAMPLE Three inputs are specified 4 as years of loan in cell A5 200 as monthly payment in cell A6 and 8000 as amount of loan in cell A7 The answer is 00924176 or 924 75 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icrnsuft Excel Lecture jiscuuntlnterest Eile Edit Eiew insert Fgrmat eels gate indnw elp Adn e F39DF FinanEi lfuntti ns 539 atav m Eva eve 39JL 391 a SLIM r X J 5 FEATEIIAE12 EMJII 4 E C D E F 1 2 RATE RHTEmperpmtpvfutypeguess 3 4 Data Descriptin 5 at Year39s f the Ian 5 200 Mnthly payment r BUUUAmunt f the Ian 3 ReTE 512A6A7 1 g 0092 Pmnual rate f the Ian 00924175 r 924 1 76 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 9 Matrix and its dimension Types of matrix OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 8 o Matrices QUESTIONS Every student wonders why he or she should study matrices There are manty important ques ons Where can we use Matrices Typical applications What is a Matrix What are Matrix operations Excel Matrix Functions There are many applications of matrices in business and industry especially where large amounts of data are processed daily TYPICAL APPLICATIONS Practical questions in modern business and economic management can be answered with the help of matrix representation in Econometrics Network Analysis Decision Networks Optimization Linear Programming Analysis of data Computer graphics WHAT IS A MATRIX A Matrix is a rectangular array of numbers The plural of matrix is matrices Matrices are usually represented with capital letters such as Matrix A B C For example 1 1 El 1 52 FE SE A El II III 3 El 52 33 EB 15 5 2 The numbers in a matrix are often arranged in a meaningful way For example the order for school clothing in September is illustrated in the table as well as in the corresponding matrix 4 3 Size Youth S M L XL Sweat Pants 0 10 34 40 12 Sweat Shirts 18 25 29 21 7 Shorts 19 13 48 36 9 Tshirts 27 7 10 24 14 The data in the above table can be entered in the shape of a matrix as follows 77 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU III ll 34 ill 12 13 25 23 21 F 13 13 EH3 3B 3 2 F ll 24 H DIMENSION Dimension or Order of a Matrix Number of Rows x Number of Columns Example Matrix T has dimensions of 2x3 or the order of matrix T is 2x3 X is just the notation it do not mean to multiply both of them T E 2 l eruwl all III F frmWE 3 l 393 call EDIE cn3 ROW COLUMN AND SQUARE MATRIX Suppose n 1234 A matrix with dimensions 1xn is referred to as a row matrix For example matrix A is a 1x4 row matrix A matrix with dimensions nx1 is referred to as a column matrix For example matrix B in the middle is a 2x1 column matrix A matrix with dimensions nxn is referred to as a square matrix For example matrix C is a 3x3 square matrix 2 2 3 A12 i ll 9 El E a i 5 u IDENTITY MATRIX An identity matrix is a square matrix with 139s on the main diagonal from the upper left to the lower right and 039s off the main diagonal An identity matrix is denoted as Some examples of identity matrices are shown below The subscript indicates the size of the identity matrix For example I represents an identity matrix with dimensions ni i n Di f Di iii iii CIDDH EDI II l fl fgziill Equotiii i101 MULTIPLICATIVE IDENTITY With real numbers the number 1 is referred to as a multiplicative identity because it has the unique property that the product a real number and 1 is that real number In other words 1 is called a multiplicative identity because for any real number n 1 n n and n 1n With matrices the identity matrix shares the same 78 Copyright Virtual University of Pakistan Business Mathematics amp unique property as the number 1 In other words a 2iquot 2 identity matrix is a multiplicative Statistics MTH 302 inverse because for any 2K 2 matrix A I 39 A A and A39 I A Examgle Given the 23quot 2 matrix A 10 2 1 32 11 3 4 2 11 11 Nib 3 4 01 Work r1c1 12 03 2 r2c1 02 13 3 r1c1 21 10 2 r2c1 31 40 3 r1c2 11 04 1 r2c2 01 14 4 r1c2 20 11 1 r2c2 30 41 4 392 1 3 4 ii 15 ii iii where r is for row and c is for column Copyright Virtual University of Pakistan VU 79 Business Mathematics amp Statistics MTH 302 VU LECTURE 10 MATRICES OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 9 o Matrices EXAMPLE 1 An athletic clothing company manufactures Tshirts and sweat shirts in four differents sizes small medium large and xlarge The company supplies two major universities the U of R and the U of S The tables below show September39s clothing order for each university University of S39s September Clothing Order II S l M l L l XL Tshirts 1oo 3oo 5oo 3oo sweat shirts quot150 quot400 quot450 quot250 University of R39s September Clothing Order II 3 II M l L l XL Tshirts 60 250 4oo 250 sweat shirts quot100 quot200 quot350 quot200 Mtrix Representation The above information can be given by two matrices S and R as shown below 1m 30C 50C 30C 8 150 400 450 250 El 250 400 250 R 100 200 350 220 MATRIX OPERATIONS The matrix operations can be summarized as under Organize and interpret data using matrices Use matrices in business applications Add and subtract two matrices Multiply a matrix by a scalar Multiply two matrices Interpret the meaning of the elements within a product matrix 80 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PRODUCTION The clothing company production in preparation for the universities39 September orders is shown by the table and corresponding matrix P below II 8 II M l L l XL Tshirts 3oo 7oo 9oo 5oo sweat shirts quot300 quot700 quot900 quot500 EDD TDD EDD EDD P EDD TDD EDD EDD ADDITION AND SUBTRACTION OF MATRICES The sum or difference of two matrices is calculated by adding or subtracting the corresponding elements of the matrices To add or subtract matrices they must have the same dimensions PRODUCTION REQUIREMENT Since the U of S ordered 100 small Tshirts and the U of R ordered 60 then altogether 160 small Tshirts are required to supply both universities Thus to calculate the total number of Tshirts and sweat shirts required to supply both universities add the corresponding elements of the two order matrices as shown below 1ED 4DD 4ED EED 1DD EDD EED 22D 15D EED E39DD EED IDD EDD EDD EDD 5D EED 4DD ED 4 EED EDD EDD 4TD OVERPRODUCTION Since the company produced 300 small Tshirts and the received orders for only 160 small Tshirts then the company produced 140 small Tshirts too many Thus to determine the company39s overproduction subtract the corresponding elements of the total order matrix from the production matrix as shown below EDD TDD EDD EDD 16D EED E39DD EED 14D IED D ED EDD TDD EDD EDD EED EDD EDD 4TD ED IDD IDD ED MULTIPLY A MATRIX BY A SCALAR Given a matrix A and a number 0 the scalar multiplication cA is computed by multiplying the scalar c by every element of A For example 3 2 1n n V 26 3 4 21 a 3 2 4 2 T 2 1 4 81 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MULTIPLICATION OF MATRICES To understand the reasoning behind the definition of matrix multiplication let us consider the following example Competing Companies A and B sell juice in 591 mL 1 L and 2 L plastic bottles at prices of Rs160 Rs230 and Rs310 respectively The table below summarizes the sales for the two companies during the month of July 591mL 1 1L 1 2L 1 quotCompany A 20000 5500 quot10600 quotCompany B quot18250 7000 11000 What is total revenue of Company A What is total revenue of Company B Matrices may be used to illustrate the above information As shown at the right the sales can be written as a 2X3 matrix S the selling prices can be written as a column matrix P and the total revenue for each company can be expressed as a column matrix R 150 00000 5500 10500 230 00510 10050 1000 11000 39 00400 3 510 R P Since revenue is calculated by multiplying the number of sales by the selling price the total revenue for each company is found by taking the product of the sales matrix and the price matrix 100 20000 5500 10000 230 22510 K 10250 2000 11000 20400 Consider how the first row of matrix S and the single column P lead to the first entry of R 150 5000001150155001050110500r51015 00 000 5500 10 500 00 510 0 050 t L t 18250 000 11000 20400 310 Pr0du0t 0f Pr00LIOt 0f Pr0000t 0f First Errtri00 0000110 Entri00 Third Entri00 With the above in mind we define the product of a row and a column to be the number obtained by multiplying corresponding entries first by first second by second and so on and adding the results MULTIPLICATION RULES 82 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If matrix A is a m K n matrix and matrix B is a n K p matrix then the product AB is the m 3quot p matrix whose entry in the ith row and the jth column is the product of the ith row of matrix A and the jth row of matrix B The product of a row and a column is the number obtained by multiplying corresponding elements first by first second by second and so on To multiply matrices the number of columns ofA must equal the number of rows of B W Given the matrices below decide if the indicated product exists And if the product exists determine the dimensions of the product matrix quot 5 ti 4 A 12 3 iii 15 1 391 4 3 9 i 5 T MULTIPLICATION CHECKS The table below gives a summary whether it is possible to multiply two matrices It may be noticed that the product of matrix A and matrix B is possible as the number of columns ofA are equal to the number of rows of B The product BA is not possible as the number of columns of b are not equal to rows of A Does a product exist Product Dimensions of Is it possible to multiply Dimensions of the Matrices the given lProduct Matrix matrices in this order A 31 3 B 31 2 Yes the product exists T T since the AB inner dimensions match 3quot 2 inner dimen MS of columns of A of rows of B No the product does not B33ii2 A3gtlt3 exist I 1 since the inner BA dimensions do na inner dimena39ons not match of columns of B ti of rows of A MULTIPLICATIVE INVERSES Real Numbers Two nonzero real numbers are multiplicative inverses of each other if their products in both orders is 1 Thus 1 1 1 1 the multiplicative inverse ofa real number x is I or 539 Since x 39 I 1 and I 39 x 1 83 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 1 The multiplicative inverse of 5 is 5 since 1 1 539 51and 53951 Matrices Two 2 2 matrices are inverses of each other if their products in both orders is a 23quot 2 identity matrix Thus the multiplicative inverse of a 2i 2 matrix A is 31 1 since A39 111 1 f2 and 111 1 quot A 32 Example 2 1 3 1 The multiplicative inverse of a matrix is 5 3 5 2 2 1 3 1 1 0 5 3 5 2 o 1 3 1 2 1 1 0 5 2 5 3 0 1 84 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 11 MATRICES OBJECTIVES The objectives of the lecture are to learn about o Review Lecture 10 o Matrix functions in Excel 0 Set up and manipulate ratios 0 Allocate an amount on a prorata basis using proportions MATRIX FUNCTIONS IN MS EXCEL The Matrix Functions in Microsoft Excel are as follows 1 MINVERSE 2 MDETERM 3 MMULT MINVERSE Returns the inverse matrix for the matrix stored in an array Syntax MINVERSEarray array is a numeric array with an equal number of rows and columns Remarks 0 Array can be given as a cell range such as A1 03 as an array constant such as 1 23456789 or as a name for either of these 0 If any cells in array are empty or contain text MINVERSE returns the VALUE error value 0 MINVERSE also returns the VALUE error value if array does not have an equal number of rows and columns 0 Formulas that return arrays must be entered as array formulas o Inverse matrices like determinants are generally used for solving systems of mathematical equations involving several variables The product of a matrix and its inverse is the identity matrix the square array in which the diagonal values equal 1 and all other values equal 0 0 As an example of how a tworow twocolumn matrix is calculated suppose that the range A1 82 contains the letters a b c and d that represent any four numbers The following table shows the inverse of the matrix A1 82 Column A Column B Row 1 dadbc bbcad Row 2 cbcad aadbc o MINVERSE is calculated with an accuracy of approximately 16 digits which may lead to a small numeric error when the cancellation is not complete 0 Some square matrices cannot be inverted and will return the NUM error value with MINVERSE The determinant for a noninvertable matrix is 0 85 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MINVERSE M INVE REE array rray i5 a nun merit arragyr with an equal IuIInIJer sf raw5 f I39IiI39I c lulnna luvBrae 0f the In ri 3E1 132 C llll39l39ll39l C lllll39lll E FE W quotI nilquotiatilIJ HE IiiquotIJ J WC iil Raw illI39l 391 a quotIi aquotIa 39I39I I quotC rray f rl39l39illl 1 F2 2 Enter fuzannula 3 Etrl Shift Enter MlNVERSEEXAMPLE Find the inversion or multiplicative inverse of following matrix 4 1 2 0 The excel formula in the example must be entered as an array formula Otherwise a single result 0 will appear Please note that formulas that return mvs must be entered as array formulas The steps of finding multiplicative inverse of above matrix is as follows 1 Enter data of array to be inverted in Cells A4B ie in cells A4 B4 A5 BS 2 Click on cell A6 3 Keeping left mouse button pressed drag it to cell B7Four cells A6 BG A7 B7 will be selected 4 Press F2 from your keyboard Key F2 is selected to enter Edit Mode in the active cell It s a keyboard shortcut Even if you don39t press F2 you can write the formula 5 Type the formula MINVERSEA4B5 This will appear in cell A6 6 Press Ctrl Shift Enter keys simultaneously from your keyboard 7 Multiplicative inverse of the matrix will appear in cells A6 A7 BB B7 Now click any cell among A6B7 in the formula bar you can see curly brackets round the formula That is MINVERSEA4BS This shows that you have gone through the right procedure of entering array formula 86 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU k lillicrnanft Excel Lecture1lllllatricea Elle Edit iew insert Fgrmat Innls Qata indnw Help D l g y E Eiiii amm39 aa mnaaeeeaaaalaltaa Java 12 v r A 2 MINVERSE a Data Data 4 4 1 5 2 0 U 05 MINVERSA435 1 2 T EEI IT Cl El MDETERM Returns the matrix determinant of an array Syntax MDETERMarray array is a numeric array with an equal number of rows and columns Remarks 0 Array can be given as a cell range for example A1 03 as an array constant such as 1 23456789 or as a name to either of these o If any cells in array are empty or contain text MDETERM returns the VALUE error value 0 MDETERM also returns VALUE if array does not have an equal number of rows and columns 0 The matrix determinant is a number derived from the values in array For a three row threecolumn array A1 C3 the determinant is defined as MDETERMA1C3 A1 BZC3B3CZ A2B3C1B1C3 A3B1CZBZC1 0 Matrix determinants are generally used for solving systems of mathematical equations that involve several variables 0 MDETERM is calculated with an accuracy of approximately 16 digits which may lead to a small numeric error when the calculation is not complete For example the determinant of a singular matrix may differ from zero by 1E16 87 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MDETERMEXAMPLE The example shows an array of dimension 4 x 4 in cell range A14D17 The formula was entered in cell A18 The result of this calculation is 88 Microsoft Excel Book l Eile Edit iew insert Formal Iools Eata indow Help Fuder PDF El 2 it v a v 2 v v s L a T SLIM v x or 151 MDETERMAHD1 A E r D E F 11 MDETERM 12 13 Data Data Data Data 11 1 5 5 1 E 1 5 1 1 1 I 1 Iquot 10 2 IE MDETERMA1 ID1 19 Determinant f the matrix abide 88 There are other ways also for using this function For example you can enter the matrix as an array constant MDETERM3611103102 Determinant of the matrix as an array constant 1 You can calculate the determinant of the matrix in the array constant MDETERM3611 Determinant of the matrix in the array constant 3 Unequal number of rows and columns results in an error MDETERM13851361 Returns an error because the array does not have an equal number of rows and columns VALUE MMULT Returns the matrix product of two arrays The result is an array with the same number of rows as array1 and the same number of columns as array2 Syntax MMULTarray1 array2 Array1 array2 are the arrays you want to multiply Remarks 88 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o The number of columns in array1 must be the same as the number of rows in array2 and both arrays must contain only numbers 0 Array1 and array2 can be given as cell ranges array constants or references o If any cells are empty or contain text or if the number of columns in array1 is different from the number of rows in array2 MMULT returns the VALUE error value 0 The matrir product array a of two arrays b and c is at Ebece 31 where i is the row number andj is the column number o Formulas that return arrays must be entered as array formulas MMULTEXAMPLE Let 1 3 2 0 A B Find AB 7 2 0 2 To find the product AB follow these steps 7 1 Enter arra 1 in cell range A25B26 and arra 2 in cell range D25E26 I Miereseft Excel Beet1 Elle Edit Elev I lnserl Fgrmel leels gate window Help i l ldel le vze lev I a De III E i riai til I E H lg E 39 13 v i e I a I I E I F I 23 MMULT 24 array1 array2 25 1 3 2 El 25 T 2 El 2 2 2E Preduet ef array1 and array2 29 2 E MMULTA25 BEEDEE E25 3D 14 4 91 2 Find the dimension of AB matrix Here as A and B are 2X2 matrices so AB is also a 2X2 matrix Click on cell A29 Keeping left mouse button pressed drag it to cell B30Four cells A29 B29 A30 B30 will be selected Press F2 from your keyboard Key F2 is selected to enter Edit Mode in the active cell It s a keyboard shortcut Even if you don39t press F2 you can write the formula Type the formula MMULT Select array1 Put comma Select array2 590 0quot 99 51 89 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 10 Close bracket 11 Press Ctrl Shift Enter keys simultaneously from your keyboard 12 Product AB will appear in cells A29 A30 829 B30 RATIO A Ratio is a comparison between things If in a room there are 30 men and 15 women then the ratio of men to women is 2 to 1 This is written as 21 and read as two is to one is the notation for a ratio Be careful order matters A ratio of 21 is not the same as 12 In the form of fraction we can write 21 as 21 The method of calculating ratios is as under 1 Find the minimum value 2 Divide all the values by the smallest value In the above example the smallest value was 15 Division gives 30 15 2 for men and 15 15 1 for women The ratio is therefore 21 for men and women RATIOEXAMPLE Three friends Ali Fawad and Tanveer are doing business together To set up the business Ali invested Rs 7800 Fawad Rs 5200 and Tanveer Rs 6500 What is the ratio of their investments As discussed above the smallest value is 5200Rs All values are divided by 5200 The results are 15 for Ali 1 for Fawad and 125 for Tanveer The answer is 15 l 125 90 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE Micrusuft Excel Lectu re1 Ratiu5P ru pa rtiu n5 E Elle Edit Eiew insert Fgrmat Innls Qatar indnw Help Adobe PDF SEE v Ev jf 1 LIME3 T v E J EIEBIEIEE A E I D I 52 53 54 CALCULATING RETIRE 55 55 RptTIW 5 Ni 15 SE Fawad 5200 1 59 Tanveer 353l353 an 125 El This example can be solved in Excel The formula is as under Cell D57 B57 B58 Cell D58 B58 B58 Cell D59 B59 B58 The result for cell D59 is shown in cell D60 because the cell D59 is used to display the formula PROPORTION A proportion is an equation with a ratio on each side It is a statement that two ratios are equal 34 68 OR 34 68 is an example of a proportion When one of the four numbers in a proportion is unknown cross products may be used to find the unknown number This is called solving the proportion or ESTIMATING USING RATIO EXAMPLE Ratio of sales of Product X to sales of Product Y is 43 The sales of product X is forecasted at Rs 180000 What should be the Sales of product Y to maintain the ratio of sales between the two products CALCULATION Ratio sales X Y 4 3 Insert the value for forecasted sale for X 180000 Y 4 3 It can be rewritten as 180000Y 43 91 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cross multiply 180000x 3 4 x Y Rewrite to bring the unknown to the left of the equality 4xY180000x3 Solve Y 180000 x 34 Y 135000 Rs Qlculations usinq EXCEL In cells B70 and B71 the ratios of Product X and Y were entered The value of forecast of product X was entered in cell D70 Before writing down the formula in excel it was derived as follows 1 Ratio of X cell B70 2 Ratio of y cell B71 3 Sale of X cell D70 4 Sale on cell D71 Now Ratio X Y cell B70 cell B71 Ratio of sales cell D70 cell D71 Crossmultiply cell B70 x cell D71 cell B71 x cell D70 Cell D71 is unknown Hence cell D71 cell B71 x cell D70 cell B70 Or cell D71 cell B71 cell B70 cell D70 Thus the formula was B71B70D70 Please note that actually we are using the ratio Y to X as it is easier to think of ratio of unknown to the known Microsoft Excel Lecture39l Ratin5l3rnpnrtinn Eile Edit EiEl n39 insert Fgrmat Idols gate window elp littler F39DF El rm Ev 3 L a a i SLIM r K vquot r EI1IEIDDEI A E C D E SE a ESTIMATING RATIS EE ES RIBITI SIDILES m Prduct 14 4 I 180000 1 Prduct t I BT1ITUiiD70 2 1135000 F3 ESTIMATING USING RATIOEXAMPLE 2 In a 500 bed hospital there are 200 nurses and 150 other staff If the hospital extends by a new wing for 100 beds then what additional staff is needed 92 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let 500 beds B1 and 100 beds 32 Staff nurses N1 is 200 and other staff O1 is 150 What is the value of N2 and O for 32 Obviously the ratio of beds will be used As pointed out above think of the ratio of unknown to known In other words ratio 3231 or 3231 Ratio of nurses would be N2N1 Ratio of other staff would be 0201 Now N2N1 3231 Or N2 3231N1 or N2 Nurses 0201 3231 Or N2 3231O1 or Oz 100500150 30 other staff Calculation Beds Nurses Other staff 500 200 150 100 X Y Nurses 500 200 100 X 500 X 200 x 100 X 200 x 100500 40 Other staff Y 150 x 100500 30 Qlculation usinq EXCEL The calculation using EXCEL was done in a similar fashion as the previous example The calculation is selfexplanatory Hicrnsuft Excel Lectur21 atiusrupurtinns Eile Edit Eiew insert Fgrmal Tools gate window Help adage F39DF a t 133 1 H a SLIM 1 X of f3 DFID5EEI A a t D E LI 3 H 1 2 R lTIS 3 4 Hspital dditin 5 Beds 500 100 5 Nurses 200 9 quot jtherstaff 150 IDE EEI a El ESTIMATING USING RATIOEXAMPLE 3 A Fruit Punch recipe requires mango juice apple juice and orange juice in the ratio of 321 To make 2 liters of punch calculate quantity of other ingredients Again we shall use the ratio of unknown to the unknown The unknowns are mango and apple juice Consider first ratio of required mango juice 3 to total quantity of punch 6 This was calculated from 321 Now the quantity of 93 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU required mango for 2 liter would simply be 362 Similarly the required quantity of apple juice is 262 m Mango juice Apple juice Orange juice 3 2 1 Total 6 X Y 2 Total 2 litre Mango juice X 3 6 2 1 litre Apple juice Y 2 6 2 067 litre Orange juice 2 1 6 2 033 liter Calculation usinq EXCEL Here also the similar ratios were used Mango BZOBZ3D23 Apple BZ1BZ3D23 Orange BZZBZ3D23 Mir rusuft Excel Lent ure1 atiu5P ru pa rtiu I15 Eile Edit iew insert Fgrmat IDEIIS Qatar induw Help adage F DF 539 i E v TL a T T SUM 139 K 39J 5 EIEEIEIEED23 A El 3 D E F 3 IE 1 USING RATIS IE 19 RATI 2n Mang juice 239 1 21 Apple juice 2 2 arrange juice I 1 I 3221323mz3 23 Ttal Litre E 2 24 94 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 12 RATIO AND PROPORTION MERCHANDISING OBJECTIVES The objectives of the lecture are to learn about 0 Module 3 Review Lecture 11 Ratio and Proportions Merchandising Assignment 1A and 1B MODULE 3 Module 3 has the following content 0 Ratio and Proportions o Merchandising Lectures 12 0 Mathematics of Merchandising Lectures 1316 ESTIMATING USING RATIOSEXAMPLE1 In the previous lecture we studied how ratios can be used to determine unknowns Here is another example with a slightly different approach Here ratios between the quantitiesand data of only one quantity is known We will estimate the total quantity that can be made It is the quantity of orange juice that will determine the total quantity that can be made Again the method is to use the ratio of the unknown to the known W In a punch the ratio of mangojuice applejuice and orangejuice is 321 If you have 15 liters of orange juice how much punch can you make m Mangojuice Applejuice Orangejuice 3 2 39 1 Total 321 6 X say Ysay Zsay 15litres Total litre Mangojuice X 31gtlt15 45 litre Applejuice Y 21gtlt15 30 litre Orangejuice Z 15 litre Punch 45 30 15 9 litres EXCEL calculation The method used is the same as in previous examples 95 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Mic rueuft Excel Lect ure1 Ratiue ru pa rtiu I15 Elle Edit iew insert Fgrmat Idols gate window Help littler F39DF E Iii iiiF ri quot E quot if L H a 39 SLIM T E uquot f EIEMEIEE 1333 A E I D E F G 2 23 RATIMS 29 an RATIW 31 Mang juice 39Pquot 32 pple juice 2 39Pquot 33 range juice 1 15 15 Ttal Litre 35 BE ESTIMATING USING RATIOSEXAMPLE2 In a punch ratio of mango juice apple juice and orange juice is 3 2 15 If you have 500 litres of orange juice find how much mango and apple juices are required to make the punch ME The ratio of mangojuice applejuice and orangejuice is 3 2 15 If you have 500 milliliters of orange juice how much mango juice and apple juice is needed Mangojuice Applejuice Orangejuice 3 2 1 5 Total 65 X say Ysay Z say 9 39 39 500 litres Total litres Mangojuice X 315500 1000 litre Apple juice Y 21 5500 667 litre Orange juice Z 500 litre Punch 1000 667 500 2167 litre EXCEL Calculation Here also ratios were used Mango juice B45B47D47 Apple juice B46B47D47 Orange juice D47 96 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU micrnenft Excel Lecture l Ratiu5Prnpurtinne E Elle Edit ew insert Fgrmat Innle Qatar indnw elp Adnge F39DF n it m 2 v L H a SLIM r X d it El lEfEl lFD l A El C D E I F 41 42 USING RnTIS 43 44 Rth 1000 Mang juice B45lB47i Df l7 e ipple juice 2 007 1 range juice 15 500 500 18 Ttal MI 05 2107 39 EXAMPLE In a certain class the ratio of passing grades to failing grades is 7 to 5 How many of the 36 students failed the course The ratio quot7 to 5quot or 7 5 or 75 tells you that of every 7 5 12 students five failed That is 512 of the class failed Then 512 36 15 students failed PROPORTION ab old the values in the quotbquot and quotcquot positions are called the quotmeansquot of the proportion while the values in the quotaquot and quotdquot positions are called the quotextremesquot of the proportion A basic defining property of a proportion is that the product of the means is equal to the product of the extremes In other words given ab old it is a fact that ad bc PROPORTIONEXAMPQ ls 24140 proportional to 30176 M 140gtlt30 4200 24X176 4224 So the answer is that given ratios are not proportional PROPORTION EXAMPLE 1 Find the unknown value in the proportion 2 x 3 9 2 x 3 9 First convert the colonnotation ratios to fractions 2x 39 Cross multiply 18 3x 6 x 97 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PROPORTION EXAMPLE 2 Find the unknown value in the proportion 2x 1 2 x 2 5 2x12x25 First convert the colonnotation ratios to fractions 2x 12 x 25 Then solve 52x 1 2x 2 10x 5 2x 4 8x 1 x 18 MERCHANDISING What does merchandising cover Understand the ordinary dating notation for the terms of payment of an invoice Solve merchandise pricing problems involving mark ups and markdowns Calculate the net price of an item after single or multiple trade discounts Calculate a single discount rate that is equivalent to a series of multiple discounts 0 Calculate the amount of the cash discount for which a payment qualifies STAKEHOLDERS IN Merchandising Who are the stakeholders in merchandising The main players are Manufacturer Middleman Retailer Consumer There are discounts at all levels in the above chain 0 O O O MIDDLEMAN A middle man is a person who buys a product directly from the manufacturer and then either sells the product at retail prices to the public or sells the product at wholesale prices to a distributor There can often be more than one middle man when the latter practice is adopted A middle man can purchase from the manufacturer and then work with another middle man who buys for the distributor The manufacturer often views the middle man as the alternative to direct distribution gst price or Retail price List price refers to the manufacturer39s suggested retail pricing It may or may not be the price asked of the consumer Much depends on 1 the product itself 2 the builtin profit margin 3 Supply and demand A product that is in high demand with low availability will sometimes sell higher than the list price though this is less common than the reverse Virtually all products have a suggested retail or list price Resellers middleman retailer buy products in bulk and get a substantial discount in order to be able to get profit from selling the product at or below list price 98 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Trade Discount Let L is the list price then amount of trade discount is some percentage of this price List price less amount of discount is the net price In mathematical terms we can write Amount ofdiscount d x L Where d Percentage of Discount L List Price Net Price L Ld L1 d Net Price List Price Amount of Discount Copyright Virtual University of Pakistan VU 99 Business Mathematics amp Statistics MTH 302 VU LECTURE 13 MATHEMATICS OF MERCHANDISING OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 12 o Solve merchandising pricing problems involving markup and markdown MARKUP Markup is an amount added to a cost price while calculating a selling price Especially an amount that takes into account of overhead and profit Markup can be expressed as 1 Percentage of cost 2 Percentage of sale 3 Rs Markup Markup as Percentage of Cost MUC Here markup is some percentage of cost price For simplicity it is also named as Markup on cost The relation between markup on cost cost price and selling price is Selling Price Cost price Cost price x Markup on cost Cost price 1 Markup on cost Markup as Percentage of Sale price MUS Here markup is some percentage of selling price For simplicity it is also named as Markup on sale The relation between markup on sale cost price and selling price is Selling Price Cost price Selling price x Markup on sale Cost price Selling price Selling price x Markup on sale Selling price 1 Markup on sale Rs Markup Markup in terms of rupees is called Rs markup The relations between Rs markup cost price and selling price are 1 Selling Price Cost price Rs Markup 2 Rs Markup Markup on cost x Cost price 3 Rs Markup Markup on sale x Selling price Any of the above formula can be used to find Rs Markup For example The cost price of certain item is 80Rs and its selling price is 100Rs Then Rs Markup Selling price Cost price 100 80 20 Rs Remember If some percentage is given as markup without mentioning that whether it is markup on cost or markup on sale it is evident that markup on cost is under consideration EAMPLE 1 A golf shop pays its wholesaler 2400Rs for a certain club and then sells it for 4500Rs What is the markup rate Calculation of Markup 100 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cost price 2400Rs Selling price 4500Rs Selling Price Cost price Cost price x Markup on cost Markup on cost Sellinq price Cost price X100 Cost price Since Rs Markup Selling Price Cost price Markup on cost Rs markup X100 Cost price 1 First calculate Rs markup Rs markup 4500 2400 2100Rs Then markup on cost Markup on cost Rs markup X100 Cost price 21002400 X100 875 Remember to convert this fraction value to a percent The markup rate is 875 Qlculgtion usinq EXCEL Enter wholesale price 2400 in cell B5 Enter sale price 4500 in cell BG Enter formula for Rs Markup BGB5 in cell B7and press enter The answer is 2100 Enter formula for markup B7B5100 in cell B8 and press Enter The answer is 875 shown in cell BQ Microsoft Excel Bool Eile Edit Eiew insert Fgrmat Tools gate window Help Atler F39DF Q 53935 r 2 1 3939 In T SLIM v x of f ElF39IElE1IIIEI A E I C D l l 2 MARKUP 3 4 5 Whlesale price ErrI00 a Sale price 1500 Markup Rs 2100 3 Markup Brro t1oo 9 85 IEI Copyright Virtual University of Pakistan 101 Business Mathematics amp Statistics MTH 302 VU MARKUPEXAMPLE 2 A computer software retailer used a markup rate of 40 Find the selling price of a computer game that cost the retailer Rs 1500 Markup The markup is 40 on the cost so as Rs Markup Markup on cost x Cost price Rs Markup 040 1500 Rs 600 Selling Price Then the selling price being the cost plus markup that is Selling Price Cost price Rs Markup 1 500 600 Rs 2100 The item sold for Rs 2100 Qlculgtion usinq EXCEL Here we use the following formula to show an alternate method of solving the above problem Selling Price Cost price 1 Markup on cost Enter wholesale price 1500 in cell B17 Enter Markup in cell B18 Enter formula 1B18100B17 in cell B19 Here the term 1B18100 is the multiplication factor B18100 is the markup in fraction The answer 2100 is shown in cell B20 We could have calculated the multiplication factor separately But as you see it is not necessary as the entire calculation can be done in one line E Micrusu Excel Baum Elle Edit ew insert Fgrmat pols gate window Help adage PDF is at 3 e v E a 4 in 39L 3939 If T SLIM v E I f 1B1BI1EIEITEI I A E If D 39 12 13 M MARKUP 15 Example 2 15 1 Whlesale price 1500 18 sh Markup 1e1r1oora1rv 2 2100 21 MARKDOWN Markdown means a reduction from the original sale price to l 02 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 1 stimulate demand or 2 take advantage of reduced costs or 3 force competitors out of the market Markdown can be expressed as 9 v Percentage of current selling price 00 Rs markdown Markdown as Percentage of current selling price Here markdown is some percentage of current selling price For simplicity it is also named as percent markdown markdown The relation between current selling price markdown and new selling price is New selling price Current selling price Current selling price x markdown Current selling price 1 markdown Rs Markdown Markdown in terms of rupees is called Rs markdown 1 New selling price Current selling price Rs Markdown 2 Rs markdown Current selling price x markdown Let us look at an example to understand how markdown is calculated MARKDOWNEXAMPLE1 An item originally priced at 3300 Rs is marked 25 off What is the sale price Markdown First find the Rs markdown The markdown is 25 of the original price or current selling price as Rs markdown original price x markdown 0253300 825Rs Selling Price Then calculate the sale price by subtracting the markdown from the original price New Selling price 3300 825 2475Rs The sale price is 2475 Rs Qlculgtion usinq EXCEL Enter original price 3300 in cell B28 Enter Markdown 25 in cell B29 Enter formula for Rs Markdown B29100B28 in cell B30 Here the term B29100 is the markdown in fraction The result of this part of the calculation is 825 Enter formula for new sale price B28 B30 in cell B31 This formula is not shown in the slide We could have calculated the new sale price directly also by writing just one formula 1B29100B28 by using following formula New selling price Current selling price 1 markdown In other words the multiplication factor is calculated as 1025 075 and multiplied with the original price 3300 The answer would be the same By breaking the calculation in parts you can check the intermediate result and avoid errors But if you become very conversant with formulas then you may wish to reduce the number of unnecessary steps in the calculations 103 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejj icrusuft Excel Boom Elle Edit ElevJ insert Fgrmal Ippls gate window Help littler PDF 1 are m 2 ve 1quot Fe SLIM v x 4quot f ElEEiflIIIIIIElEE A 25 MARKDWN 25 Example 1 2 EB riginal price 29 We BEBI1DWBZB 31 Sale price 2MB 32 DISCOUNT Discount is a reduction in price which the seller offers to the buyer There are different types of discount 1 Trade discount 2 Cash discount 3 Seasonal discount etc TRADE DISCOUNT When a manufacturer or wholesaler offers goods for sale a list price or retail price is set for each item This is the suggestion price to be charged from the ultimate consumer A discount on the list price granted by a manufacturer or wholesaler to buyers in the same trade is called trade discount Trade discount represents a reduction in list price in return for quantity purchases Thus Rs Trade discount list price x discount rate Net price list price Rs trade discount There are two main types of trade discounts 1 Single trade discount 2 Multiple or series trade discount 104 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SINGE TRADE DISCOUNTEXAMPL 1 The price of office equipment is 3000 Rs The manufacturer offers a 30 trade discount Find the net price and the trade discount amount Diseeunt Net Price L1 d 30001 03 300007 2100 Rs Amount of discount dL 03 x 3000 900 Rs Qlculgtion usinq EXCEL Enter price of equipment 3000 in cell B39 Enter trade Discount 30 in cell B40 Enter formula for Rs Discount B40100B3 in cell B41 Here the term B40100 is the discount in fraction The result of this part of the calculation is 900 Enter formula for net price B39B41 in cell B42 This formula is not shown in the slide The result is 2100 as shown in cell B42 Ejjdicrusuft Excel Baum Elle Edit Elevi insert Fgrmal Duels Qatar induw Help Filippa F39DF 259 3 iv e v 2 v v e 1 3939 FE T SLIM r K J F EldelIIIIIIEl39 A E C D 35 35 DISCUNT 3 Example 1 33 39 Price f equipment 4n 111 Trade discunt e4ur1ocreae 42 Net price 2100 43 1 05 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 14 MATHEMATICS OF MERCHANDISING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 13 Financial Mathematics Part 1 SERIES TRADE DISCOUNT This refers to the giving of further discounts as incentives for more sales Usually such discount is offered for selling product in bulk lf series discount of 15 10 5 are offered on list price say L of an item then net price is calculated as follows Subtract 15 of L from L Let the new price is L1 L1L L X15 Then subtract 10 of L1 from L1 Let the new price is L2 L2 L1L1x10 Then subtract 5 of L2 from L2 The new price is net price of an item N L2 sz 5 Or alternatively N L 1 15 1 10 1 5 Let d1 15 d2 10 d3 5 then above formula becomes N L 1 di1 d21 d3 Remember total discount is not 15 10 5 30 SERIES TRADE DISCOUNTEXAMPLE The price of of ce furniture is Rs 20000 The series discounts are 2010 5 What is the net price For series trade Discount Net price 1d1 ldz 1d3 Here d1 20 d2 10 d3 5 So Net price 2000010210101005 200000809095 2000006840 13680 Rs 106 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU H39i39emmefl lEamEI bloom 333 Fr L 39 iiiaw lrncri ngrma Ind gate 135mm Align FDF r Em En an 2 v El Te 39239 quot39 7 lil39i l J j39i EFF39Ei39EFEHDZIE 39EW1U jmm uui 5 E 1 5395 IF39E TRl rlEl E 39DlllElGLl39MT h 3 Grass price R5 2mm 3 First EE unlt 3 9 Next discount Eta ln first 1 a Herrt discount quot3 train ti r51 7 5 all earrtnaarar 3339 a tutti lETEHDEI 3 Net prime E5 1350 LEBH 35 LIST PRICE An order for power tools has a Rs 2100 net price after a 30 trade discount What is the list price Net Price Net Price L1 d L N 1 d 21001 03 2100 07 3000 Rs EXCEL Calculation EXCEL formula for list price was based on the calculation 21001 03 The net price was entered in cell 867 Trade discount was entered in B69 The formula for list price was entered in cell B71 as BG7 1869100 The answer is shown in cell C72 as 3000 1 07 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Elle Edit ew ll39lSE rt Fgrmal IDDIS Qatar window elp Adnge PDF 33 it a v w v E v T e 1 3939 Fe SLIM r K d 15quot EIEF It1ElEBI1EIEI A E C D 53 54 LIST PRICE 55 55 a Net price 2100 EB 59 Trade discunt 39an m eerrr1 59 oo 2 73 TRADE DISCOUNTEXAMPLE 2 Find the single discount rate that is equivalent to the series 15 10 and 5 Trade Discount Apply the multiple discount to a list price of Rs 100 Net price 1d11d21d3 1001151101 5 100085 09 095 10007268 7268 Discount 100 7268 2762 EXCEL calculation EXCEL formula for net price was based on the calculation 10010151011005 The formula for net price was entered in cell F8 The formula is shown in cell F8 The answer is shown in cell F12 108 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Bunk1 Elle Edit ElemI Lnsert Fermet leels gate indew elp Fidelge F39DF it a v m w E v Q 1 TL 39JLJZ Fe SLIM 1quot X J 5 llFfli ll1F5leDjI1FEI1EIDIII1IJD A E C D E F I G H J K 1 2 SERIES DISCUNTS 3 4 First Discth 15 ill 5 Next Diecunt 10 a 5 Next Discunt 5 le F ne price r1F4r1normFsiworhFEI in manned 11 12 727 13 In the following slide the net price was calculated in cell F8 Then the discount was calculated assuming the list price was 100 This is a common method to assume 100 as the list price when no price is given but you are required to calculate the net discount Micrusuft Excel Bunl Elle Edit iew insert Fgrmet eels gate indnw Help delge F39DF cit IE v an v E v E 3 TL 391 FerE EUM 1quot H 39439 1UElFE A e e e E 4 3 H 2 SERIES DISCUNTS 3 4 Firet Dieeunt 15 quote 5 Next Diseunt 10 In 5 Next Diseunt 5 quota F e Net price 72T Dieeunt 1 TRADE DISCOUNTEXAMPLE3 The price of car parts is Rs 20000 The series discounts are 20 8 2 What is the single equivalent discount rate Also find Rs discount 1 09 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let Rs 100 is the list price then Net price 1001 021 0081 002 10008092098 10007213 7213 Single equivalent discount rate 100 7213 2787 Rs Discount 0278720000 5574 Rs EXCEL calculation EXCEL formula for net price was based on the calculation 10010210081002 The formula for net price was entered in cell F21 The formula is not shown Price of car parts was entered in cell F23 Formula for discount was based on 0278720000 and is shown in cell F24 The answer is shown in cell F26 as 5574 Microsoft Excel Baum Elle Edit ew insert Fgrmat Innis Data window Help Adobe PDF clquot LE v E v E v 3 1 3939 quotEa SLIM v E u 75 F23F2271IIIIZI A El I3 D E F G H 15 SINGLE EUIVALENT DISCUNT RATE IE 1 First Discount 20 3937 IE Next Diecunt 8 7a 19 Next Diecunt 2 3917 ED 21 Net price 721 3937 22 Discunt 279 7 23 Price of car parts Rs Diecunt F23F22l 100i 25 25 5574 27 CASH DISCOUNT A seller always desires to be paid by the buyer as soon as possible A discount given for the prompt payment of the dues is called Cash Discount Such a discount is an advantage to both the seller and the buyer The buyer has a saving of money while the seller has funds at his disposal Cash Discount is allowed on Invoices Returned Goods Freight Sales Tax and A common business phrase for a cash discount is quot310 net30quot meaning that a 3 discount is offered if the amount due is paid within 10 days otherwise 100 of the amount due is payable in 30 days For example if the amount due is 100Rs the buyer may pay 97Rs within 10 days or 100 Rs within 30 days 110 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU DISCOUNT PERIODS Discount Periods are periods for the buyer to take advantage of Discount Terms CREDIT PERIODS Credit Periods are periods for the buyers to pay invoices within specified times CASH plSCOUNT EXAMPE Invoice was dated May 1st The terms 210 mean that 2 discount is offered if invoice is paid up to 10thMay What is the net payment for invoice value of Rs 50000 if paid up to 10th May Cash Discount N L1 d 500001002 50000098 49000 Rs EXCEL Calculation EXCEL formula for net price was based on the calculation 500001002 However here an IF condition was applied that means that if the payment date in cell D31 sign is put in front of row and column to fix its location is less than or equal to 10 May then the discount will be as given in cell D30 Here also sign was used to fix the location of the cell In cell D38 the date was changed to 11 May and the same formula was applied again The result as shown in cell D39 and D40 are 0 for discount and 0 Rs for Rs iscount 39 Miereenft Excel Bunk Eile Edit yew insert Fgrrnat IDDIS gate indnaI Help in lgeniliaaliii itlasel vailev Avlazvt Emil rm vII a IE 2 E aal 1 a at at me v a a e e a E F e 29 CASH DISCDUNT an Dieeeunt 2 quotii 31 Last date 10 W39layF 32 aa lnveiee value 50000 Re 34 35 Payment date 0 allayF a3 Dieeeunt IFD3539D31D300 a 1000 R5 3a Payment date 11 allayF ae Dieeeunt 0 quotit an 0 R5 1 1 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 15 MATHEMATICS OF MERCHANDISING PART 3 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 14 0 Financial Mathematics Part 2 PARTIAL PAYMENTS When you buy on credit and have cash discount terms part of the invoice may be paid within the specified time These part payments are called Partial Payments Let us look at an example You owe Rs 40000 Your terms were 310 3 discount by 10th day Within 10 days you sent in a payment of Rs 10000 Rs 10000 was a part payment How much is your new balance First we will find the amount that if 3 discount is given on it the net amount is 10000Rs Let that amount is tThen 10000 t 1 003 This implies t 10000 1 003 Thus t 10309Rs This means that although you pay 10000Rs due to 3 cash discount 10309Rs among 40000Rs is paid Hence the new balance 40000 10309 29691 Rs MARKETING TERMS There are a number of marketing terms First of these is the Manufacturer Cost This is the cost of manufacturing Next is the price charged to middlemen in The Distribution Chain The DistributorgtWholesalergtRetailer is a chain The next term is the Selling Price This is the price charged to Consumers by Retailers It may or may not be the same as list price MARKETING OPERATING EXPENSES AND SELLING PRICE Gross Sales less Cost of Goods sold gives the Gross Profit The gross Profit less the Operating Expenses gives the Net Profit Operating Expenses Expenses the company incurs in operating the business eg rent wages and utilities is called operating Expenses Selling Price Selling Price is composed of Cost and Rs Markup Selling Price S Cost C Rs Markup M MARGIN While determining Sale Price a company includes the operating expenses and profit to their own cost This amount is called the margin of the company It is usually calculated 112 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU as percentage but can also be expressed as rupees It is also named as markup on sale Margin or markup on sale Selling price Cost price X100 Selling Price Selling price Cost price Rs Margin Margin and markup confuse many By margin company evaluates that for every rupee generated in sales how much is left over to cover basic operating costs and profit Markup represents the amount added to a cost to arrive at a selling price Markup on cost Selling price Cost price X100 Cost price For example an item costs 50Rs and is sold for 100Rs Markup 100 50 x100 100 50 Margin 100 50 X100 50 100 Note Remember unless it is mentioned that markup is on sale simple markup means markup on cost Example A computer s cost is 9000Rs An amount of Rs 3000 was added to this cost by the retailer to determine the sale price for the consumer Thus the selling price 9000 3000 Rs 12000 Rs Rs 3000 is Margin available to meet Expenses and make a Pro t MARKUP If the Markup on cost 33 then Selling Price S Cost C Cost C X Markup on cost MUC S C C X MUC MARKUPEXAMPLE You buy candles for Rs 10 You plan to sell them for Rs15 What is your Rs Markup What is your percent Markup on cost Rs Markup Selling price cost price Selling price Cost 15 10 Rs Markup Rs 5 Markup on cost Selling price Cost price X100 Cost price Markup 510100 50 SELLING PRICE ll3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fawad s Appliances bought a sewing machine for Rs 1500 To make the desired profit he needs a 60 Markup on Cost What is Fawad s Rs Markup What is his Selling price Selling Price Rs Markup Cost price x Markup on cost Rs Markup 1500 x 06 900 Rs Selling Price S Cost C Rs Markup M Selling Price 1500 900 2400 Rs 9r Alternatively since Selling Price S Cost C Cost C X Markup on cost MUC SCCgtltMUC C1MUC Selling price 15oo x 106 1500 X 16 2400 Rs EXCEL Calculation Here 1500 is the Sewing machine cost in cell F4 and 60 is the Percent Markup on cost in cell F5 EXCEL formula in cell F6 for Rs Markup was based on the calculation 601001500 The Selling price was calculated in cell F7 by using the formula F4F6 The answer 2400 is shown in cell F7 Micrnenft Excel Ennk1 Elle Edit iew insert Fgrmel Icicle gate indnw elp ureter eerieleela Jam were initial 310 r l E I H I E E 3 We 539 TEE13933 I 39 FiE v e A e r D E t i l 2 MARKUP SELLING PRICE 3 4 Sewing machine ceet 1500 Re 5 A Markup an east 60 5 Re Markup F5I100F4 Re r Selling price 2400 Re 8 114 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU RS MARKUP AND PERCENT ON COST Tanveer s flower business sells floral arrangements for Rs 35 To make his desired profit Tanveer needs a 40 Markup on cost What do the flower arrangements cost Tanveer What is the Rs Markup Rs Markup and Percent Markup on Cost Sale price S Cost C C XMarkup on cost MUC S C 040C 35 1 40C C 3514 25 Rs Rs Markup 25 x 04 10 Rs EXCEL Calculation Here 35 is the Selling pricefloral arrangement in cell H15 Markup on cost is in cell H16 EXCEL formula in cell H18 for Cost was based on the calculation 3514 The Rs Markup was calculated in cell H19 by using the formula H18H16100 The answer as shown in cell H19 was 10 El Hii f39i ILIEIEIEII Llllitlrljflil39l LiliaI ELEElilli lil iJ ilrliil rli i39 5 Elli FigJP rillmg lam Eng all Fl 35quot Fl F Ea E Ell Z 7111 1a rm 4 tr 3 iingllariur at E t n E F 3 H l I l l 11 IE 13 R5 MAEHUF AIME FEEEEHT on EDET H 15 Sellng priceaflnral arrangement I EEIH E 1E llllrltup en EDE E 4 til 1quot ttest Ltll15fltlll1 l1 ll m 5 Martin 1 E5 33 21 Formula 5 E 4quot IM 2 E rr s l 23 1 Jill I 15 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MARKUP AgAIN You buy candles for 2 Rs You plan to sell them for 250 Rs What is your Rs Markup What is your Percent Markup on Selling Price Rs Markup Rs Markup 25 2 05 Rs Percent Markup on Sellinq Price As explained in lecture 13 Cost price Selling price 1 Markup on sale Markup on selling price Selling price Cost price X100 Selling price Markup on Selling Price 0525 X100 20 EXCEL calculation Here 2 is the Purchase price in cell E30 Sale price is entered in cell E31 Rs Markup on Purchase Price was calculated by using the formula E31E30 in cell E32 The Markup on sale price was calculated in cell E33 by using the formula E32E31100 The answer as shown in cell E35 was 49 20 116 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Mir rusuft Excel Lectu re14Marl up i5cuu ntMarl duwn Elle Edit Eiew insert Fgrn39lal Innls Qatar window Help Atler PDF El a it T m T 2 T 1T n a SLIP391 r K J f E32fE311EIEI A E C D E I F G H 2 23 MARKUP AGAIN 29 an Purchase price 2 RS 31 Sale price 25 R5 32 R5 Markup 05 R5 3 Markup n sale price E32fE31f100 3T1 39 35 20 SE SELLING PRICE Fawad s Appliances bought a sewing machine for Rs 1500 To make the desired profit he needs a 60 Markup on Selling price What is Fawad s Rs Markup What is his Selling Price Selling Price As explained in lecture 13 Selling Price Cost price Selling price x Markup on sale Selling Price S 1500 068 S 068 1500 Rs Or 048 1500 3750 Rs Rs Markup Rs Markup 3750 x 06 2250 Rs EXCEL calculation Here 1500 is the Purchase price in cell E39 Markup on Sale Price is entered as 60 in cell E40 Sale Price was calculated by using the formula E391E40100 The result 3750 is shown in cell D41 EXCEL formula in cell E42 for Rs Markup was E41E39 The result 2250 is shown in cell E42 Basic formula SC06S is shown in cell A44 In cell A45 it was simplified to 04C In cell A46 it is rewritten as SC04SC1mucC04 Here muc is the Markup on cost 117 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture14liilar up i5cnunt larkdnwn Eile Edit ew insert Fgrmat Innls Qatar indnw Help ndnlge PDF El quot3 it a v 2 v 1 n n SLIM T K I f E3Em lE4IZII1IZIEIJ A E c n E F G 3 33 SELLING PRICE AGAIN 39 Purchase price 1500 Rs 4 Markup n sale price 50 A Sale price 3750 E39l1E40l100 42 R5 Markup 2250 Rs 43 14 45 45 SCi04SCI1mu Cl 04 4 RS MARKUP AND PERCENT MARKUP ON COST Tanveer s flower business sells floral arrangements for Rs 35 To make his desired profit Tanveer needs a 40 Markup on Selling Price What do the flower arrangements Cost Tanveer What is the Rs Markup Selling Price Selling Price Cost price Selling price x Markup on sale Selling Price 35 C 04 x 35 35 C 14 C 35 14 21 Rs Or alternatively C S 04 S 06 S 06 x 35 21 Rs Rs Markup Rs Markup 35 x 04 14 Rs EXCEL Calculation Here 35 is the Sale price in cell E50 Markup on Sale Price is entered as 40 in cell E51 Cost was calculated by using the formula E501E51100 The result 21 is shown in cell D52 EXCEL formula in cell E53 for Rs Markup was E50E52 The result 14 is shown in cell E53 Basic formula SC04S is shown in cell A55 In cell A56 it was simplified to 06SCS1mus 1 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture14Mar up i5cuuntiuiar duwn Eile Edit new insert Fgrmat Iouls gate window Help Fldtl e PDF El 2 it e v 2 v 1L 39 w SUM v in uquot is E5III1E51I1IIIIII A El 3 u E F G 48 49 Rs MARKUP AND PERCENT N CST an Sale price 135 Rs 51 le Markup n sale price 40 We 52 Est 21 E501E51i100 53 Rs Markup M Rs 54 55 UESCS1mus 5 CONVERTING MARKUPS Convert 50 Markup MU on Cost to MU on Sale Formula for converting Markup on Sale mus to Markup on Cost Price muc is Markup on Selling Price mus Markup on Cost 1 Markup on Cost mus muc1muc Solution Markup on Sale mus 05 105 0515 mus 03333 3333 Convertinq Markups Converting 3333 MU on Sale to MU on C M Convert Markup on Cost muc to Markup on selling price mus Markup on cost Markup on S l Markup on S muc mus lmus Markup on cost 033331 0333 0333306666 05 50 EXCEL Calculation 119 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Here 333 is the Markup on sale in cell E61 EXCEL formula in cell E62 for Markup on cost was E611001E61100100 The result 50 is shown in cell E64 Basic formula mucmus 1mus is shown in cell A65 Micrnanft Excel Lecture l4lllarltupi5cnuntlllarlnlwrn E Elle Edit ew insert Fgrmat nulls Qata window Help H Eyeiir 51 x 39039 LE 3 i sum v x J r EElllDDllEElllDDlIJD A E C D E F G H J K l 59 CNVERTING MARKUP N SALE T MARKUP N CST El E1 Markup n sale gMarkup n est EE1I100l1E61l100100 EH El 50 ES mucmus11 mus100 55 EE ES 1 20 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 16 MATHEMATICS OF MERCHANDISING PART 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 15 o Markup and Markdown Financial Mathematics Part 3 MARKDOWN Reduction from original selling Price is called Markdown M Markdown Rs Markdown Selling Price original x100 MARKDOWNEXAMPLE 1 Store A marked down a Rs 500 shirt to Rs 360 What is the Rs Markdown What is the markdown Rs Markdown Let S Sale price Rs Markdown Old S New S Rs 500 Rs 360 Rs 140 Markdown Markdown Markdown Markdown x100 Old S Markdown mx100 500 028x100 28 EXCEL calculation Here 500 is the Original price in cell E73 Price after Markdown is entered as 360 in cell E74 Rs Markdown was calculated in cell E75 by using the formula E73E74 The result 140 is shown in cell D75 121 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU XCEL formula in cell E76 for Markdown was E75E73 100 The result 28 is shown Hicrnsuft Excel Lecture14Marltup i5cnunt tarkdnwn Elle Edit Eiew insert Fgrmat Innis Qatar indnw elp Adobe PDF SLIM r X a 3 3 E3 Ei r39Ji El I i E I F I3 H F EI r1 MHRKDWN 72 r3 riginal price I SHORE r4 Price after Marde 350 Rs r5 Rs Markdwn 140 E7 3E74l n3 n Wu F m 2 MARKDOWNEXAMPLE2 A variety of plastic jugs that was bought for Rs 5775 was marked up 45 of the SellingPrice When the jugs went out of production they were marked down 40 What was the Sale Price after the 40 markdown Here there are two parts to this problem First we must find the original sale price so that markdown can be calculated on that price Original Sale Price Let Selling price 100 Markup on selling price 45 Cost 100 45 55 Thus Original Sale price 10055 x 5775 105Rs Rs Markdown Markdown 40 04 Rs Markdown 105 x 04 42 Sale price after markdown Sale price after markdown 105 42 63 Rs EXCEL calculation Here 5775 is the purchase price in cell F83 Selling price is entered as 100in cell F84 Rs Markup was calculated in cell F85 using the formula F84F83 The result is shown as 45 in cell F85 Original Sale Price was calculated in cell F87 by using the formula F84F86F83 The result 105 is shown in cell E87 Markdown was entered as 40 in cell F88 The Rs Markdown was calculated using the formula F87F88100 in cell F89 The result 42 is shown in cell F89 122 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The reduced price was calculated by using the formula F87F89 in cell F90 The result is shown as 63 in cell F90 Micrusu Excel Lecture14Mar upi5uuntMarI duwn Elle Edit Elevi Insert Fgrmal Incl Qatar indnw Help Ado e F39DF site m 2 e 1quot Fe SLIM r K a fair FEWFEEFI33 A E c n E F l 3 H a MARKDWN 33 Purchase price 34 Let Selling price 35 Markup BE ESEquot at riginel Sale price as Merkdwn 39 RS Markdwn en Reduced Price Ell PROJECT FINANCIAL ANALYSIS Financial analysis is the analysis of the accounts and the economic prospects of a firm which can be used to monitor and evaluate the firm39s financial position to plan future financing and to designate the size of the firm and its rate of growth When you carry out Project Financial analysis a number of Financial Calculations are required The important ones are summarized below Cost estimates Revenue estimates Forecasts of costs Forecasts of revenues Net cash flows Benefit cost analysis Internal Rate of Return BreakEven Analysis 123 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COST ESTIMATES In every project you will be required to prepare a cost estimate Generally such cost estimates cover calculations based on quantities and unit rates Such calculations are done in the form of tabular worksheets In large projects there may be a number of separate calculations for part projects Such component costs are then combined to calculate total cost These are simple worksheet calculations unless conditional processing is required Such conditional processing is useful if unit prices are to be found for a specific model from a large database REVENUE ESTIMATES Along with costs even revenues are calculated These calculations are similar to component costs FORECASTS OF COSTS Forecasting requires a technique for projections One of such technique Time Series Analysis will be covered later in this course Forecasting techniques vary from case to case The applicable method should be determined first Calculation of future forecasts can then be done through worksheets FORECASTS OF REVENUES These will be done similar to the forecast of costs Here also the method must be determined first Once the methodology is clear the worksheets can be prepared easily ET CASH FIOWS The difference between Revenue and Cost is called the Net Cash flow This is an important calculation as the entire Project Operation and Performance is based on its cash flows BENEFIT COST ANAIYSIS This is the end result of the Project Analysis The ratio between Present Worth of Benefits and Costs is called the Benefit Cost BC ratio For a project to be viable without profit or loss the BC Ratio must be 1 or more Generally a BC Ratio of 12 is considered acceptable For Public projects even lesser BC ratio may be accepted for social reasons MRNAI RATE OF RETURN Internal Rate of Return or lRR is that Discount Rate at which the Present Worth of Costs is equal to the Present Worth of Benefits lRR is the most important parameter in Financial and Economic Analysis There are a number of functions in EXCEL for calculation of lRR BREAKEVEN ANALYSIS In every project where investment is made it is important to know how long it takes to recover the investment It is also important to find the breakeven point where the Cash lnflow becomes equal to Cash Outflow After that point the company has a positive cash flow ie there is surplus cash after meeting expenses 124 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 17 MATHEMATICS FINANCIAL MATHEMATICS INTRODUCTION TO SIMULTANEOUS EQUATIONS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 16 0 Financial Mathematics 0 Introduction to Linear Equations MARKDOWN Module 4 Module 4 covers the following 0 Financial Mathematics Lecture 17 0 Applications of Linear Equations 0 Lecture 1718 0 Breakeven Analysis 0 Lectures 1922 0 MidTerm Examination PROJECT FINANCIAL ANALYSIS Project Financial Analysis covers the following 0 Cost estimates Revenue estimates Forecasts of costs Forecasts of revenues Net cash flows Benefit cost analysis Internal Rate of Return BreakEven Analysis EXCEL FUNCTIONS FOR FINANCIAL ANALYSIS List of Excel Financial functions is as under The name and utility of each function is given below AMORDEGRC Returns the depreciation for each accounting period If an asset is purchased in the middle of the accounting period the prorated depreciation is taken into account The function is similar to AMORLINC except that a depreciation coefficient is applied in the calculation depending on the life of the assets Syntax AMORDEGRCcostdatepurchasedfirstperiodsalvageperiodratebasis Important Dates should be entered by using the DATE function or as results of other formulas or functions For example use DATE2008523 for the 23rd day of May 2008 Problems can occur if dates are entered as text Cost cost of the asset Datepurchased date of the purchase of the asset Firstperiod date of the end of the first period Salvage salvage value at the end of the life of the asset Period period 125 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Rate rate of depreciation Basis year basis to be used Basis Date system 0 or omitted 360 days NASD method 1 Actual 3 365 days in a year 4 360 days in a year European method Remarks Microsoft Excel stores dates as sequential serial numbers so they can be used in calculations By default January 1 1900 is serial number 1 and January 1 2008 is serial number 39448 because it is 39448 days after January 1 1900 This function will return the depreciation until the last period of the life of the assets or until the cumulated value of depreciation is greater than the cost of the assets minus the salvage value The life of the asset is calculated by 1 quotratequot The depreciation coefficient depends on the life of the asset If the life of the asset is between 3 and 4 years the coefficient is 15 If the life of the asset is between 5 and 6 years then the coefficient is 2 If the life is the asset is greater than 6 years then the coefficient is 25 The depreciation rate will grow to 50 percent for the period preceding the last period and will grow to 100 percent for the last period If the life of assets is between 0 zero and 1 1 and 2 2 and 3 or 4 and 5 the NUM error value is returned See its example in next lecture AMORLINC Returns the depreciation for each accounting period If an asset is purchased in the middle of the accounting period the prorated depreciation is taken into account Syntax AMORLINCcostdatepurchasedfirstperiodsalvageperiodratebasis Cost cost of the asset Datepurchased date of the purchase of the asset Firstperiod date of the end of the first period Salvage salvage value at the end of the life of the asset Period period Rate rate of depreciation Basis year basis to be used Basis Date system 0 or omitted 360 days NASD method 1 3 4 Actual 365 days in a year 360 days in a year European method AMORQNCEXAMPE A2 A3 A4 Data Description 2400 Cost 8192008 Date purchased 12312008 End of the first period 126 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU A5 300 Salvage value A6 1 Period A7 15 Depreciation rate A8 1 Actual basis see above Formula Result Description AMOFiLNCA2A3A4A5A6A 7A8 First period depreciation 360 m Returns the cumulative interest paid between two periods For description see lecture 8 CUMPRINC Returns the cumulative principal paid on a loan between two periods For description see lecture 8 DB Returns the depreciation of an asset for a specified period using the fixeddeclining balance method Syntax DBcostsalvagelifeperiodmonth Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is being depreciated sometimes called the useful life of the asset Period is the period for which you want to calculate the depreciation Period must use the same units as life Month is the number of months in the first year If month is omitted it is assumed to be 12 Remarks 0 The fixeddeclining balance method computes depreciation at a fixed rate DB uses the following formulas to calculate depreciation for a period cost total depreciation from prior periods rate where rate 1 salvage cost quot 1 ife rounded to three decimal places 0 Depreciation for the first and last periods is a special case For the first period DB uses this formula cost rate month 12 o For the last period DB uses this formula cost total depreciation from prior periods rate 12 month 12 DDB Returns the depreciation of an asset for a specified period using the doubledeclining balance method or some other method you specify Syntax DDBcostsalvagelifeperiodfactor Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset This value can be 0 Life is the number of periods over which the asset is being depreciated sometimes called the useful life of the asset Period is the period for which you want to calculate the depreciation Period must use the same units as life Factor is the rate at which the balance declines lf factor is omitted it is assumed to be 2 the doubledeclining balance method 1 27 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Important All five arguments must be positive numbers Remarks 0 The doubledeclining balance method computes depreciation at an accelerated rate Depreciation is highest in the first period and decreases in successive periods DDB uses the following formula to calculate depreciation for a period Min cost total depreciation from prior periods factorlife cost salvage total depreciation from prior periods 0 Change factor if you do not want to use the doubledeclining balance method MIRR Returns the modified internal rate of return for a series of periodic cash flows MIRR considers both the cost of the investment and the interest received on reinvestment of cash Syntax MIRRvaluesfinanceratereinvestrate Values is an array or a reference to cells that contain numbers These numbers represent a series of payments negative values and income positive values occurring at regular periods Values must contain at least one positive value and one negative value to calculate the modified internal rate of return Otherwise MIRR returns the DlVO error value If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included Financerate is the interest rate you pay on the money used in the cash flows Reinvestrate is the interest rate you receive on the cash flows as you reinvest them INTERNAI RATE OF RETURN IRR Returns the internal rate of return for a series of cash flows o lf IRR can39t find a result that works after 20 tries the NUM error value is returned 0 In most cases you do not need to provide guess for the IRR calculation lf guess is omitted it is assumed to be 01 10 percent o If IRR gives the NUM error value or if the result is not close to what you expected try again with a different value for guess IRREXAMPLE In the slide the Excel worksheet is shown 128 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In cell A97 the investment of 70000 is entered with minus sign to denote negative cash flow In cell A98 to A102 revenue per year 449 5 is entered IRRA97A101 formula in cell A103 only years 1 to 4 were selected from the revenue stream The IRR is 2 in this case In the next formula in cell A105 the entire revenue stream was considered The IRR improved to 9 Next only first 2 years of revenue stream were considered with an initial guess of 10 The result was 44 r J Microsoft Excel Eoolt1 Eile Edit Elevr insert Fgrmet Idols gate window elp seller v sflev as E slil Esrial vmvlsso Es3s rfsEs39iEIEEEE 390 END v H a i s I E l I l D as lRRyaluesguess as Description sr 7oooo Initial cost of a business as 12 Net income for the rst year as 1sooo Net income for the second year 1o 1sooo Net income for the third year an 21ooo Net income for the fourth year 1o2 zsooo Net income for the fth year its ms lRRfAQA1 1 IRR after 4 years 2 ice IRRA9A1DE IRR after 5 years 9 me lRRfAQAQQ1 IRR after 2 years in include a guess 44 391I39lll 129 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 18 MATHEMATICS FINANCIAL MATHEMATICS SOLVE TWO LINEAR EQUATIONS WITH TWO UNKNOWNS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 17 o Solve two linear equations with two unknowns AMORDEGRCEXAMPLE itirm EamlLmture1 IFinzamialFum nna E A I L Eli Ejlt fallow lnaort Egmat Elma Eater lrdaw Ijolpi ningol iE39F W Shaitiiu l im l f etr m E H ELM 1 a a a aaaaIaaacaamaararamaazuaaii a a 13 AMDHDEGRCcoatdatapurchaaadfiratpariadaavagapariodratabaais 14 EData a DE EE39Irilpti ll I 13 tg Data piu rchaaad 1104231 End of the first perind 1 300 Salvage valua 1a a uii39l Depreciation rate 21 1 mammal baaia AMQRDEGRGIM 5IA15I First ariad de 39reeiation WE MEWS quot AMORLINCEXAMPLE 1 3 0 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft IEmal Lecturej EF39i39lil i lFl Il1 l rl E IEIIE Edit ew insert Format Ionls Esta lndnw liens 41der FDF mrkshee tfunttionsbgrtat w i sea 39 a a vevar EUN 1 X r KRMDRLIIHICEA1EAIEA1Fgma i g em A I E 13 Data Description 15 2400 Cost IE 030319 Date purchased 1 031231 End of the first period 13 300 Salvage value 19 1 Period 33 015 lDepreciation rate 21 391 Actual basis AMORLINC A15II 6 22 A17A131 AZUmZ ill 23 First period depreciation 350i DBEXAMPLE Microsoft Excel Lecture1EFinancialFunctiune Elle Edit iew insert Format eels Data window elp Ado eF DF worksheet functions hr tat v E gt 39 f l 11s it 303 3 a o g 3 i SLIM r X o a DEII EF QE EEIIFI a a 25 DBcostsalvagelifeperiodmonth 25 Data Description 2 1000000 Initial cost 23 100000 Salvage value 29 0 quotLifetime in years so DBA27023A2017 Depreciation in first year 31 39Dims 39 I39i39iii with with only 7 months calculated a 13303333 33 ADDITIONAL DB EXAMPLES Look at the following examples to see how the DB function can be used in different ways DBA27A28A2917 Depreciation in first year with only 7 months calculated 18608333 13 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU DBA27A28A2927 Depreciation in second year 25963942 DBA27A28A2937 Depreciation in third year 17681444 DBA27A28A2947 Depreciation in fourth year 12041064 DBA27A28A2957 Depreciation in fifth year 8199964 DBA27A28A2967 Depreciation in sixth year 5584176 DBA27A28A2975 Depreciation in seventh year with only 5 months calculated 1584510 Returns the present value of an investment Syntax PVratenperpmtfvtype Rate interest rate per period Nper total number of payment periods in an annuity Pmt payment made each period and cannot change over the life of the annuity Fv future value or a cash balance you want to attain after the last payment is made Type number 0 or 1 and indicates when payments are due Microsoft Excel Lecture1EFinancialFunctinns Elle Edit ElevJ insert Format Idols Data window elp adobe F39DF worksheet functions by tat v 539 X s a 2 v a a v a v a v SLIM v X J 8 F39leEf12 12M9MF Ilj a a E 45 PWrate nper pmtfvtype 15 Data Description Money paid out of an insurance annuity at 500 1 the end of every month 008 Interest rate earned on the money paid out 48 19 20 Years the money will be paid out 5 s pvm412 Present value of an annuity with the terms i12m49m47 above 5977715 52 2 0 m Returns the net present value of an investment based on a series of periodic cash flows and a discount rate Syntax NPVratevalue1value2 Rate rate of discount over the length of one period Value1 value2 1 to 29 arguments representing the payments and income 1 32 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Lecture1EFinancialFunctinns Elle Edit Eiew insert Fgrrnat eels gate indew elp adage F39DF tt v m 239 a SLIM v X J f3 NPViAEAEEA5Ei EDAEij a E aNPWratejualueWalueZ aEiMMPLE 1 55 Data Description 5 annual disceunt rate Initial cast of investment cine year from 55 teday 55 Return from first year 55 4200 Return tram secend year 55 Return from third year a NPVM57A53A59 60ai rt 55 118844 XNPV Returns the net present value for a schedule of cash flows that is not necessarily periodic Syntax XNPVratevaluesdates Rate discount rate to apply to the cash flows Values series of cash flows that corresponds to a schedule of payments in dates ates schedule of payment dates that corresponds to the cash flow pamnts Microsoft Excel Eeelt1 5 Elle Edit EielaI insert Fgrmat leels gate indew elp are e are areaia aavaia v via E at are gala am all e u l E E l s as 939 tat 433 is E 7 Eli 139 a XNPVrate5ualues5dates Example 1 rate 9 Values Dates 1 i ii ii i January 15 2mm 25 Marsh 1 EMS 425D Deteher 3D 2W3 325i February 155 2009 25l April 15 zone XHPVBSA5A9B BQ Net present value 2 85 5 1 3 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 19 PERFORM BREAKEVEN ANALYSIS EXCEL FUNCTIONS FINANCIAL ANALYSIS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Perform breakeven analysis 0 MS EXCEL Financial Functions SLN Returns the straightline depreciation of an asset for one period Syntax SLNcostsalvagelife Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset Miereeeft Eaeel Leeture1EFineneielFunetiene Elle Edit iew insert Fgrmat eels gate indew Help adage F39DF WDFkShEEtFUI39IEtiDI39IS E 9 lt v a v E v E E a v a v a SUM v X a a eulraaaaaaarm a a a3 SLN cestsalvaelife 5 Data Descriptin Ea Cat 59 7500 Salvage value a 10 Years f useful life SLNA68A69MD a The depreciatin allwance fr each year 2250 F3 i39 39 SYD Returns the sumofyears39 digits depreciation of an asset for a specified period Syntax SYDcostsalvagelifeper Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset 1 34 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Per is the period and must use the same units as life Remark 0 SYD is calculated as follows STE 2 east aetvage 139 Ef e per39 lth ii jtil 1 Micrusnft Excel Lecture1EFinancialFunctiun5 Elle Edit ew insert Fgrmat eels Data indnw elp adage F39DF WDrkShEEHUI39IEtiEII39IS h39r Eat 139 E quot9 at E r E at E 1r 6 Ir a 139 T SUM r X J a S rD ED 31 EE1 a a as SYDcst salvage life per rr SYD CsatSalvage ifeper12llifelife1 FE ra Data Descriptin a 300 Initial est a 7500 Salvage value a 10 quotLifespan in years aa SYD lUA31 A21 a S TP F tquot at 1W3 iation allowance for the first year 409091 as I VDB Returns the depreciation of an asset for any period you specify including partial periods using the doubledeclining balance method or some other method you specify VDB stands for variable declining balance Syntax VDBcostsalvagelifestartperiodendperiodfactornoswitch Cost is the initial cost of the asset Salvage is the value at the end of the depreciation sometimes called the salvage value of the asset Life is the number of periods over which the asset is depreciated sometimes called the useful life of the asset Startperiod is the starting period for which you want to calculate the depreciation Startperiod must use the same units as life Endperiod is the ending period for which you want to calculate the depreciation Endperiod must use the same units as life Factor is the rate at which the balance declines lf factor is omitted it is assumed to be 2 the doubledeclining balance method Change factor if you do not want to use the doubledeclining balance method For a description of the doubledeclining balance method see DDB Noswitch is a logical value specifying whether to switch to straightline depreciation when depreciation is greater than the declining balance calculation o If noswitch is TRUE Microsoft Excel does not switch to straightline depreciation even when the depreciation is greater than the declining balance calculation 135 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o If noswitch is FALSE or omitted Excel switches to straightline depreciation when depreciation is greater than the declining balance calculation All arguments except noswitch must be positive numbers All arguments except noswitch must be positive numbers Hicrusuft Excel Lecture1EFinancialFunctiuns Elle Edit ew insert Format sols Qatar window elp adage F39DF W rksheetfuntti nst El 3 a v o v E v E if E quotl39 quotl39 a 139 sum 139 x or a vuaraaaaauaairsaaaaau1j a a as VDBCcost salvage life startperid endperiod a factor noswitch 33 Data Description 39 2400 Initial cost 9 i300 Salvage value 9 10 Lifetime in years a VDBM39MUA91365 A9501 as First day39s depreciation a Excel automatically assumes that f39actr is 2 132 as E Returns the internal rate of return for a series of cash flows represented by the numbers in values These cash flows do not have to be even as they would be for an annuity However the cash flows must occur at regular intervals such as monthly or annually The internal rate of return is the interest rate received for an investment consisting of payments negative values and income positive values that occur at regular periods 0 If IRR can39t find a result that works after 20 tries the NUM error value is returned 0 In most cases you do not need to provide guess for the IRR calculation If guess is omitted it is assumed to be 01 10 percent 0 If IRR gives the NUM error value or if the result is not close to what you expected try again with a different value for guess 136 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Remarks IRR is closely related to NPV the net present value function The rate of return calculated by IRR is the interest rate corresponding to a 0 zero net present value The following formula demonstrates how NPV and IRR are related NPVRRB1B6B1B6 equals 360E08 Within the accuracy of the IRR calculation the value 360E08 is effectively 0 zero IRREXAMPLE In the slide the Excel worksheet is shown In cell A97 the investment of 70000 is entered with minus sign to denote negative cash flow In cell A98 to A102 revenue per year 1 to 5 is entered In the first formula in cell A103 IRRA97A101 only years 1 to 4 were selected for the revenue stream The HR is 2 in this case In the next formula in cell A105 the entire revenue stream was considered The IRR improved to 9 Next only first 2 years of revenue stream were considered with an initial guess of 10 not shown in slide The result was 44 Microsoft Excel Eoolt1 Elle Edit ew insert Format Iools gate windowI elp t sold cv Elev E vsltl isrial m vls s ol Elia a s assists L Eno v a s B C D E l as lRRyaluesguess as Description a JDEIIIDD Initial cost of a business as 12 Net income for the rst year as 15 Net income for the second year 1o 13 Net income for the third year 1o1 21 Net income for the fourth year 1oz E Net income for the fth year 103 ms IRRAQA1D1 IRR after 4 years 2 ms IRRA9A102 IRR after 5 years 9 ins IRRAQA991 IRR after 2 years 1o include a guess 44 XIRR Returns the internal rate of return for a schedule of cash flows that is not necessarily periodic To calculate the internal rate of return for a series of periodic cash flows use the IRR function 137 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If this function is not available and returns the NAME error install and load the Analysis ToolPak addin To do that 1 On the Tools menu click AddIns 2 In the AddIns available list select the Analysis ToolPak box and then click OK 3 If necessary follow the instructions in the setup program Syntax XIRRvaluesdatesguess Values is a series of cash flows that corresponds to a schedule of payments in dates The first payment is optional and corresponds to a cost or payment that occurs at the beginning of the investment If the first value is a cost or payment it must be a negative value All succeeding payments are discounted based on a 365day year The series of values must contain at least one positive and one negative value Dates is a schedule of payment dates that corresponds to the cash flow payments The first payment date indicates the beginning of the schedule of payments All other dates must be later than this date but they may occur in any order Dates should be entered by using the DATE function or as results of other formulas or functions For example use DATE2008523 for the 23rd day of May 2008 Problems can occur if dates are entered as text Guess is a number that you guess is close to the result oleRR Remarks 0 Microsoft Excel stores dates as sequential serial numbers so they can be used in calculations By default January 1 1900 is serial number 1 and January 1 2008 is serial number 39448 because it is 39448 days after January 1 1900 Microsoft Excel for the Macintosh uses a different date system as its default 0 Numbers in dates are truncated to integers o XIRR expects at least one positive cash flow and one negative cash flow otherwise XIRR returns the NUM error value o If any number in dates is not a valid date XIRR returns the VALUE error value o If any number in dates precedes the starting date XIRR returns the NUM error value o If values and dates contain a different number of values XIRR returns the NUM error value 0 In most cases you do not need to provide guess for the XIRR calculation lf omitted guess is assumed to be 01 10 percent 0 XIRR is closely related to XNPV the net present value function The rate of return calculated by XIRR is the interest rate corresponding to XNPV 0 0 Excel uses an iterative technique for calculating XIRR Using a changing rate starting with guess XIRR cycles through the calculation until the result is accurate within 0000001 percent lf XIRR can39t find a result that works after 100 tries the NUM error value is returned The rate is changed until where P r I39 92 tar ail quot3 1 rate 355 di the ith or last payment date d1 the 0th payment date Pi the ith or last payment XIRR EXAMPLE Here the investment is in cell A111 The revenue stream is in cells A112 to a115 The dates for each investment or revenue are given in cells B111 to B115 Please note that the dates are in European format yearmonthday On your computer you may not have this format 138 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ei iernsnft Excel Lecture1 EFinaneialFunctiens E Eile Edit Eiew Lnsert Fgrmet Inels gate lndew elp Ade e PDF werksheetfu 9 3 CE v 2 110 e 55 sum x r r KIRHIIA I I IA115EllllEli 5El1j 2 El 109 MERWalues atesguess 10 Values Dates 111 10000 20080101 112 250 20080301 113 r4250 20081 0130 1 31250 20090215 2750 20000401 E XIRRM111 KIRR 033352535 r 3731 11 1153111 123311201 1 After entering these days in Excel you can right click on the cell You see a short cut menu as shown below E iereseft Excel Leeture1EFinaneialFunetinns Elle Edit EleniI Lnsert Fgrrnet eels gate indew Help Dg g y l rr Ev l jmws nrial 20IQEE 3EE iii E quot3quot Ellll fr DATEEUUE11II E mazegueee 11 Dates 11I20080101 3 Cut 112 20080301 113 20031030 Earl E15 Paste eeciel H 20000215 1 20000401 Clear Ientents V r a H men 0373325 34 r llr HE EermatCells 119 PiclgFrem List 1239 Field match imid tES re rperlink When you will select Format Cells the Format Cells Dialog Box appears as shown below You can then choose the desired format for the date 1 3 9 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fu rrnat Eells Filignment 1 Fent 1 Etercler 1 Patterns 1 F39retectien 1 Qategcnry Sample General J 20080101 Number Currency Type Ficccnuntini clen 14 mars EIIIIIII T39me secures14 Percentage scarce14 133u Fractlen 0143314 Scientific 010314 1330 Text 5 Etial 143 2001 51311 J Lecale lecaticunIH 15weclish Date Fermats clisplay39 clate ancl time serial numbers as clate values Except FcIr items that have an asterislt II applied Fermats 2ch net switch clate erclers Iwith the eperating system 1 Di 1 Cancel 1 In cell A116 the formula XRRA111A115B111B11501 the range A111A115 is the cost and revenue stream The range B111B115 is the stream for dates The third term 01 is the initial guess for XIRR The answer in fraction or is given in cell B1163734 MAR QUATIONS Linear equations have following applications in Merchandising Mathematics 0 Solve two linear equations with two variables Solve problems that require setting up linear equations with two variables Perform linear CostVolumeProfit and breakeven analysis employing The contribution margin approach The algebraic approach of solving the cost and revenue functions SOL VING LINEAR EQUATIONS Here is an example of solving simultaneous linear equations 2x 3y 6 x y 2 Solve for y 2x 3y 6 2x 2y 4 5y 10 y 105 y 2 140 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Let us look at the same equations again 2x 3y 6 x y 2 We solved for x Now let us substitute y by 2 2x 32 6 2x 6 6 2x 0 x 0 Check your answer By substituting the values into each of the equations Eguation 1 2x 3y 6 x 0 y 2 LHS 2x 3y 2032 6 RHS x y 2 LHSX y022RHS The right side is equal to left hand side Hence the answer is correct 141 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 20 PERFORM BREAKEVEN ANALYSIS EXCEL FUNCTIONS FOR FINANCIAL ANALYSIS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 MS EXCEL Financial Functions 0 Perform BreakEven Analysis LINEAR QUATIONS Zain purchases the same amount of commodity 1 and 2 each week After price increases from Rs 110 to Rs 115 per item of commodity 1 and from Rs 098 to Rs 114 per item of commodity 2 the weekly bill rose from Rs 8440 to Rs 9170 How many items of commodity 1 and 2 are purchased each week Let x of commodity 1 Let y of commodity 2 Settinq up Linitr Equations Eguation 1 110x 098y 8440 1 Eliminate x in 1 by Dividing both sides by 110 1 10x 098y110 8440110 x 08909y 7673 M 115x114y917 2 Eliminate x in 2 by Dividing both sides by 115 115x 114y115 9170115 x 09913y 7974 Result 1 x 08909y 7673 3 x 09913y 7974 4 Next Subtract 4 from 3 w 01004y 301 y 30101004 Or y2998 approximantely 30 30 items of commodity 2 are purchased each week 110X 098y 8440 Substitution Substitute value of y in 1 Result 110x 0982998 8440 Solve 110x 2938 8440 110x 8440 2938 110x 5502 Resu x 5002 approximately 50 50 items of commodity 1 are purchased each week 1 42 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Check your answer New weeklv cost Commodity 1 50 x 115 5750Rs Commodity 2 30 x 114 3420 Total cost 91 70Rs TERMINOLOGY There are either Business Costs or Expenses Mk Even Analysis Break Even Analysis refers to the calculation to determine how much product a company must sell in order to get break even point The point at which no profit is made and no losses are incurred on that product Breakeven analysis provides insight into whether or not revenue from a product or service has the ability to cover the relevant costs of production of that product or service Managers can use this information in making a wide range of business decisions including setting prices preparing competitive bids and applying for loans CostVolume Profit Analysis Costvolumeprofit CVP analysis expands the use of information provided by breakeven analysis It deals with how profits and costs change with a change in volume More specifically it looks at the effects on profits by changes in such factors as variable costs fixed costs selling prices volume By studying the relationships of costs sales and net income management is better able to cope with many planning decisions For example CVP analysis attempts to answer the following questions 1 What sales volume is required to break even 2 What sales volume is necessary in order to earn a desired target profit 3 What profit can be expected on a given sales volume 4 How would changes in selling price variable costs fixed costs and output affect profits Fixed Costs FC Fixed Costs are such costs that do not change if sales increase or decrease eg rent property taxes some forms of depreciation Variable Costs VC Variable costs do change in direct proportion to sales volume eg material costs and direct labor costs Production Ca citv PC It is the number of units a firm can make in a given period Break Even Point Break Even point is a point at which neither a profit nor loss is made Revenue is exactly equal to costs Break even point can be expressed as 1 units 2 Sales or Rupees Rs 3 Percent of capacity BEP in units calculates how many units should be sold to break even lfthe product is sold in a quantity greater than this the firm will makes a profit below this point a loss BEP in units Fixed Costs Contribution Margin per unit 1 43 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BEP in Rs calculates the revenue that must be obtained to reach break even point BEP in Rs Fixed Costs x Net Sales Contribution Margin BEP in Rs Fixed costs x Selling Price per unit Contribution Margin per unit BEP as percent of capacity calculates what percent of production capacity will be utilized to produce the number of units required to reach break even point BEP as of capacity BEP in units x 100 Production capacity Contribution Margin Contribution Margin is the Rs amount that is found by deducting Variable Costs from Sales or revenues and contributes to meeting Fixed Costs and making a Net Profit It can be calculated on a total or per unit basis Contribution Margin Net Sales Variable Cost S VC Contribution margin per unit CM Sale price per unit Variable cost per unit Contribution Rgte CR Contribution rate Contribution Margin X 100 CM X 100 Net sales S Contribution rate Contribution Margin per unit X 100 CM X 100 Sale price per unit S A CONTRIBUTION MARGIN STATEMENT Rs Net Sales Price x Units Sold x 100 Less Variable Costs x x Contribution Margin x x Less Fixed Costs x x Net Income x x The net sales are calculated by multiplying price per unit with number of units sold Net Sales Sale price per unit x number of units sold This figure is treated as 100 Next variable costs are specified and deducted from the Net sales to obtain the Contribution Margin Next Fixed costs are deducted from the contribution Margin The result is Net Income Net Income Contribution Margin Fixed Costs Under the column percentage of each item is calculated with respect to the Net Sales 1 44 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Here39s an example Net Sales 462452 Rs 100 Less Variable Costs Cost of goods sold 230934 Rs 50 Sales Commissions 58852 Rs 127 Delivery Charges 13984 Rs 3 Total Variable Costs 230934 58852 13984 303770 Rs 657 Contribution Margin 462452 303770 158682 Rs 343 Less Fixed Costs Advertising 1850 Rs 04 Depreciation 13250 Rs 28 Insurance 5400 Rs 12 Payroll Taxes 8200 Rs 18 Rent 9600 Rs 21 Utilities 17801 Rs 38 Wages 40000 Rs 86 Total Fixed Costs 18501325054008200 96001780140000 96101 Rs 208 Net Income 158682 96101 62581 Rs 135 m A firm is planning to add a new item in its product line Market research indicates that the new product can be sold at Rs 50 per unit Cost analysis provides the following information Fixed Costs FC per period Rs 8640 Variable Costs VC Rs 30 per unit Production Capacity per period 900 units How much does the sale of an additional unit of a firm s new product contribute towards increasing its net income M Contribution Margin per unit CM S VC Contribution Rate CR CMS x 100 Break Even Point BEP in Units x Rs x FC CM in Sales Rs Rs x FC CM S in of Capacity BEP in UnitsPC x 100 At Break Even Net Profit or Loss 0 Scenario 1 Summary Selling price per unit S 50 Rs Fixed Costs per period FC Rs 8640 Variable Costs VC Rs 30 per unit 145 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Production Capacity per period PC 900 units Solution of this problem is in the next lecture 146 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 21 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS USING THE CONTRIBUTION MARGIN APPROACH OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Perform BreakEven Analysis 0 MS EXCEL Financial Functions 0 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 20 0 Perform linear costvolume profit and breakeven analysis 0 Using the contribution margin approach SCENARIO 1 Contribution Margin per unit CM S VC 50 30 20 Rs Contribution rate CR CMS x 100 Rs 205OX100 40 Break Even Point In Units x FC CM 864020 432 Units In Rs x FCCM S Rs 8640Rs20 Rs50 Rs21600 of Capacity BEP in units PC X100 432 900gtlt100 48 Thus by selling more than 432 units of its new product a firm can make profit SCENARIO 2 The Lighting Division of A Lighting Fitting Manufacturer plans to introduce a new street light based on the following accounting information FC per period Rs 3136 VC Rs157 per unit 8 Rs 185 per unit Production Capacity per period 320 units Calculate the break even point BEP in units in rupees as a percent of capacity Break Even Point in units FC CM CM S VC Rs185 157 Rs28 147 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BEP in units 313628 112 Units Break Even Point in Rupees FC CM S 313628 185 20720 Rs Break Even Point as a percent of cayacitv BEP in unitsPCX100 112320gtlt100 35 Capacity SCENARIO 21 FC Rs3136 VC Rs157per unit 8 Rs185 per unit Production Capacity 320 units Determine the BEP as a of capacity if FC are reduced to Rs2688 Formula BEP as a of capacity BEP in unitsPCX100 Step1 Find CM Step 2 Find BEP in units Step 3 Find of Capacity Step 1 Find CM per unit S 185Rs per unit VC 157Rs per unit CM S VC 185 157 Rs 28 Step 2 Find EP in units BEP in units FCCM Rs 2688 Rs28 96 Units Step 3 Find of Capacity BEP as a of capacity BEP in units PCX100 96320gtlt100 30 of Capacity SCENARIO 22 FC Rs3136 VC Rs157 per unit S Rs185 per unit Production Capacity 320 units per period Determine the BEP as a of capacity if FC are increased to Rs4588 and VC reduced to 80 of S BEP as a of capacity BEP in unitsPCX100 NewVC S x 80 185 x 08 Rs148 New FC 4588 Rs 148 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Find CM per unit S 185 Rs per unit VC 148Rs per unit CM S VC Rs 37 Step 2 Find EP in units BEP in units FCCM Rs 4588 Rs 37 124 Units Step 3 Find of Capacity BEP as a of capacity BEP in units PCgtlt100 124320gtlt100 39 of Capacity SCENARIO 2 3 FC Rs 3136 VG Rs157 per unit 8 Rs185 per unit Production Capacity 320 units per period Determine the BEP as a of capacity if S is reduced to Rs171 BEP as a of capacity BEP in unitsPCX100 Step 1 Find CM per unit S 171Rs per unit VC 157Rs per unit CM S VC Rs 14 Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 14 224 Units Step 3 Find BEP as a of Capacity BEP as a of capacity BEP in units PCgtlt100 224320gtlt100 70 of Capacity 149 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 22 PERFORM LINEAR COSTVOLUME PROFIT AND BREAKEVEN ANALYSIS OBJECTIVES The objectives of the lecture are to learn about I Review Lecture 21 I Perform Linear CostVolume Profit and BreakEven analysis Using Microsoft Excel SCENARIO 1 Let us look at different scenarios for calculation of contribution margin and net profit The explanations are given in the slides The Break Even Point in Rs is 21600 The break Even point as a of capacity is 48 iiz i 39s ll EIIEEII LEEIUTIEJ I5IquotE quotl ri iir1 link the rm Ila15 tutz Lta39li e hem Le Ensignatria a Eriftl l 39w rrmvnry 1 L EEF39tI mammal 39 i El I n E F 3 H I t i Scenar 1 3 Frtluctnh Galactle E i Eale 5 a t 5 Variable eets tilt 31 a Eentrihutien Illlariri i Elli Eluln Ell HliIHS r FiltEEI oat 1 Ft E 364m 3 in Units l FE l Ellll ll32 lllTlFlE 3 EP in HE E FE l EMF 5 Et EHlEtHi l in Ean units lPEf39l lll 4 4E EllEfllitt39lll39l SCENARIOZ The Break Even Point in Rsis 20720 The break Even Point as a of capacity is 35 150 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E H39Ermirl39 EEG LeliaLi i TiErnrrrs1 EJEWWLWFWWWeWmnn mm 5i i3EEiJEE iii Iingir EIEM I f gEIi if aquot r12 33 F FE H IIJ R if a I III E Scenario 2 i Preductien Sagacity 32 15 Sale E il iEi llTiarialzle eets VIE iii a Ennirihutinn Margin em S iii EH HI15Hfi 13 Fine East Fl area 5 El in Unita 2r FE i EM 112 Hi l i Hi EF in Rs 3 FE i EMF 5 Elii39 ii HHTHT IE N 31 EEFiin unite iFEiii 35 iiI IEi39Ii39ii iii 23 SCENARIO 21 The Break Even Point in Rs is 17760 The break Even Point as a of capacity is 30 Mitremm Ham Leelmejljtmarieg jr H 53mm L1H Ei DIESELEEIT39E39 E i i w viii ri uiy i itr r 13 j amalgam Scenarln 21 25 F Tn lmiinin Erapanzity 3 20 23 Sallie E E E 39il 2 variable GizaSite iiE iiiT na Enntrilutian Margin E Eliili Siiii 2E EHEEmiliili39ii 29 Fixed Enst Fri EF in Units in FE i EM BE HHEEJHEE 31 EP in IRS iIFC i Eiiili E 39iii39iiiliv HEilii iHEE JEF in mite ilFCiilii Elli EHE iHE iil 13 F G H I J L SCENARIO 22 The Break Even Point in Rs is 22940 The break Even Point as of capacity is 39 15 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I39I39Iifn 39iJIE Jil Einf l L z lijfLlljtn iill i39lilfi51 39il EJEIH at am 1w Em ism ram mim rah gamer 4331 D i f evw i Emu wh fye ri r 39 I j i5 E 392 El E i Senarm mFrrndiuctiicrn apaeity F a 32G ilam 5 quot135 a Haraicile Errata iii that a 14 13 Eenirihuti ni Iiilrgin DIM F E Iuii 3 H4iilHi42 dai Fixe ail IFE 1533 5 E in Units at 1 FE i EM quot124 HaidiHai EIEF in R5 iFCi i EMF 223i EH iH l 4r iEFini units iFEi i 39 H i i39H i iii 13 F I H i I lIi SCENARIO 23 The Break Even Point in Rsis 38304 The break Even Point as of capacity is 70 E i ielrmi EEEI u Lwiurgjii juMiaiHJJ Em eii i aiw Efg Erasg rrg m gfugi 122 ii j HI I ID E iF 1 eii i J a IF39IrIirueiinm capacity EM 3 Eaie a E iii a iuiarriahile casts hit 5 L15 Gienirillzmiieri Margin 5 cm a id EHIEEeH d Fittea East 3135 EF in Limits 5 FE i EM H55giH55 EIF in 1R5 F6 i came 3331M i I iilli53 EEFEIHJ iLlll it39S iFEi i iii mg iii H5WH52WEIE ED 152 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Net Income NI or Profit Net incomeNlNumber of units sale above BEP in unitsgtltContribution Margin per unit SCENARIO 24 FC Rs 3136 VC Rs 157 8 Rs 185 Capacity 320units Determine the Net income NI if 134 units are sold Formula for Net Income NI Nl Number of units sold above BEP x CM Step 1 Find CM per unit S 185Rs per unit VC 157Rs per unit CM S VC 185 157 Rs 28 CM of Rs28 per unit Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units sold over BEP Units Sold 134 units BEP in units 112 units Number of units sold above BEP in units 134 112 22 Hence ompan had a net income NI of 22 Rs 28 Rs 616 I Microsoft Excel Book1 i Eile Edit ew insert Format Iools gate indow Help tale acne alaal a ale via 2 tltlll ENEl le IE I H IEEEEI 5 a 9 TaEe39iEIEE iil 39 F15 v a i a e c o E E e 39 1 Scenario 24 2 Production capacityr 320 units 3 SaleS 135 Re 4 Variable CoeteVC 15 Re 5 Contribution Margin CM 3 VC 23 F3F4 5 Fixed Coats FE 3135 Re F BEP in units FE i CM 112 FEIF5 3 Units eold 134 9 Units over BEFin uniteiUnite eoldBEFiin units 22 FEFT in HI unite over BEFin units a CM 515 FEF5 SCENARIO 25 FC Rs 3136 VC Rs157 S Rs185 Capacity 320 units How many units must be sold to generate Nl of Rs 2000 Formula for Net Income Number of Units sold above BEP in units Nl CM 1 5 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Step 1 Find CM per unit S 185Rs per unit VC 1 Rs per unit CM Rs 28 CM of Rs28 per unit Step 2 Find BEP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units over m Number of units above BEP Nl ICM Rs 2000Rs 28 71 Units Total units sold 71 Units above BEP 112 BEP Units 183 units A Hioroioll ml t loro 5ioooiooii EllI sologm BEBE Elli l Tj oa n u gg g h 39 1 In 3quot TE or 3 zrtzl Hivl A E E II E F E H J h in Production Eooooity 33D H Solo o 2 too liloriolhlo illooto E E Iii o Contribution lllllari rl Ellll ENE Eii HEEJH 3 o Fixed Eot E Fl 31 o EF in Units or 2 lF l l39 Elli 112 HEEllilE o EFl in Ho FE lC llllF E E i HE39Eil Il o EFlirl onito iFlCi1 2 till 345 HEEll ll 1li l1 so llllEUnltoooorEEiF iEMl Elilil Lini llo WET BEEP 1 llilllll il39ll Ti 9 Total oniioE EF is oooo EFFquot onito 133 EHEHHEE Net loss Net lossNLNumber of units sale below BEP in units x Contribution Margin per unit Alternatively Negative sign with Net Income means Net loss That is Net loss Net Income Number of units sale below BEP in units Number of units sale above BEP in units Thus Net loss Net income Number of units sale above BEP in unitsgtltContribution Margin per unit SCENARIO 26 1 5 4 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FC Rs 3136 VC Rs157 S Rs185 Capacity 320 units Find the number of units sold if there is a Net Loss NL of Rs 336 M Net loss Number of units sale below BEP in units x Contribution Margin per unit Number of Units below BEP in units NLCM Step 1 Find CM per unit S 185 Rs per unit VC 157 Rs per unit CM 28 Rs CM of Rs28 per unit Step 2 Find EP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units below BEPNLCM Number of unit sales below BEP Rs 336Rs 28 per Unit 12 Units Hence Total Sold Units BEP in units Number of units sale below BEP 112 12 100 Alternate Method Net loss Net income Number of units sale above BEP in unitsgtltContribution Margin per unit Number of units sale above BEP in units Net loss Contribution Margin per unit 33628 12 Total sold units BEP in units Number of units sale above BEP in units 112 12 100 155 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hittersift HEEL LII itiIQJQHEIE Eitii glaii are 512 we Meat Tami we Mil3w Dita Mia r lial 3 ea 1131 Haiti 1quot I H 3 mm EVVLBVF EIWLE F lg Eoenarlo 2 Pram31in Eapeeity EEIE e Variant Eats WE 7 Ennirihuti n Mryirn fax3211M EmitE Fired eet Fl 1 me E in Units in Fl 1quot EM E ER in Re FE i39 EZIFiin units iPC39i iiill it m me Hi Unis ever iElFiiiilii WE Limits irter iEF Hlii i i TelWEI units EEIP as above Eli units SCENARIO 27 FC Rs 3136 VC Rs157per unit 8 Rs185 per unit Production Capacity 320 units The company operates at 85 of its capacity Find the Profit or Loss M Number of units above BEP in units x CM NI Step 1 Find CM per unit S 185 Rs per unit VC 1 Rs per unit CM Rs 28 CM of Rs28 per unit Step 2 Find BEP in units BEP in units FCCM Rs 3136 Rs 28 112 Units Step 3 Find units over BEP Units produced 320085 272 Units BEP in units 112 units Number of units over BEP in units 272 112 160 Hence Net income N 160 Units 28 4480 Rs ii i H 11 quot E 333 1155 HE H EITHEH 31 11112 ETH1JDDEH ETQ E H1i 1H E 3 55 THW1JHEEWHG eiEH1iii rH 9 H39Ii EslaH il39i Copyright Virtual University of Pakistan 156 Business Mathematics amp Statistics MTH 302 VU EttaElf ml miniature iii immin fi if Ella 55 ram L39 a Briam 51 3 E E if tl Iiii 15 1 E I E E m it Iquot 53 in El I ll Scenario 2 m Preduetien Espaaei tye PI eEEtt Capaeit ETE m Elle E E E 15 m varrlahle Costs tit 1 115 an Eantrilh tttiion Mriirt EM EB Hl39i iEth 13 we Fitted East an a 31135 E in Unlts it FE mitt 1112 HHEIHHM t inE l9 at EFF 1 Es FIE i Chi E E i l111til till1 il mg EEZPirl units liF39E39i39tl ii MATE liill ltIEIH ltl1ltti mg m Units eater HEP FEEEP39 Units t li ll1tl39ileH11E Profit i Units tarer EF EM itED lill1E il ll1tl4 Company A s year end operating results were as follows Total Sales of Rs 375000 Operated at 75 of capacity Total Variable Costs were Rs 150000 Total Fixed Costs were Rs 180000 What was Company A s BEP expressed in rupees of sales 1 5 7 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 23 STATISTICAL DATA REPRESENTATION OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 22 0 Statistical Data Representation MODULE 5 Statistical data representation Lecture 23 Measures of central tendency Lectures 2425 Measures of dispersion and skewness Lectures 2627 MODULE 6 Correlation Lecture 2829 Line Fitting Lectures 3031 Time Series and Exponential Smoothing Lectures 3233 MODULE 7 Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 3536 ChiSquare Lectures 37 Binomial Distribution Lectures 38 MODULE 8 Patterns of probability Binomial Poisson and Normal Distributions Lecture 3941 Estimating from Samples Inference Lectures 4243 Hypothesis testing ChiSquare Distribution Lectures 4445 EndTerm Examination STATISTICAL DATA Information is collected by government departments market researchers opinion pollsters and others Information then has to be organized and presented in a way that is easy to understand EASIS FOR CLASSIFICATION 1 Qualitative Attributes sex religion 2 Quantitative Characteristics Heights weights incomes etc 3 Geographical Regions Provinces divisions etc 4 Chronological or Temporal 5 By time of occurrence Time series 1 5 8 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TYPES OF CLASSIFICATION There are different types of classifications Oneway One characteristic Population Twoway Two characteristics at a time Threeway Three characteristics at a time METHODS OF PRESENTATION Different methods of representation are Text quotThe majority of population of Punjab is located in rural areas Semi tabular Data in rows Tabular Tables with rows and columns Graphic Charts and graphs TYPES OF GRAPHS O O O O O O O O PICTU Picture graph Column Graphs Line Graphs Circle Graphs Sector Graphs Conversion Graphs Travel Graphs Histograms Frequency Polygon Cumulative Polygon or Ogive RE GRAPH or PICTOGRAPH ln picture graph or pictograph each value is represented by a proportional number of pictures In the example below one car represents 10 ca l39S PICTURE GRAPHS Ears passin school it during the day 3 112m 3 3 511 lE r L PEEL w 1amp2 11 am i E11511 Ii ELEM goron GRAPHS 159 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Sector graphs use the division of a circle into different sectors The full circle is 360 degrees For each percentage degees are calculated and sectors plotted SECTOR GRAPHS Distributien ef employment injuries by eccupatien him lithemine and al quot 39 flan 23955 i39tPJ Ether Iii1 1 a5 39t I Trudi ntju Ii Ii herj LI1rquotI FailEEEJL an m m Example of Sector Graph Leeel e Liih eri 2 W F39IeeeLi re 2 39ii Residential Tit Diner 1 Cemmerciel 31 Agriculture 37 20 COLUMN AND BAR GRAPHS The following slide gives the Proportion of households by size in the form of a Column and Bar graph 160 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Fi f HEEHIE EiEE E E quotquot quot39 rr j I 39i 3 EEJgt r a Him inquot v r i I11 Mam Hm 2131mm Famous pietl i7 LINE GRAPHS Line graphs are the most commonly used graphs A line graph plots data as points and then joins them with a line LINE GRAFHS A student39s height mm a 1m mam EISEI m tlf T a it 1quot ith 1 1 al 9 Ht 493 11 lll lll llll ll 311231SE E EE EQMLEHHLE ig fm r Hr ig i aw Example 161 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Time Needed tn heatE luitahlle Lihra r Hateriallg El Number Inf 539 Emmi13 4D AIJIIE En Lamina 3 Eu ahl Materials 3393 39 IIII I I I I I I I I 5 ID 15 El 25 3 39II39II39IIITIE iim I39I39ll nuum Line chart with multiple series 4 35 an 4 3E 25 2D 15 1D 5 III New Dec Jan Feb Mar Apr 2005 EDGE EDI EDI EDI EDI itItaly EElanain 162 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 24 STATISTICAL REPRESENTATION MEASURES OF CENTRAL TENDENCY PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 18 0 Statistical Representation 0 Measures of Central Tendency LINE GRAPHS Line graphs are the most commonly used graphs In the following graph you can see the occurrence of causes of death due to cancer in males and females You can see that after the age of 40 the occurrence of cancer is much greater in the case of males The line graph of heart diseases also shows that the disease is more prominent in the case of males As you see line graphs help us to understand the trends in data very clearly LINE GRAPHS anges of death 539 nice Hurt the e I m 1H m mn 3 Hale1 iraia E EH39jifzl T k 5 a 39 r m u at 22 3391 4393 ti E39ti fl tiquot i ii 3quot at Ilil 5quot ts fit at quotIf 3951 i 39 WM Another line graph of temperature in 4 cities A B C and D shows that although the general pattern is similar the temperature in city A is lowest followed by D B and C In city C the highest temperature is close to 30 whereas in city A and B it is about 25 The highest temperature in city D is about 28 degrees 1 63 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LINE GRAPHS Temperature efcities ill E t and i939II39I 3812143 rquot v 39 n H 3 JE l 3 1 quot Ir If 1 an W El r39 fl Jib 15311 JAE JWESI JFZHAEdTJAEElWEI t 5r 39 n 41 p 5 I 5 15 l 35 39 E 1 739 a 1n s g 33 FIEAEIHJ JAEPERI ILiquot F51ALd J JAE UWE Centrgl Tendencv The term central tendency refers to the middle value sometime a typical value of the data Measures of central tendency are measures of the location of the middle or the center of a distribution The Mean is the most commonly used measure of central tendency MEAN Also known as the arithmetic mean the mean is typically What is meant by the word average The mean is perhaps the most common measure of central tendency The mean of a variable is given by the sum of all its valuesthe number of values For example the mean of 4 8 and 9 is 4 8 93 7 Example 58 69 73 67 76 88 91 and 74 8 marks Sum 596 Mean 5968 745 Please note that the mean is affected by extreme values MEDIAN Another typical value is the median To find the median of a number of values first arrange the data in ascending or descending order then locate the middle value If there are odd number of data points then median is the middle values lfthere are even number of data points then median is mean of the two middle values Median is easier to find than the mean and unlike the mean it is not affected by values that are unusually high or low 1 64 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 3 611 141919 21 24 31 9 values In the above data there are 9 values So median is The middle value ie 19 MODE The most common score in a set of scores is called the mode There may be more than one mode or no mode at all 2212032114111220321 The mode or most common value is 1 DescriptionExplanation Advantages Disadvantages ihe sum of all the results Quick and easy to May not be representative of the Mean 1nc1uded 1n the sample d1v1ded calculate whole sample by the number of observatlons More tedious to calculate than Median the middle value of all the Takes all numbers into the other two numbers in the sample account equally Can be affected by a few very large or very small numbers the most frequently observed value of the measurements in the sample There can be more than one mode or no mode 0 for an even number of Fairly easy to calculate Mode values the median is Half of the sample Tedious to nd for a large the average of the middle two values 0 for an odd number of values the median is the middle of the all of the values normally lies above the median sample which is not in order 165 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 ORGANISING DATA There are many different ways of organizing data Orqanizinq Numerical Data Grganizing Numariaal Data I llumariml Data drama2252121 at 3821 Fratpamy li rilmliara lnlllttiaa i rilniiana I 2121 21 2321 2130223341 2 Milt 3 I133 4 1 lTalIla I Numerical data can be organized in any of the following forms o The Ordered Array and Stemleaf Display 0 Tabulating and Graphing Numerical Data 0 Frequency Distributions Tables Histograms Polygons 0 Cumulative Distributions Tables the Ogive ta Stern and Leaf Display A stem and leaf display also called a stem and leaf plot is particularly useful when the data is not too numerous NUMERICAL Data in Raw farm aa aallaatatl at as 224 a 21 2 an an 32 3 Mia rtlararl f ra IquotIquotI Sm laat ta Largaat 2124 24 5 2 a a 32 an 41 Stern LE af g V N V 14451 Stem an Laaf 3 quot3 diaplay 4 1 Copyright Virtual University of Pakistan VU 166 Business Mathematics amp Statistics MTH 302 VU Since2120110gtlt21 This is represented in the plot as a stem of 2 and a leaf of 1 The digit at the tenth place is taken as stem and the digit at units place is taken as leaf Similarly 26 is represented in the plot as a stem of 2 and a leaf of 6 Remember a stem is displayed once and the leaf can take on the values from 0 to 9 Example Consider Figure 1 It shows the number of touchdown TD passes thrown by each of the 31 teams in the National Football League in the 2000 season 37quot 33 33 32 29 EB 28 23 22 22 22 21 2121ED2C1919181E1E1E1Ei15 1414141212EEi Figure 1 Number of touchdown passes A stem and leaf display of the data is shown in the Table 1 below The left portion of the table contains the stems They are the numbers 3 2 l and 0 arranged as a column to the left of the bars As in 34 3 is stem and 4 is leaf In 16 l is stem and 6 is leaf Stem and leaf display showing the number of passing touchdowns 32337 2001112223889 l2244456888899 069 Table 1 To make this clear let us examine this Table l more closely In the top row the four leaves to the right of stem 3 are 2 3 3 and 7 Combined with the stem these leaves represent the numbers 32 33 33 and 37 which are the numbers of TD passes for the first four teams in the table The next row has a stem of 2 and 12 leaves Together they represent 12 data points We leave it to you to figure out what the third row represents The fourth row has a stem of 0 and two leaves One purpose of a stem and leaf display is to clarify the shape of the distribution You can see many facts about TD passes more easily in Figure 1 than in the Table 1 For example by looking at the stems and the shape of the plot you can tell that most of the teams had between 10 and 29 passing TDs with a few haVing more and a few haVing less The precise numbers of TD passes can be determined by examining the leaves Tabulatinq gnd Grgphinq UnivaLiate Cgteqorical Data There are different ways of organizing univariate categorical data 0 The Summary Table 0 Bar and Pie Charts the Pareto Diagram 167 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Tabulatinq and Graphing Bivariate categorical Data Bivariate categorical data can be organized as o Contingency Tables 0 Side by Side Bar charts III GRAPHICAL EXCELLENCE AND COMMON ERRORS IN PRESENTING DATA It is important that data is organized in a professional manner and graphical excellence is achieved in its presentation High quality and attractive graphs can be used to explain and highlight facts which otherwise may go unnoticed in descriptive presentations That is why all companies in their annual reports use different types of graphs to present data TABULATING NUMERICAL DATA Group data into classes In some cases it is necessary to group the values of the data to summarize the data properly The process is described below Step 1 Sort Raw Data in Ascendinq Order Data 12 13 17 21 24 24 26 27 27 30 32 35 37 38 41 43 44 46 53 58 Step 2 Find Range Range Maximum value Minimum Value Thus Range 58 12 46 Step 3 Select Number of Classes Select the number of classes The classes are usually selected between 5 and 15 In our example let us make 5 classes Step 4 Compute Class width Find the class width by dividing the range by the number of classes and rounding up Be careful of two things a You must round up not off Normally 32 would round to be 3 but in rounding up it becomes 4 b lfthe range divided by the number of classes gives an integer value no remainder then you can either add one to the number of classes or add one to the class width In our example Class width Range 92 Number of classes 5 Round up 92 to 10 Step 5 Determine Class Boundaries limits Pick a suitable starting point less than or equal to the minimum value You will be able to cover quotthe class width times the number of classesquot values Your starting point is the lower limit of the first class Continue to add the class width to this lower limit to get the lower limit of other classes 1 68 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In this example if we start with 10 we will cover 10 x 5 50 values which is close to our range So let 10 be the lower limit of the first class Continue to add 10 to this lower limit to get the lower limit of other classes 10 201010 302010 403010 504010 To find the upper limit of the first class subtract one from the lower limit of the second class Then continue to add the class width to this upper limit to find the rest of the upper limits Upper limit of first class is 20 1 19 Rest upper limits are 29 1910 39 2910 49 3910 Step 6 Compute Class Midpoints Class Midpoint Lower limit Upper limit 2 First midpoint is 10192 145 Other midpoints are 20292 245 30392 345 40492 445 50592 545 Depending on what you39re trying to accomplish it may not be necessary to find the midpoint Step 7 Compute Class Intervals First class Lower limit is 10 Higher limit is 19 We can write first class interval as 10 to 19 or 10 19 or 10 but under 20quot In 10 but under 20quot a value greater than 195 will be treated as above 20 Similarly other 4 class intervals are 20 29 30 39 40 49 50 59 Important points to remember 1 There should be between 5 and 15 classes 2 Choose an odd number as a class width if you want to have classes midpoints as an integer instead of decimals 3 The classes must be mutually exclusive This means that no data value can fall into two different classes 4 The classes must be all inclusive or exhaustive This means that all data values must be included 5 The classes must be continuous There should be no gaps in a frequency distribution Classes that have no values in them must be included unless it39s the first or last class es that could be dropped 6 The classes must be equal in width The exception here is the first or last class It is possible to have a quotbelow as a first class or and abovequot as a last class Frequency Distribution Count Observations amp Assiqn to Class lnterv Looking through the data shows that there are three values between 10 and 19 Hence frequency is 3 Similarly frequency of other class intervals can be found as follows 10 19 3 169 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 202916 30 39 5 40 49 4 50 59 2 Total frequency 3 6 5 4 2 20 Relative frequency Relative Frequency of a class Frequency of the class interval Total Frequency There are 3 observations in first class interval 10 19 The relative frequency is 320 015 Similarly relative frequency for other class intervals are calculated Percent Relative Frequency If we multiply 015 by 100 then the Relative Frequency 15 is obtained iFFIE HENevuI39smaunuus 1313112124 e26EtEtt tttteemme 5353 Class Frequerwy we 3993 we PM FratHt 113mm EH 3 IE 15 hut 3 ii in iiquot Eillhrtt mide1 till 5 35 5 i hut made 5E 1 3923 If hmtm39uier Eff I l H Tnhl II 1 llll Cumulative Frequency If we add frequency of the second interval to the frequency of the first interval then the cumulative frequency for the second interval is obtained Cumulative frequency of each class interval is calculated below 10 19 3 20 29 3 6 9 30 3936514 40 49365418 50 593654220 Percent cumulative relative frequency This can be calculated same as cumulative frequency except now percent relative frequency for each class interval is considered The percent cumulative relative frequency of the last class interval is 100 as all observations have been added Percent cumulative relative frequency of each class interval is calculated below 10 19 15 20 29153045 30 40 153025 70 40 501530252090 50 60 1530252010100 1 70 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 25 STATISTICAL REPRESENTATION MEASURES OF CENTRAL TENDENCY PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 24 0 Statistical Representation 0 Measures of Central Tendency Part 2 GRAPHING NUMERICAL DATA THE HISTOGRAM Histogram is a bar graph of a frequency distribution in which the widths of the bars are proportional to the classes into which the variable has been divided and the heights of the bars are proportional to the class frequencies Histogram of the example of frequency distribution discussed at the end of lecture 24 is given below GRAPHING NUMERIEAL DATA THE HISTDERAM 313 quot1 I dEI39E 3912 5 a 3quot 34a 2 4 Eli 3 23 3 321 35 3t 33 41 43 443 rm a i I39Iazr m i E39 I I I I I I I I Initlhzriltg MEASURES OF CENTRAL TENDENCY Measures of central tendency can be summarized as under 1 Arithmetic Mean a Arithmetic mean for discrete data i Sample Mean ii Population Mean b Arithmetic Mean for grouped data 2 Geometric Mean 3 Harmonic Mean 4 Weighted Mean 5 Truncated Mean or Trimmed Mean 6 Winsorized Mean 7 Median l 7 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU a Median for grouped data b Median for discrete data 8 Mode a Mode for grouped data b Mode for discrete data 9 Midrange 10 Midhinge As you see it is a long list However if you look closely you will find that the main measures are Arithmetic Mean Median Mode All the above measures are used in different situations to understand the behavior of data for decision making It may be interesting to know the average median or mode salary in an organization before the company decides to increase the salary level Comparisons with other companies are also important The above measures provide a useful summary measure to consolidate large volumes of data Without such summaries it is not possible to compare large selections of data EXCEL has a number of useful functions for calculating different measures of central tendency Some of these are explained below You are encouraged to go through EXCEL Help file for detailed descriptions of different functions For selected functions the help file has been included in the handouts The examples are also from the help files AVERAGE Returns the average arithmetic mean of the arguments Syntax AVERAGEnumber1number2 Number1 number2 are 1 to 30 numeric arguments for which you want the average Remarks 0 The arguments must either be numbers or be names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included Example An example of AVERAGE is shown below Data was entered in cells A4 to A8 The formula was AVERAGEA4A8 The 11 is shown in cell A10 1 72 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Buul Elle Edit ElevJ insert Fgrrnat Innls Eata indnw Help Fading3 F39DF H at n v E I 3 v lt3 E E El SLIP391 j x J AVEHAGEMampEI W E L D E F 3 2 3 AVERAGEmumbeH numberE 4 1 I 5 F a Q r 2 2 AUERAG EA lA3 1 1 AVERAGEA Calculates the average arithmetic mean of the values in the list of arguments In addition to numbers text and logical values such as TRUE and FALSE are included in the calculation Syntax AVERAGEAvaIue1value2 Value1 value2 are 1 to 30 cells ranges of cells or values for which you want the average Remarks 0 The arguments must be numbers names arrays or references 0 Array or reference arguments that contain text evaluate as 0 zero Empty text 39quot39 evaluates as 0 zero If the calculation must not include text values in the average use the AVERAGE function 0 Arguments that contain TRUE evaluate as 1 arguments that contain FALSE evaluate as 0 zero Example A 1 Data 2 10 3 7 4 9 5 2 1 73 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 6 Not available 7 Formula Description Result Average of the numbers above and the text AVERAGEAA2A6 quotNot Availablequot The cell with the text quotNot availablequot is used in the calculation 56 AVERAGEAA2A5A7 Average of the numbers above and the empty cell 7 MEDIAN Returns the median of the given numbers The median is the number in the middle of a set of numbers that is half the numbers have values that are greater than the median and half have values that are less Syntax MEDIANnumber1number2 Number1 number2 are 1 to 30 numbers for which you want the median Remarks 0 The arguments should be either numbers or names arrays or references that contain numbers Microsoft Excel examines all the numbers in each reference or array argument o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If there is an even number of numbers in the set then MEDIAN calculates the average of the two numbers in the middle See the second formula in the example Example The numbers are entered in cells A14 to A19 In the first formula MEDAN12345 the actual values are specified The median as you see is 3 in the middle In the next formula MEDANA14A19 the entire series was specified There is no middle value in the middle Therefore the average of the two values 3 and 4 in the middle was used as the median 35 Copyright Virtual University of Pakistan 174 Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Buuk1 Elle Edit ElevJ insert Fgrrnat IDDIS Eata irtjaw elp ACIDQE F39DF Eggn nvz ml fIDTB l g EEE E15 3 A E L D E F G 12 MEDIANmumber numberz n 13 Ir1 1 15 2 TE 3 I I IE IEI MEDIANH 2 1 5 5 MEDIANA14A1S El 21 22 MODE Returns the most frequently occurring or repetitive value in an array or range of data Like MEDIAN MODE is a location measure Syntax MODEnumber1number2 Number1 number2 are 1 to 30 arguments for which you want to calculate the mode You can also use a single array or a reference to an array instead of arguments separated by commas Remarks 0 The arguments should be numbers names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If the data set contains no duplicate data points MODE returns the NA error value In a set of values the mode is the most frequently occurring value the median o is the middle value and the mean is the average value No single measure of central tendency provides a complete picture of the data Suppose data is clustered in three areas half around a single low value and half around two large values Both AVERAGE and MEDIAN may return a value in the relatively empty middle and MODE may return the dominant low value Example The data was entered in cells A27 to A32 The formula was MODEA27A32 The answer 4 is the most frequently occurring value 1 75 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU crusuft Excel Buuk1 Elle Edit ew insert Fgrrnal Innls Eata irtjaw Help adage F39DF DEE nvE lf1oTBEamp EEE I 535 j 24 A C D E F G 25 MDEfnumber39ir iLm berz 25 2 55 28 29 3 31 32 33 1 MDEA2A32 34 COUNT FUNCTION Counts the number of cells that contain numbers and also numbers within the list of arguments Use COUNT to get the number of entries in a number field that39s in a range or array of numbers Syntax COUNTvalue1value2 Value1 value2 are 1 to 30 arguments that can contain or refer to a variety of different types of data but only numbers are counted Remarks 0 Arguments that are numbers dates or text representations of numbers are counted arguments that are error values or text that cannot be translated into numbers are ignored o If an argument is an array or reference only numbers in that array or reference are counted Empty cells logical values text or error values in the array or reference are ignored If you need to count logical values text or error values use the COUNTA function Example A 1 Data 2 Sales 3 1282008 4 5 19 6 2224 7 TRUE 176 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 8 DIVO Formula COUNTA2A8 COUNTA5A8 COUNTA2A82 39 FREQUENCY Description Result Counts the number of cells that contain numbers in the list above 3 Counts the number of cells that contain numbers in the last 4 rows of the list 2 Counts the number of cells that contain numbers in the list and the value 2 4 Calculates how often values occur within a range of values and then returns a vertical array of numbers For example use FREQUENCY to count the number of test scores that fall within ranges of scores Remember FREQUENCY returns an array it must be entered as an array formula Syntax FR EQUENCYdataarray binsarray Dataarray is an array of or reference to a set of values for which you want to count frequencies f dataarray contains no values FREQUENCY returns an array of zeros Binsarray is an array of or reference to intervals into which you want to group the values in dataarray f binsarray contains no values FREQUENCY returns the number of elements in dataarray Remarks 0 FREQUENCY is entered as an array formula after you select a range of adjacent cells into which you want the returned distribution to appear 0 The number of elements in the returned array is one more than the number of elements in binsarray The extra element in the returned array returns the count of any values above the highest interval For example when counting three ranges of values intervals that are entered into three cells be sure to enter FREQUENCY into four cells for the results The extra cell returns the number of values in dataarray that are greater than the third interval value 0 FREQUENCY ignores blank cells and text 0 Formulas that return arrays must be entered as array formulas A B Scores Bins 79 70 85 79 78 89 85 50 81 95 88 ACOW IQUIhWNl 97 177 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Formula Description Result FREQUENCYA2A1 Number of scores 0 3235 ess than or equal to 39 7o 1 Number of scores in the bin 7179 2 Number of scores in the bin 8089 4 Number of scores greater than or equal to 90 2 Example Note The formula in the example must be entered as an array formula After copying the example to a blank worksheet select the range A13A16 starting with the formula cell Press F2 and then press CTRLSHFTENTER If the formula is not entered as an array formula the single result is 1 ARITHMETIC MEAN GROUPED DATA Below is an example of calculating arithmetic mean of grouped data Here the marks Classes and frequency are given The class marks are the class mid points calculated as average of lower and higher limits For example the average of 20 and 24 is 22 The frequency f is multiplied by the class mark to obtain the total number In first row the value of fx is 1 x 22 22 The sum of all fx is 1950 The total number of observations is 50 Hence the arithmetic mean is 195050 39 Marks Frequency Class Marks fX 2024 1 22 22 2529 4 27 108 3034 8 32 256 3539 1 1 37 407 4044 15 42 630 4549 9 47 423 5054 2 52 104 TOTAL 50 1950 n 50 SumfX 1950 Mean 195050 39 Marks EXCEL Calculation The above calculation would be common in business life Let us see how we can do it using EXCEL The basic data of lower limits is entered in cell range A54A60 The data of higher limit is entered in cells B54B60 Frequency is given in cell range D54D60 Class midpoints or class marks were calculated in cells F54F60 ln cell F54 the formula A54B542 was used to calculate the class mark This formula was copied in other cells F55 to F60 The value of fx was calculated in cell H54 using the formula D54F54 This formula was copied to other cells H55 to H60 Total frequency was calculated in cell D61 using the formula SUMD54D60 Sum of fx was calculated in cell H61 using the formula SUMH54H60 Mean was calculated in cell H62 using the formula ROUNDH61D610 178 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsn Excel Lecturej ile Edit Eiew insert Fgrmat Innls gate indnw elp ndings F39DF Dusss 45 23 33444 TEUvBIHEEEE h v s s s H53 3 4 E r U E F G L I J K 53 Marks Frequencym Class Markm fit 54 20 24 1 22 22 D545F54 55 25 2B 4 27 108 55 30 8 132 256 5 35 39 1 1 37 40 as 40 44 15 42 5130 59 45 4B 9 47 4213 an 50 54 2 52 104 E1 TTAL 50 1950 SUMH54HEU j Mean 33 HE1IDE1 53 179 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 26 STATISTICAL REPRESENTATION MEASURES OF DISPERSION AND SKEWNESS PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 25 0 Statistical Representation 0 Measures of Dispersion and Skewness FREQUENCYEXAMPLE FREQUENCY Function calculates how often values occur within a range of values and then returns a vertical array of numbers For details see handout for lecture 25 The syntax is FREQUENCYdataarraybinsarray Microsoft Excel Frequency Elle Edit ew insert Fgrmat Innis gate window Help all5 1SST1317 Ti 139 539 X D i 3139 ci lb r SE39ESiEi i mmtr ArieI mm are E EEJEEEE Javen DH r A E C D I E F G H J H LT i FREUENCYidetearraybineerrey 2 Scere Bins 3 70 Step 1 Select the range A13 16 starting 4 85 with the f rmula cell 5 89 Step 2 Press F2 e 85 Step Fees CTRLSHIFTENTER i 50 1 FeEUENCYn3n11B3BE a 31 2 g 95 4 m 33 2 H 9 The data was entered in cells A3 to A11 The Bins array which gives the limits 70 79 and 89 were entered in cells B3 to B5 Select cells BC to 3010 one more than the limits Type the formula FREQUENCYA3A1B3B5 Then CTRLShiftEnter were pressed to indicate that we are entering an array formula The result is given in cells B7 to 810 We can interpret the result as the frequency oflt Less than or equal to 70 is 1 71 to 79 is 2 80 to 88 is 4 89 and above is 2 Application of FREQUENCY function in Frequency Distribution Consider the example at the end of lecture 24 1 80 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 12 mi 31 act Class Franyrenrr we 3996 we Elm FratHt RelatesHirer El 3 15 15 ifth If ii EH Elihu IdleI ill 5 15 25 43111311 5H 4 Elam nder Eff I ll 1 Tab El 1 llll Letus find Frequency distribution using Excel function FREQUENCY Microsoft Excel Bauld Elle g agile l ile 39 10 1 E I u I E24 v a a e i D E Data Class 12 Lower limit Upper Limit Edit Eiew insert Fgrmal Inels gate indew elp IIIIII 5 in Frequency M J J I Jquot LCI LEI DMbU Im 2 Total M C Write data in column A In column C and D write the class s lower and upper limits Here we will take class s upper limit as bins Select the cells E3 to E8 under Frequency heading one cell more than the number of classes Type FREQUENCYA2A21D3D7 Press CtrlShiftEnter Your Frequency column will get filled with above mentioned values FREQUENCY POLYGONS Frequency polygon is a line graph obtained from a frequency distribution by joining with straight lines points whose abscissa are the midpoints of successive class intervals and whose ordinates are the corresponding class frequencies 1 8 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GRAPHING NUMERIEAL DATA THE FREQUENle PDLVGDN Data in arderad arrayI 1313133133 33 33 3quot 3quot 33 33 35 37quot 33 31 3333335353 Elam luitlminta CUMULATIVE FREQUENCY Relative frequency can be converted into cumulative frequency by adding the current frequency to the previous total In the slide below the first interval has the relative as well as cumulative frequency as 3 In the next interval the relative frequency was 6 it was added to the previous value to arrive at 9 as cumulative frequency for interval 20 to 30 What it really means is that 9 values are equal to or less than 30 Similarly the other cumulative frequencies were calculated The total cumulative frequency 20 is the total number of observations Percent Cumulative frequency is calculated by dividing the cumulative frequency by the total number of observations and multiplying by 100 For the first interval the cumulative frequency is 320100 15 Similarly other values were calculated TAULATING NUMERIC L AT l EUMLIL tTNE FREQUENCY ata l l El E array 1213112123232321213332353133313333355333 W EEillumulatiue ch35 ime iFlplrI39tiun Frpqllpnqr l hutm 39m 3 15 mhutunl 39iil 9 5 Mllutm 39 l 14 TI Whutunla39m 13 H mhl l 39 II llll CUMULATIVE POLYGONOGIVE From the cumulative relative frequency polygon that starts from the first limit not mid point as in the case of relative frequency polygons can be drawn Such a polygon is called Ogive The maximum value in an Ogive is always 100 Ogives are determining cumulative frequencies at different values not limits 1 82 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU GRAPHING NUMERICAL ATA THE EUMULATWE a PnLveeHl Data in ordered arrayr 121317 21ELLE 128212303235333 1J3M5353 iiglw Class Elcunclaries NEE I39i39i39iu39pnints TABULATING AND GRAPHING UNIVARIATQATA Univariate data one variable can be tabulated in Summary form or in graphical form Three types of charts namely Bar Charts Pie Charts or Pareto Diagrams can be prepared TALlLATl AND ERAF HIN G EATEEDRIEAL DATA UNIVARIATE ATA Emegoricnl IIItutu I Talmliiing Data Gml li Data TI39IEI Stl39lrrrtll39jlr Tallies Pie Elmrte Diagram SUMMARY TABLE A summary table is built specifically from detailed data It contains summaries of the data and is used to speed up analysis A typical Summary Table for an investor s portfolio is given in the slide The variables such as stocks etc are the categories The table shows the amount and percentage Investment Amountin Percentage Category thousand Rs Stocks 465 4227 Bonds 32 2909 Cash 155 1409 Deposit Savings 16 1455 TOTAL 110 100 1 83 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BAR CHART The data of Investor s portfolio can be shown in the form of Bar Chart as shown below This chart was prepared using EXCEL Chart Wizard The Wizard makes it very simple to prepare such graphs You must practice with the Chart Wizard to prepare different types of graphs CHART F 119 Ml II39III quE RPS RTF Ll I39l quotilitIF39i F39IZIFTIZIIIIZI Gamngr II I B on E III Iii III 213 Elli 41339 SD mpzuntln PIE CHARTS Pie Charts are very useful charts to show percentage distribution These charts are made with the help of Chart Wizard You may notice how Stocks and bonds stand out PIE CHART fIF DR MI IH IuI E E35 RTF Ll An39hz rlllt Iwe l in H5 Savings 15 Steel5 III 1241 M39H P 39 l t fl39 r ll l l SE5 to lienaan 1 H t PARETO DIAGRAMS A Pareto diagram is a cumulative distribution with the first value as first relative frequency in this case 42 The point is drawn in the middle of bar for the first category stocks Next the category Bonds was added The total is 71 Next the savings 15 were added to 71 to obtain cumulative frequency 86 Adding the 14 for CD gives 100 Thus the Pareto diagram gives both relative and cumulative frequency 1 84 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU HAGRAM PA RETC Axis for bar chart shows EH invested in each category Axis for line Euth shears eLrnIJ atiue EH invested CONTINGENCY TABLES Another form of presentation of data is the contingency table An example is shown in the slide below The table shows a comparison of three investors along with their combined total investment TALILATING CATEGDRIEAL DATA WARIATE DATA en ngeney Table Ill39u39i l39l39 39lt in mlmmls If mum Investment huestzr P I39mester El twat3r I ctal Category Stacks 35 55 25 123 Bands 32 am 13 35 ED 155 EIII 135 13 Savings 1E E Tquot 51 113 11D la 1quot E 321 SIDE BY SIDE CHARTS The same investor data can be shown in the form of side by side charts where different colours were used to differentiate the investors This graph is a complete representation of the contingency table GRAPHING CATEGDRJEAL DATA EIVARIATE DATA SIIIEIy ERIE El rt Dem paring I1 H latrs 5 GEOMETRIC MEAN 1 85 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Geometric mean is defined as the root of product of individual values Typical syntax is as under Gx1x2x3xnquot1n Example Find GM of 130 140 160 GM 130140160 A 13 1428 HARMONIC MEAN Harmonic mean is defines as under HM n n 1x11x21xn Sum1xi Example Find HM of 10 8 6 HM 3 766 1101816 QUARTILES Quartiles divide data into 4 equal parts Syntax 1st Quartile Q1 n14 2nd Quartile Q2 2n14 3rd Quartile Q3 3n14 Grouped data Qi ith Quartile l hfSum f4i of l lower boundary h width of CI of cumulative frequency DECILES Deciles divide data into 10 equal parts Syntax 1st Decile D1n110 2nd Decile D2 2n110 9th Deciled D9 9n110 Grouped data Qi ith Decile i129 l hfSum f10i of l lower boundary h width of CI of cumulative frequency PERCENTILES Percentiles divide data into 100 equal parts Syntax 1st Percentile P1n1100 2nd Decile D2 2n1100 99th Deciled D9 99n1100 Grouped data Qi ith Decilei129 l hfSum f100i of l lower boundary h width of CI of cumulative frequency 186 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Symmetrical Distribution mean median mode Positively Skewed Distribution Tilted to left mean gt median gt mode Negatively Skewed Distribution mode lt median lt mean Tilted to right EMPIRICAL RELATIONSHIPS Moderately Skewed and Unimodal Distribution Mean Mode 3Mean Median Example mode 15 mean 18 median Median 13mode 2 mean 1315 218 15363 513 17 A trimmed or truncated mean is a measure of central tendency and is one type of modified mean In this first we sort the data Then according to the problem discard an equal number of data at both ends Most often 25 percent of the ends are discarded That is values below first quartile and above third quartile are removed Mean of the remaining data is called trimmed mean or truncated mean WINSORIZED MEAN It involves the calculation of the mean after replacing given parts of a data at the high and low end with the most extreme remaining values Most often 25 percent of the ends are replaced That is values below first quartile and above third quartile are replaced Example Find trimmed and winsorized mean 91 92 93 92 92 99 Arrange the data in ascending order 91 92 92 92 93 99 Position of Q1 614 175 Q1 2nd value approximately 92 Position of Q3 3614 525 Q3 5th value approximately 93 Trimmed Mean 92 92 92 93 4 9225 Winsorized Mean 92 92 92 92 93 93 6 9233 DISPERSION OF DATA Definition The degree to which numerical data tend to spread about an average is called the dispersion of data TYPES OF MEASURES OF DISPERSION Absolute measures Relative measures coefficients DISPERSION OF DATA Types Of Absolute Measures Range l 87 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Quartile Deviation Mean Deviation Standard Deviation or Variance Types Of Relative Measures Coefficient of Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation Copyright Virtual University of Pakistan VU 188 Business Mathematics amp Statistics MTH 302 VU LECTURE 27 STATISTICAL REPRESENTATION MEASURES OF DISPERSION AND SKEWNESS PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 26 0 Measures of Dispersion and Skewness MEASURES OF CENTRAL TENDENCY VARIATION AND SHAPE FOR A SAMPLE There are many different measures of central tendency as discussed in the last lecture handout These include Mean Median Mode Midrange Quartiles Midhinge Range lnterquartile Range Variance Standard Deviation Coefficient of Variation Rightskewed Leftskewed Symmetrical Distributions Measures of Central Tendency Variation and ShapeExploratory Data Analysis FiveNumber Summary BoxandWhisker Plot Proper Descriptive Summarisation Exploring Ethical Issues Coefficient of Correlation MEANS The most common measure of central tendency is the mean The slide below shows the Mean Arithmetic Median Mode and Geometric mean Another mean not shown is the Harmonic mean Each of these has its own significance and application The mean is the arithmetic mean and represents the overall average The median divides data in two equal parts Mode is the most common value Geometric mean is used in compounding such as investments that are accumulated over a period of time Harmonic mean is the mean of inverse values Each has its own utility The slide shows the formulas for mean and geometric mean Measures nf Central Tendeney It entral TEI IdEnW I 2 33114135 Mean iquot 24339 f f39 KIIII J quotT lHarmnnit Mean l 189 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE MEAN The formula for Arithmetic Mean is given in the slide It is the sum of all values divided by the number In the case of mean of a sample the number n is the total sample size When the sample data is to be used for estimating the value of mean then the number is reduced by 1 to improve the estimate In reality this will be a slight overestimation of the population mean This is done to avoid errors in estimation based on sample data that may not be truly represented of the population The Mean Arithmetie Average The Arithmetic Average ef data trainee ETfn m E1I2 HI fu 71 H Sample Hean SEmPIE Size F 1 3 IN E1f2 II f 1 N H Pcleaticn Mean P ma n Size EXTREME VALUES An important point to remember is that arithmetic mean is affected by extreme values In the following slide mean of 5 values 1 3 5 7 and 9 is 5 In the second case where the data values are 1 3 6 7 and 14 the value 14 is an outlier as it is considerably different from the other values In this case the mean is 6 in other words the mean increased by 1 or about 20 due to the outlier While preparing data for mean it is important to spot and eliminate outlier values The Me an The Meat Centnten Meaeure ef Central Tentleneyr Affeeteel by Eatren39ie valuee utliere 24 3 h 0E4aaam I214 39 mm 5 I Mm 190 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE MEDIAN The Median is derived after ordering the array in ascending order Ifthe number of The Median lmpertant Meaeure ef Central Tendeney In an erdered array the median ie the middle number If n ie add the median ie the middle num er If Fl ie even the median ie the average at the 2 middle numbera observations is odd it is the middle value otherwise it is the the average of the two middle values It is not affected by extreme values The Median Net Affected by Extreme Values 392 If Fee llillild lie l6EFeellil l IIquot 392 I4e39e 5 tamer THE MODE The mode is the value that occurs most frequently In the example shown on the slide 8 is the most frequently occurring value Hence the mode is 8 Mode is also not affected by extreme values 191 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The Mede A Meeeure ef Centre Ter39rrJer ieyr Value that eeure Meet then Net Affected by Extreme Values E gi gee li dEEFEEEIIIII I213 IWIE B An important point about Mode is that there may not be a Mode at all no value is occurring frequently There may be more than one mode The mode can be used for numerical or categorical data The slide shows two examples where there is no mode or there are two modes The Mede There May Net lee e Mede There lil39leyr lee Several Medee Lleed fer Either Numeri eel er Eeteerieel eta eeeeeee g g 3931 3 4139 393 l39lelutmle Tmhlerl RANGE Another measure of dispersion of data is the Range It is the difference between the largest and smallest value The slides show examples where the value of range was calculated 1 92 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU ZEIU N C F ATA Range H Largest Smallest Iul alue Example Fintl range 31 13 43 13 Elquot 32quot 1E 33 33 3 24 El Largest value Elnalleet 13 Range 13 12 31 The Range Meaaure efjtifariatien Etifferenee etween Largeat Smalleat Deeeraatiena Elan 93 Largeat Elnalleat Value Igneree Haw Data Are Dietriuteel Range12T5 RargelET ecuJean 139101111 III 5 III 0 1391111112 MIDRANGE Midrange is the average of smallest and largest value In other words it is half of a range Midrange is affected by extreme values as it is based on smallest and largest values Mid range Meaaure ef Central Teniler39ieisr Aeerae ef 3m alleat and Largeat Ubaereatian I z f I Midrange w 193 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Midrange ffeeted by Extreme value 3 ESSJ J I2Ia4ua rl2l l39l Mtlmlm 5 thngaa 3 QUARTILES Quartiles are not exclusively measures of central tendency However they are useful for dividing the data in 4 equal parts each containing 25 of the data So there are three quartiles 25 data falls below first quartile 50 below second and 75 below third quartile Quartiles t a ITIEHEUI39E Elf E tl l tendency Split l dEl E data i t 4 HUEWEFE 25 25 25 25 I 2 313 Position of ith quartile 39 n 1 4 Where i 1 2 3 and n is the number of data points To understand the procedure of finding the value of each quartile let us consider an example Example Find first second and third quartile of the following data 112217161221161318 Arrange the data in ascending order 11 12 13 16 16 17 1821 22 Here n number of data points 9 Position of first quartile Q1 M 25 4 Since we get position of first quartile as decimal fraction so we proceed as follows 01 2nd value 05 x 3rd value 2nd value 12 05 x 13 12 12 05 x 1 125 Position of second quartile Q2 2 x 9 1 5 4 1 94 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU So 02 5th value 16 Position of third quartile Q3 3 x 9 1 75 4 So 03 7th value 05 x 8th value 7th value 18 05 x 21 18 1805x3 195 MIDHINGE Midhinge is the average of the first and third quartiles Midhinge Q1 Q3 2 QUARTILE DEVIATION Quartile Deviation is the average of 1st and 3rd Quartile QD Q3 Q1l 2 Example Find QD of the following data 14 10 17 5 9 20 8 24 22 13 Here number of data points n 10 Position of Q1 n14 1014 275 So Q1 2nd value 075 x 3rd value 2nd value 8075x98 8 075 x1 875 Position of Q3 3 x 10 1 4 3 x 275 825 So Q3 8th value 0259th value 8th value 20 025 x 22 20 2050 QD 2050 875 5875 2 SHAPE OF DISTRIBUTION 1 Symmetrical Distribution 2 Asymmetrical Distribution a Rightskewed or positivelyskewed distribution b Leftskewed or negativelyskewed distribution We can find the shape of distribution using 5number summary 5NUMBER SUMMARY 5 number summary is o Smallest value 1St Quartile Q1 MedianQ2 3rd Quartile 03 Largest value 195 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 90X AND WHISKER PLOT Box and whisker plot shows 5number summary Eaplaratnry ata Analysis anatantlwhialtar Graphi al iSFtlay at data uaing numlaar auntrnaryr Madian i HEWE39EH 1 EEG i 4 E 3 ll 12 The plot gives a good idea about the shape of the distribution as detailed below Box and whisker plot for symmetrical leftskewed and rightskewed distribution are shown below iatrilautian Shape Er Baaeand whiakar Plata LeftEltawad Eymmetric ightEltawad q Matlim Malian Malian 4 l lIl I 1 Symmetrical Distribution Data is perfectly symmetrical if 0 Distance from Q1 to Median Distance from Median to Q3 0 Distance from Xsmauest to Q1 Distance from Q3 to Xlargest That is Median Midhinge Midrange 2 AsymmetricalDistribution a Rightskewed distribution Distance from Xlargest to Q3 greatly exceeds distance from Q1 to Xsmauest That is Median lt Midhinge lt Midrange b Leftskewed distribution Distance from Q1 to Xsmauest greatly exceeds distance from Xlargest to Q3 1 96 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU That is Median gt Midhinge gt Midrange Example Suppose there are nine homes valued at Rs 1500000 Rs 1400000 Rsl600000 Rs1500000 Rsl600000 Rsl700000 Rsl600000 Rs1500000 and Rsl600000 in a new area There is one small empty lot in the area and someone builds small home with a valuation of Rs 200000 Find either the frequency polygon of these values is negatively skewed or positively skewed Following sample represents annual costs in 000 Rs for attending 10 conferences 130 145 149 152 152 154 156 162 17 231 Find 5 number summary and shape of the distribution Position of Q1 10 1 4 275 Q1 145 075 x 149 145 148 Position on3 3 1o 14 825 03 162 025 x 17 62 164 Median 152 154 2 153 Thus 5 number summary is o Smallest value 130 1St Quartile Q1 148 MedianQ2 153 3rd Quartile Q3 164 Largest value 231 Midrange Largest value Smallest value 231 130 1805 2 2 Midhinge Q1 Q3 148 164 156 2 2 Find the relationship between median midhinge and midrange Median lt Midhinge lt Midrange Thus the shape of distribution is right skewed SUMMARY MEASURES The slide shows summary of measures of central tendency and variation In variation there are range Interquartile range standard deviation variance and coefficient of variation The measures of central tendency have been discussed already 197 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Summary Maaurs I lll39l39ll39l39l l39jf Measures I ICentIril Tattlency I Varia un I tmtlnrtl El I Wi ti ill Interquartile Marian Im39 ml l I II Range C ef ciant c L mi 0539 I IV ri n I if 139Ii39rifil39l tllliliIFI MEASURES OF VARIATION In measures of variation there are the sample and population standards deviation and variance the most important measures The coefficient of variation is the ratio of standard deviation to the mean in Masu res Variation iiaria nn lirarime lemma Deviat39nn Coef ciemnf E l ifaria nn ME 395quot W cr 1nn Var39nme 531111 5131113111 IIItEIqIEII39 h Rang I Darth INTERQUARTIE RANGE Interquartile range is the difference between the 1st and 3rd quartile 1 98 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lnte rquertile Range Meeeure ef erietien Alee Knewn ee Mideereed pureed in the Middle ifferer39iee etereen Thir E Firet IZZELIer39tilee Interquertile Renge Datainfh39 ugll ng39 11 1 13 15 15 139 139 11 1 E 393 1re1e55 Net A eeteel by Extreme Veluee 199 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 28 MEASURES OF DISPERSION CORRELATION PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 27 0 Measures of Dispersion 0 Correlation MODULE 6 Module 6 covers the following Correlation Lecture 2829 Line Fitting Lectures 3031 Time Series and Exponential Smoothing Lectures 3233 VARIANCE Variance is the one of the most important measures of dispersion Variance gives the average square of deviations from the mean In the case of the population the Verienee II39I I PEI NIGHT MEESU I39E Elf V311 ti SHOWS V ti b l l t tI39IE MEEI I Fer the Population a 23933 1 N Fer the Sample 5 Elia 5fl 11 1 Fer the Pepuletien uee III lF er the Sahele uee n if in the tlenemintier fl inithe tIenerniintgte31 Sum of square of deviations is divided by N the number of values in the population In the case of variance for the sample the number of observations less 1 is used STANDARD DEVIATION Standard deviation is the most important and widely used measure of dispersion The square root of square of deviations divided by the number of values for the population and number of observations less 1 gives the standard deviation 200 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example Standard emetlen Meet lmpertaht Meeeure ef Varietieh Ehewe 1I39l39r erietieh eheut the Mean Sam e Ll i39l ef n1 eeeurem em 35 the 5 hSEW ti E F X we rr Fur the Semi1e FIJIquot Fepuletieh F ch F llulatien nee H Fertiliei iaihhleiiiee hif in the lenemin m 7 in the hequotmni m rliy Let us do this for a simple dataset shown below The Number of Fatalities in Motorway Accidents in one Week Number of fatalities Day X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28 The arithmetic mean number of fatalities per day is izgz n 7 4 Taking the deviations of the Xvalues from their mean and then squaring these deviations we obtain X 4 6 2 0 3 5 8 20 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Hence 2X 02 42 is now positive and this positive value has been achieved without bending the rules of mathematics Averaging these squared deviations the variance is given by Variance The variance is frequently employed in statistical work but it should be noted that the gure achieved is in squared units of measurement In the example that we have just considered the variance has come out to be 6 squared fatalities which does not seem to make much sense In order to obtain an answer which is in the original unit of measurement we take the positive square root of the variance The result is known as the standard deviation Standard Deviation Lye W n l 265 fatalities The formulae that we have just discussed are valid in case of raw data In case of grouped data ie a frequency distribution each squared deviation round the mean must be multiplied by the appropriate frequency figure ie 202 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU COMPARLNG STANDARD DEVIATIONS In many situations it becomes necessary to calculate population standard deviation SD on the basis of SD of the sample where n1 is used for Camparing Standard amatlana Data ID 139 11 15 1397 18 18 1 143 Meanl 1 TIE dig 3 33535 N tuftind far the atantlartl HWl l ll Ia lard ar far data canairlarad aa a Sample division In the slide the same data is first treated as the sample and the value of SD is 42426 When we treat it as the population the SD is 39686 which is slightly less than the SD for the sample You can see how the sample SD will be overestimated if used for the population COMPARLNG STANDARD DEVIATIONS The slide shows three sets of data A B and C All the three datasets have the same mean 155 but different standard deviations A s3338 B s09258 and C s457 It is clear that SD is an important measure to understand how different sets of data differ from each other Mean and SD together form a complete description of the central tendency of data Camparing Standard H amatlana LEW i a a a E a a a MPH5 II I 13 III If IE IF I I3 2 2 WEE g Men 155 quot I I1 I l3 E l a a 2 2 53953 Mamii g a4 l l iii a 14 rs IE a COEFFICIENT OF VARIATION 203 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Cempering Ceeffieie rit ef Verietien Steelt ill average price laet year Re 5131 Standard deuiatien Re 5 my 1DWD age price Iaet year Re tee tandem tlevietien RS 5 lEeel fieierit ef vana en Steel t tZ lf 113 Steelt E int 5 Coefficient of variation CV shows the dispersion of the standard deviation about the mean In the slide you see two stocks A and B with CV10 and 5 respectively This comparison shows that in the case of stock A there was a much greater variation in price with reference to the mean EXAMPLE Suppose that in a particular year the mean weekly earnings of skilled factory workers in one particular country were 1950 with a standard deviation of 4 while for its neighboring country the figures were Rs 75 and Rs 28 respectively From these figures it is not immediately apparent which country has the GREATER VARIABILITY in earnings The coefficient of variation quickly provides the answer COEFFICIENT OF VARIATION For country No 1 4 5 x 100 205 per cent And for country No 2 ExlOO 373 per cent 75 From these calculations it is immediately obvious that the spread of earnings in country No 2 is greater than that in country No l and the reasons for this could then be sought MEAN DEVIATION Other useful measures are Mean Deviation about the Mean and median The mean deviation of a set of data is defined as the arithmetic mean of the deviations measured either from the mean or from the median The symbolic definition of the mean deviation about the mean is 1 MD Zx x for sample n i1 204 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 1 N MD Z x u For apopulatlon N i1 Note that first take the absolute value of difference of data point and the mean and then add those absolute values The absolute value of a number is the number without its sign Example Calculate the mean deviation from mean and median of the following set of examination marks 45 32 37 46 39 36 41 48 and 36 Solution Median 39 Q Mean 2 Mean 3609 40 marks x xi xi xi Median 32 8 8 7 36 4 4 3 36 4 4 3 37 3 3 2 39 1 1 0 41 1 1 2 45 5 5 6 46 6 6 7 48 8 8 9 Sum 330 o 40 39 lel 4 Mean Deviation from Mean 409 44 marks n lel Median Mean DeV1at10n from Med1an 399 43 marks n 205 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In excel ABS function returns the absolute value of a number Syntax ABSnumber Number is the real number of which you want the absolute value For the data organized into a grouped frequency distribution having k classes with midpoints x1 x2 x3 xk and the corresponding frequencies f1 f2 f3 fi 2 fi N the mean deviation about mean for grouped data is given by ixii MD i1 2f Similarly we can define mean deviation about median just by replacing the mean value in formula with the median REGRESSION ANALYSIS The primary objective of regression analysis is the development of a regression model to explain the association between two or more variables in the given population A regression model is the mathematical equation that provides prediction of value of dependent variable based on the known values of one or more independent variables In regression Analysis we shall encounter different types of regression models One of the main functions of regression analysis is determining the simple linear regression equation What are the different Measures of variation in regression and correlation What are the Assumptions of regression and correlation What is Residual analysis How do we make lnferences about the slope How can you estimate predicted values What are the Pitfalls in regression What are the ethical issues An important point in regression analysis is the purpose of the analysis SCATTER DIAGRAM The first step in regression analysis is to plot the values of the dependent and independent variable in the form of a scatter diagram as shown below The form of the scatter of the points indicates whether there is any degree of association between them In the scatter diagram below you can see that there seems to be a fairly distinct correlation between the two variables It appears as if the points were located around a straight line 206 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The Scatter isg rem Flet efell is s pairs El 5391 i i it I Types of Reqression Models There are two types of linear models as shown in the slide below These are positive and negative linear relationships In the positive relationship the value of the dependent variable increases as the value of the independent variable increases In the case of negative linear relationship the value of the dependent variable decreases with increase in the value of independent variable Types ef Regression Medels Pesilise linear R iel iu l39l emtise Linear Heatien in rrI h E El 1 C IIIII CI I I I quot Copyright Virtual University of Pakistan VU 207 Business Mathematics amp Statistics MTH 302 VU LECTURE 29 MEASURES OF DISPERSION CORRELATION PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 28 0 Correlation CORRELATION Correlation is a measure of the strength or the degree of relationship between two RANDOM variables When do we use correlation It will be used when we wish to establish whether there is a degree of association between two variables If this association is established then it makes sense to proceed further with regression analysis Regression analysis determines the constants of the regression You can not make any predictions with results of correlation analysis Predictions are based on regression equations CORRELATION ANALYSIS To analyze the strength of the relationship or covariation between two variables we use correlation analysis Correlation analysis contributes to the understanding of economics behavior aids in locating the critically important variables on which others depends may reveal to the economist the connections by which disturbances spread and suggest to him the path through which stabilizing forces may become effective CORRE LATION WHEI I d USE correlation DIE NEE it t lli I I I Iil TE THE strength f HSS Gi Wl i bl D312 H fUSE it ifyou WEI t predict tI39IE WEIUE f I QWEI I if lquot WEE F EI SE II Il t li I ii II 1 r pl 1 i ful i 1 395 Enrrel39amn kl Rigniacin II 1 SIMPLE LINEAR CORRELATION VERSUS SIMPLE LINEAR REGRESSION The calculations for linear correlation analysis and regression analysis are the same In correlation analysis one must sample randomly both X and Y Correlation deals with the association importance between variables whereas Regression deals with prediction intensity The slide shows three types of correlation for both positive and negative linear relationships In the first figure r 09 the data points are practically in a straight 208 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU line This kind of association or correlation is near perfect This applies to negative correlation also The graphs where r 05 the points are more scattered there is a clear association but this association is not very pronounced ln graphs where r 0 there is no association between variables TVF IE L RREL TI N CORRQATION COEFFICIM For calculation of correlation coefficient 1 A standardised transform of the covariance sxy is calculated by dividing it by the product of the standard deviations of X sx amp Y sy 2 It is called the population correlation coefficient is defined as r SXysty 01 r C0vX Y VarX VarY Where covariance of X and Y is de ned as C0vXY Y Y Y n This formula is a bit cumbersome to apply Therefore we may use the following short cut formula Short Cut Formula pry 220mm lZXZZXlzw ZYZ Zlera 209 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU It should be noted that r is a pure number that lies between 1 and l ie l lt r lt 1 Actually the mathematical expression that you have just seen is a combination of three different mathematical expressions Case 1 Positive correlation 0 lt r lt 1 In case of a positive linear relationship r lies between 0 and 1 Y n X In this case the closer the points are to the UPWARDgoing line the STRONGER is the positive linear relationship and the closer r is to 1 Case 2 No correlation r 0 2 l 0 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The extreme of dissociation zero correlation r 0 V n V In such a situation X and Y are said to be uncorrelated Case 3 Negative correlation l lt r lt 0 Warning Existence of a high correlation does not mean there is causation which means that there may be a correlation but it does not make things happen because of that There can exist spurious correlations And correlations can arise because of the action of a third unmeasured or unknown variable In many situations correlation can be high without any solid foundation 21 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EXAMPLE Suppose that the principal of a college wants to know if there exists any correlation between grades in Mathematics and grades in Statistics Suppose that he selects a random sample of 9 students out of all those who take this combination of subjects The following information is obtained 30 Marks in Statistics In order to compute the correlation coef cient we carry out the following computations 25 20 15 10 Marks in Marks in Mathematics Statistics Student Total Marks 25 Total Marks 25 X Y A 5 11 B 12 16 C 14 15 D 16 20 E 18 17 F 21 19 G 22 25 H 23 24 I 25 21 SCATTER DIAGRAM X 5 10 15 20 25 30 Marks in Mathematics 212 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SK 6 Ya Cr 5 11 25 121 55 12 16 144 256 192 14 15 196 225 210 16 20 256 400 320 18 17 324 289 306 21 19 441 361 399 22 25 484 625 550 23 24 529 576 552 25 21 625 441 525 156 168 3024 3294 3109 2 XY 2 XXX Yn lizxz z X2ni 2 Y2 z Y2ni 3109 1561689 3024 1562 9 3294 1682 9 3109 2912 3024 2704 3294 3136 197 197 088 320x158 22486 There exists a strong positive linear correlation between marks in Mathematics and marks in Statistics for these 9 students who have been taken into consideration EXCEL Tools For summary of sample statistics use Tools Data Analysis Descriptive Statistics For individual sample statistics use Insert Function Statistical and select the function you need EXCEL Functions In EXCEL use the CORREL function to calculate correlations The correlation coefficient is also given on the output from TOOLS DATA ANALYSIS CORRELATION or REGRESSION 2 l 3 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU tter Diagram Two Variables Yo Ehart Wizard Step 1 of 4 Ehart Type Standard Types Custam Types Qhart type lChart subtype M Calumn a E Ear Daughnut r F39adar Surface 539 Bubble Em Stpclt gt0 W Scatter Campares pairs at yalues iii Press and led tp ielaI Sample I Cancel HESII I Einish I The slide shows a scatter diagram of Advertisement and Sales over the years The graph was made using EXCEL chart Wizard As you can see one cannot draw any conclusions about the degree of association between advertising from this graph Microsoft Excel Lecture Eile Edit ew insert Fgrmat pols Qhart window Help 39 39 v 39 peeeeevs m perm vwvnru Chari3 V r A B C D E F G H J 8 I g SCATTER DIAGRAM TWO VARIABLES i 3 E 15000 11 E U 15 lg 10000 ADV I l l I 13 n 1 5000 39 SALES 20 NJ 06 21 gt 0 0 22 D U o O 6 23 lt3 24 1988 1988 1999 1992 25 g YEARS 28 I 29 SALES VERSUS ADVERTISEMENT The scatter diagram for sale versus advertisement shows a fairly high degree of association The relationship appears to be positive and linear 214 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micreseft Excel Lecture Eile Edit Eiew Lnsert Fgrmat eels Qhart window elp El 1 De i tttt v material ZDBIH t Chadd 139 f3 A E C 0 E F e H j 3839 I j SALES VS ADVERTISEMENT 12 33 8 15000 45 a 40 j 3 10000 4 40 2 E 5000 52 53 g I I I 2 U 200 400 600 OD ADVERTISEMENT RS 000 01 r CORRQATION COEFFICIENT USING EXCEL Correlation Coefficient for correlation between two steams of data was calculated using the formula CovxySxSy as given above The data for variable x was entered in cells A67 to A71 Data for variable y was entered in cells B67 to B71 Calculations for square of x square of y product of x and y Xm Ym and covxy were made in columns C D E F and G respectively Other calculations were made as follows Cell A72 Sum of x SUMA67A71 Cell B72 Sum ofy SUMB67B71 Ce C72 Sum of square ofx SUMC67C71 Cell D72 Sum of square ofy SUMD67D71 Cell E72 Sum of product ofx and y SUME67E71 Cell F72 Mean ofx A725 where 5 is the number of observations Cell G72 Mean ofy B725 where 5 is the number of observations Ce F73 Sx SQRTC725F72F72 Ce G73 Sy SQRTD725G72G72 Cell H73 Covxy E725F72G72 Cell H74 Correlation coefficient H73F73G73 The above formulas are in line with formulas described earlier 215 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TE Micreeeft Excel Lecture Elle Edit Eiew insert Fgrmat Innis gate window elp 39 3 3 in v E v E SLIM r X J 8 H3fF3G3 is E C n E F e H 54 CRRELPlTIN 55 as Y Y Km Ym a 2 00 4 3600 120 as 5 100 25 10000 500 59 4 70 10 4900 230 m 90 8100 540 in 80 6400 240 2 20 400 90 33000 1080 4 80 is Standard deviatin s 14 1414i ml 03 r H73ilF7339iiGT3 r5 39 CORREL Returns the correlation coefficient of the array1 and array2 cell ranges Use the correlation coefficient to determine the relationship between two properties For example you can examine the relationship between a location39s average temperature and the use of air conditioners Syntax CORRELarray1 array2 Array1 is a cell range of values Array2 is a second cell range of values Remarks 0 The arguments must be numbers or they must be names arrays or references that contain numbers o If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included o If array1 and array2 have a different number of data points CORREL returns the NA error value If either array1 or array2 is empty or ifs the standard deviation of their values equals zero CORREL returns the DlVO error value EXCEL calculation The X and Y arrays are in cells A79 to A83 and B79 to B83 respectively The formula for correlation coefficient was entered in cell D84 as CORREA79A83B79B83 The value or r 08 is shown in cell C86 216 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture25 Eile Edit Eiew insert Fgrmat IDEIIS Eata indnw Help as La m 2 v SLIM v 2 4 CDRHELEA9AEEEHTBEEESJ F5 A E C D E F IE r5 CRRELarray1 arrEyZ F re 1 r9 2 an 5 100 31 4 32 E Era 34 r CRRELHBH33 3395 B9333 SE 18 2 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 30 Measures of Dispersion LINE FITTING PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 29 0 Line Fitting EXCEL SUMMARY OF SAMPLE STATISTICS For summary of sample statistics use Tools gt Data Analysis gt Descriptive Statistics For individual sample statistics use Insert gt Function gt Statistical and select the function you need EXCEL STATISTICAL ANALYSIS TOOL You can use EXCEL to perform a statistical analysis On the Tools menu click Data Analysis If Data Analysis is not available load the Analysis ToolPak In the Data Analysis dialog box click the name of the analysis tool you want to use and then click OK In the dialog box for the tool you selected set the analysis options you want You can use the Help button on the dialog box to get more information about the op ons LOAD THE ANALYSIS TOOIPAK You can load the EXCEL Analysis ToolPak as follows On the Tools menu click AddIns In the AddIns available list select the Analysis ToolPak box and then click OK If necessary follow the instructions in the setup program Copyright Virtual University of Pakistan 218 Business Mathematics amp Statistics MTH 302 VU Eile Eclit iew Lnsert Fgrmat leels Qata inclew Help Arial EB 139 r A E C D E I F G H 1 2 3 4 AddIns g clcl Ins a3939ailalle F 39 39 l Analysis TeelPak 39u39ElFI H I CDI39IClithII39IEII Sum Wizard Cancel El IE Eure Currency Teels vquot Internet Assistant 39u39ElFI ll I LcncIltLIp Wizard 1 12 l Selver FiclclIn Agtumatiunl H l 1 3 1 4 1 5 1 E 1 Fquot 1 El 19 J 20 Analysis TcIcIIP39alt 21 Previcles Functicns ancl interfaces Fcr Financial ancl 22 scientific clata analysis 23 24 9F Add Iris Add Ins a3939aialle 1F Anal3939sis Tdle39alc 39 quot I Conditional Sum Wizard Cancel I EurI Curranty Tddls I Intarnat Assistant 3939EIA I Lacme Wizard l Salwar Add in EFDWSE Agtdrnatidn Analysis Tdleals 3939EiA 3939EIA Functidns Far Anal3939sis Tdle39alc 2 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Micrnft Excel Baum Eile Edit ew insert Fgrmat Innls Qatar indnw Help D E 1 g 3933 IE 3539 Queuing F m M l 3 Mal E3 u f1 quot 21 Error Checking 395 E C Share Wnrkhnnk 393 H 1 2 Erntectlnn Ir 3 Euro Conversion g Unline Cnllahnratinn Ir 5 Fnrmula auditing Ir I Tnnls nnthe Wag g Addins 1 gustnmize 11 gptlnns 12 13 gnnditinnal Sum 1411 Lnnkup 15 Data analysis IE v 1 Copyright Virtual University of Pakistan VU 220 Business Mathematics amp Statistics MTH 302 VU iata Analyeie analysis Tnnle Elli E39rrelatin Eaneel Cnrarianee Descriptive Statistics Eannnential Smnnthinn FTest TernSample lnr ilariancee Fnurier analysis Hietnnram liming average Flandnm Number Generatinn Help Ir 391 Data Analyeia analysis Tnnls m Hietnnram lrlnrrinn average gal3 Flandnm Number Eeneratinn Rank and Percentile iFaniaEliaIIIIIIIIIIIIIIIIXXIIIIIIIIIIIIIIIIE HE39F39 Samnlin tTeet Paired Tran Samnle lnr Means tTeet TwnSamnle assuming Enual ilarianees tTeet TwnSamnle assuming Unequal ilarianees aTest Tran Samnle lnr lleane quot39 221 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Regressiun I l npu K Input 1 Range I Cancel Input 5 Range Help M l lalIels l ICenstanl is gen l IZenfitlennze Level 3995 We C IuIu eptiens P Qutpul Range l F quoteI39I 393939erlshee Ely l P quoteI39I erkteelt Residuals l Residuals l Resi ual F39lets l Standardized Residuals l Line Fit F39lets Nermal Fretability l ermal Pratability F39lets SLOPE Returns the slope of the linear regression line through data points in knowny39s and knownx39s The slope is the vertical distance divided by the horisontal distance between any two points on the line which is the rate of change along the regression line Syntax SLOPEknowny39sknownx39s Knowny39s is an array or cell range of numeric dependent data points Knownx39s is the set of independent data points Remarks The arguments must be SLOPE returns the NA error value The equation for the slope of the regression line is 5112 tIthllZIl HZxEExli Example The known yvalues and xvalues were entered in cells A4 to A10 and B4 to B10 respectively The formula SLOPEA4A10B4B10 was entered in cell A11 The result 0305556 is the value of slope in cell B12 222 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E Micrusu Excel Lecture3 EiIE Edit Eiew insert Fgrrnat IncIs Eata irtjaw elp at a a m T E T E SLIM 1quot K J J SLDF39EIZMSA1EIEMSEHDII 395 I E 393 D E F 2 SLPEknwny39Sknwnx39S 3 ann y ann x 4 5 E 7 El 9 mum tween hhmH Imm lIII J SLiPE 12 Adlm l ems1 ml INTERCEPT Calculates the point at which a line will intersect the yaxis by using existing xvalues and yvalues The intercept point is based on a bestfit regression line plotted through the known xvalues and known yvalues Use the INTERCEPT function when you want to determine the value of the dependent variable when the independent variable is 0 zero For example you can use the INTERCEPT function to predict a metal39s electrical resistance at 0 C when your data points were taken at room temperature and higher Syntax INTERCEPTknowny39sknownx39s Knowny39s is the dependent set of observations or data Knownx39s is the independent set of observations or data Remarks The arguments should be either numbers or names arrays or references that contain numbers If an array or reference argument contains text logical values or empty cells those values are ignored however cells with the value zero are included If knowny39s and knownx39s contain a different number of data points or contain no data points INTERCEPT returns the NA error value The equation for the intercept of the regression line is 1 Iquot EzX where the slope is calculated as b HER EEIEF FEEit 2ij Example The data for yvalues was entered in cells A18 to A22 The data for xvalues was entered in cells B18 to B22 The formula NTERCEPTA18A22B18B22 was entered in cell A24 The answer 0048387 is shown in cell B25 223 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture3 Eile Edit ew insert Fgrmalz Innls Eata indnw Help 3 IE E r a 1 SLIP391 T x 3 F NTERCEPTEA1Eamp22E1E522 A El 1 D E F 1a INTERCEPTImwny39EJImwnz395 1 anny ann x 1a 2 5 1g 5 23 Q 11 21 1 r 21 8 5 E 24 INTERCEP 25 313322 224 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 31 LINE FITTING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 30 0 Line Fitting Types of Reqression Models There are different types of regression models The simplest is the Simple Linear Regression Model or a relationship between variables that can be represented by a straight line equation To determine whether a linear relationship exists a Scatter Diagram is developed first The Scatter iag ram Plot sofaquot mi vi pair quotr ll El i i 4 I El I I E I I i I i I i I 2 4 5 ln linear regression two types of models are considered The first one is the Population Linear Regression that represents the linear relationship between the variables of the entire population ie all the data It is quite customary to carry out sample surveys and determine linear relationship between two variables on the basis of sample data Such regression analysis is called Sample Linear Regression Copyright Virtual University of Pakistan 225 Business Mathematics amp Statistics MTH 302 VU TYPEE of Regressiun Models H atim liu l39l lIIIIT Linmr Nu Eeh nmlii IIIu39Jl39HSILEJIU EIII39JIIIC EIIIII Relationship between Variables is described by a Linear Function The change of one variable causes the other variable to change The relationship describes the dependency of one variable on the other If the relationship between the variables is exactly linear then the mathematical equation describing the liner relation is written as YabX Where Y represents the dependent variable X represents the independent variable a represents the Yintercept ie the value of Y when X is equal to zero b represents the slope of the line ie the value of the tan 9 where 9 represents the angle between the line and the horizontal axis 226 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Interpretation of a and b A very important point to note is that MANY lines can be drawn through the same scatter diagram In contrast to the above the liner relationship in some situations is not exact For this we add an unknown random error variable as Wab a We assume that the liner relationship between the dependent variable Y and the value X of the regressor X is Foouletion Lineer Regreeeion D Population Regreeeion Line le r Straight Line that IIIeeorihee The Dependence of The ovenme 1w rlue of ne lli erinlzile on The Either P animation Harmonquot Population 3 IDIE Error il 39 H IEo oient P71 n 81 r 39t 39 D reitlentf magma53 Pmrlle oly l llttl rentl t anel g Retiree e E itnlmetory line lur l39itille The slide below shows the graphical representation of the population regression equation It may be seen that the distance of the points from the regression line C Copyright Virtual University of Pakistan VU 227 Business Mathematics amp Statistics MTH 302 VU obtained by inserting values of X in the equation is the random error The intercept is shown on the Yaxis Flooulation Linear Regreaaion fwr nue E lr If a ll 381 z z ab rwd Value W o Iquot kaaaaa al C39i SEl 39Ed Value 15 Hi The slide below shows the regression equation for the sample Note that the intercept in this case has a notation Bo The slope is B1 The random error is e1 Different notations are used to distinguish between population regression and sample regression Sarnole Linear egreaaion Sample Fteg reeaion Line Prouitlee an Eatilnate of The Fionulation Reg region Line as well no a Pretliotetl Value of 1 Emma Saner Elmo lllta39ee t Era 5 ref Eoeiafieieli I a v F quot1 39 l39 I 1 bl bl l IE H Ha tllal quot r quot Ean39q e 15h ul wulee an eetllnate ot TE 4 lawman Line bl urouitlea an eatilnate of 31 REGRESSION EQUATION EXAMPLE Computer the least square regression equation of Y on X for the following data X 5 6 8 10 12 13 15 16 17 Y 16 19 23 28 36 41 44 45 50 Solution 228 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The estimated regression line of Y on X is YabX In the previous lecture we learn that the equation for the slope of the regression line is b HZ 12012 HZ 3 Z x 3 And the equation Er the intercept of the regression line is 1IrquotEf X Now from the given data X Z 1029 1133 n Y n b nZXY 2ZXZY 22831 nZX ZX a i7 49 335628311183 147 Hence the desired estimated regression line is Y 2147 2831X REGRESSION EXAMPLE Regression Analysis can be carried out easily using EXCEL Regression Tool Let us see how it can be done We chose to carry out regression on data given in the slide below Yvalues are 60 100 70 90 and 80 Xvalues are 2 5 4 6 Copyright Virtual University of Pakistan VU 229 Business Mathematics amp Statistics MTH 302 and 3 E il ierosoft Excel Lecture31 Eile Edit Eiew Lnsert Format Iools gate inookI elp Firial D51 139 J15 1101 A El 393 a REGRESSION as EXAMPLE 1 39 4 1 JH 12 J13 I14 15 J15 We start the regression analysis by going to the Tools menu and selecting the Data Analysis menu as shown below Tools I Qata window Help 23 Sjelling F Share Workbook Erotection tr Euro lConversion Tools on the Wep adolns gate analysis h H quotis The Regression dialog box opens as shown in the following slide You click the Regression analysis tool and then OK Data Analysis analysis Tools K Histogram a Moving Fwerage Random Number lSeneration Ftank and Percentile Cancel Lil Help Sampling h tTest Paired Two Sample For Means tTest TwoSample assuming Equal 39u39ariances tTest TwoSample assuming Unequal 39u39ariances eTest Two Sample For Means quotquot 230 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 The regression dialog box opens as shown below In this dialog box Input range forX and Y is required One can specify labels confidence level and output etc Regressiu n Input DH Input 1 Range 3 Cancel I Input 3 Range 3 Hel Labels I IIenstant is gen l Cpn idence Level 3995 We lElutput pptipns Ii F gutput Range i lquot MelaI Wprksheet Ely I F MelaI prklapplt Residuals I Residuals F Resiglual Flats Standardized Residuals I Line Fit F39Ipts Nermal F39relzualzuilitsi l ermal F39relzualzuilitsi Plpts For the sample data the input Y range was selected by clicking in the text box for input y range data first and then selecting the Y range A85A89 The regression tool adds the sign in front of the column and row number to fix its location The input range for X was specified in a similar fashion No labels were chosen The default value of 95 confidence interval was accepted The output range was also selected in an arbitrary fashion All you need to do is to select a range of cells for the output tables and the graphs The range A91F124 was selected as output range by selecting cell A91 and then dragging the mouse in such a manner that the last cell selected on the right was F124 Copyright Virtual University of Pakistan 231 Business Mathematics amp Statistics MTH 302 VU iiigiiiii F39 77777 7 7739s Lquot E Criirriirrrrgi iriiiii lF39mrm ull ll runiiiiiiiira i I r Eile Edit EieliiI Insert Fgrmat Tools Data indpw elp v 539 it w v a a 91 v 5 A El C D E F El e1 REG RESSIDN RNRLYSIS e2 EXRMPLE 1 Regression 33 Input Y Input i Range a35pgg 2 84 l E Input 5 Range B35BEg E ES I l Labels r Constant is gerd il 35 1 l Con dence Level 3995 as er 70 ea 30 as 30 Output options l7 gutput Range R91F124 1 F New Worksheet Ely l NewI prkbppk Resduab SD M F EESldllals l RESlgU l F39lDtS 91 l I F Standardized Residuals l7 Line Fit Hats 92 I 93 Normal Probability 94 F mermal Probability F39Iots as I an The Regression dialog box with data is shown below for clarity Ragrassiun l I Input 1 Range nescies s 39 Cancel I Input 5 Range EIBEBBE lil Hal I l Lalzlels l IIdnstant is gen p l Confidence Level 3995 quot lClutput options 3 Qutput Range ln91l124l P New Wdrksheat Ely l P New mprkbpplct Residuals IF Residuals l Residual Pldts l Standardized Residuals I Line Fit F39ldts Narmal Pratability I drmal Pratability Plats When you click OK on the Regression tool box a detailed SUMMARY OUTPUT is generated by the Regression Tool This output is shown in parts below 232 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 SUMMARY iIUTPUT Ra urassidn Statistics Multiple R Square 054 Adjusted R Snare 052 Standard Err bser39vatizia 103541512 5 ANVA df Rarassin Rasiual Ttal 1 64B 64 361 12 4 10110 Coef cients Standard Error tStat Intercat Variable 1 48 1439593846 32559 8 3464101615 23119 Copyright Virtual University of Pakistan 233 Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture31 E Eile Edit iew insert Format Innls Data window elp 1 539 Digggg g ngv m fmial 1nv f E IEE 1quot i5 E F G H 5 53 F Signi cance F e 0104088039 El LT I Ei BE 53 Pvalue Lewer Upper Lewer Upper 51 00409 1227738032 9477220 122773853 947722914 55 01041 3024327728 19024327 EE E Hicrnsnft Excel Lecture31 file Edit iew insert Fgrrnat Innls gala indnw Help v i Deal emexweeewe vlnv rua IEE v 7 B C D E 39 RESIDUAL UTPUT 7 1 Observatien Predicted Residuals i2 1 E4 4 73 2 88 12 74 10 75 4 90 6 FE 5 72 3 77 The regression Tool also generates a normal probability plot and Line Fit Plot 234 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrusuft Excel Lecture31 Elle Edit iew lneert Fgrmet leele gate indew Help g rgEe l EfAriel r liDD 1 5 J H L M N CI F39 D H 3395 Nerrnel ereeaeility Plet BE EDD El 1DD EB D 2D 4D ED DD 1DD 59 Sample Pereent ile DD 91 t l v I lquot 92 I Verleble1 Llne Flt Plet 93 E14 EDD 95 nmnu u e f BE I Predicted 1 D 39 39 39 93 D 2 4 E 3 99 it U erielile 1 1DD 13911 13932 EXCEL REGRESSION TOOI OUTPUT In the regression Tool output there are a number of outputs for detailed analysis including Analysis of Variance ANOVA that is not part of this course The main points of our interest for simple linear regression are Multiple R Correlation Coefficient R Square Coefficient of determination STEMStandard Error of mean Standard deviation of populationsample size TStatistic sample slope population slope Standard error RSQ There is a separate function RSQ in EXCEL to calculate the coefficient of determination square of r Description of this function is as follows Returns the square of the Pearson product moment correlation coefficient through data points in knowny39s and knownx39s For more information see PEARSON The r squared value can be interpreted as the proportion of the variance in y attributable to the variance in x Syntax RSQknowny39sknownx39s Knowny39s is an array or range of data points Knownx39s is an array or range of data points Remarks The arguments must be either RSQ returns the NA error value The equation for the r value of the regression line is 23 5 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU r arm39139 mm 233 EXJEHHEFE Eff Example A B 1 Known y Known x 2 2 6 3 3 5 4 9 1 1 5 1 7 6 8 5 7 7 4 8 5 4 Formula Description Result RSQA2A8BZBS Square of the Pearson product moment correlation coefficient through data points above 005795 236 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PVALUE In the EXCEL regression Tool the PValue is defined as under Pvalue is the Probability of not getting a sample slope as high as the calculated value Smaller the value more significant the result In our example Pvalue0000133 It means that slope is very significantly different from zero Conclusion X and y are strongly associated SAMPHNG DISTRIBUTION IN r It is possible to construct a sampling distribution for r similar to those for sampling distributions for means and percentages Tables at the end of books give minimum values of r ignoring sign for a given sample size to demonstrate a significant nonzero correlation at various significance levels 01 005 002 001 and 0001 and degrees of freedom 1 to 100 It is to be noted that v degrees of freedom n 2 in all these calculations SAMPHNG DISTRIBUTION IN rEXAMPE Look at a sample size n 5 Null hypothesis r 0 Calculated coefficient 08 Test the significance at 5 confidence level m Look in the table at row with vn2 3 and column headed by 005 Pearson ProductMoment Correlation Coefficient Table of Critical Values df N2 Level of signi cance for twotailed test N number of pairs of data 10 05 02 01 1 988 997 9995 9999 2 900 950 980 990 3 805 878 934 959 4 729 811 882 917 5 669 754 833 874 6 622 707 789 834 7 582 666 750 798 8 549 632 716 765 9 521 602 685 735 237 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 10 497 576 658 708 11 476 553 634 684 12 458 532 612 661 13 441 514 592 641 14 426 497 574 628 15 412 482 558 606 16 400 468 542 590 17 389 456 528 575 18 378 444 516 561 19 369 433 503 549 20 360 423 492 537 21 352 413 482 526 22 344 404 472 515 23 337 396 462 505 24 330 388 453 495 25 323 381 445 487 26 317 374 437 479 27 311 367 430 471 28 306 361 423 463 29 301 355 416 456 30 296 349 409 449 35 275 325 381 418 40 257 304 358 393 45 243 288 338 372 50 231 273 322 354 60 211 250 295 325 70 195 232 274 302 80 183 217 256 284 238 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 90 173 205 242 100 164 195 230 You will find the Tabulated value 0878 Sample value of 08 is less than 0878 Conclusion Correlation is not significantly different from zero at 5 level Variables are not strongly associated SAMPLING DISTRIBUTION IN rEXAMPLE 2 Look at a smple size n 5 Null hypothesis r 0 Calculated coefficient 095 Test the significance at 5 confidence level m Look at row with v 3 and column headed by 005 Tabulated value 0878 Sample value of 095 ignoring sign is greater than 08783 Conclusion Correlation is significantly different from zero at 5 level Variables are strongly associated 267 254 239 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 32 TIME SERIES AND EXPONENTIAL SMOOTHING PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 31 0 Time Series and Exponential Smoothing SIMPLE LINEAR REGRESSION EQUATION EXAMPLE The slide below shows the data from 7 stores covering square ft and annual sales The question is whether there is a relationship between the area and the sale for these stores It is required to find the regression equation that best fits the data Simple Lineer Regreeeien Equetien Example utlulal Stere Siguare Sels Re Hit 391 39tJlJ 3631 quotlieu me te eten39ine the 2 them 33925 r atim tie linetweet the 3 Elm E 55553 aware feetege ef mettlee i Hera and their eliulel 4 555 ghee Senute late fer T 5 it 35313 anHE me el iinetl Fintl 5 33 55553 the t l ll ef the mi tt T t 3391 3 EJ alIl line that te the late Ire First of all a scatter diagram is prepared using EXCEL Chart Wizard as shown below The points on the scatter diagram clearly show a positive linear relationship between the annual sale and the area of store It means that it will make sense to proceed further with regression analysis The estimated regression line of Y on X is Y a bX In the previous lecture we learn that the equation for the slope of the regression line is b HZ tIlZIllZIl HZ x 3 Z x 3 And the equation for the intercept of the regression line is 240 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Stores X Y XY X2 1 1726 3681 6353406 2979076 2 1542 3395 5235090 2377764 3 2816 6653 18734848 7929856 4 5555 9543 53011365 30858025 5 1292 3318 4286856 1669264 6 2208 5563 12283104 4875264 7 1313 3760 4936880 1723969 2 ZX16452 ZY35913 ZXY 104841549 2X252413218 Now from the given data X X n M 164527 235029 Y Y Z 359137 5130429 n b quot2 XY EXXZY 14866 nZXZ ZX2 a 17 49 513043 14866235029 1636489 Hence the desired estimated regression line is Y 2 1636489 14866X Copyright Virtual University of Pakistan 241 Business Mathematics amp Statistics MTH 302 VU Seetter iegrem Eeemple III IIZI39III EIIIZI BIIIEI HZIIII39ZI SDIII EIIIZI quere Fe e1 Eeeel uent Using the EXCEL Regression Tool the regression equation was derived as given below The graph of the regression line was prepared using the regression Tool The result shows the data points regression line and text showing the equation As you see it is possible to carry out linear regression very easily using Excel s Regression Tool Greph ef the Semple Regreeeien Line IEIIIIEII E Item 39 elm EH EIIIIII m we E EIIIIII E I I I I I I I2 lIIiII EIIIIIIZI 3IIiII lJIIIZI SEIIIIZI EIZIIII Supera Feet lnterpretinq the Results The slide below gives the main points namely that for every increase of 1 sq ft there is a sale of 1487 units or 1407 Rs As each unit was equal to 1000 Now that the equation has been developed we can estimate sale of stores of other sizes using this equation 242 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Interpreting the Reeulte H e teeeete death The slope at 148 means that each hcreaee at me ahitlh A we prem39ctthe average at Fte increase t5 ah eathhded 148 hath The medeieathhatea that far each incna e ef1 mature feet hr the size at the attire the expected almaai39aaiea are predhaed te hrcreaee try Re t Interpreting the Reeulte er teeeete death The atepe at 148 mean that each hrcreaee at me ahitlh A we precb39ctthe average at l ta increase hf ah eathhded 143 quotme The medeleathhatea that far each incree e ef1 aware feet h the size at the attire the expected arhaal39aaiea are predhaed te increase tar Rm tl CHART WIZARD Let us look at how we can use the Chart Wizard We wish to study the problem shown in the slide below Copyright Virtual University of Pakistan 243 Business Mathematics amp Statistics MTH 302 VU Elmicrneuft Excel Baul Eile Edit iew insert Fgrmat Inels gate window Help lt35 gEv i fnrial TEUTIHE A2 1 f3 r ear A E t D E F Wl H I J K 1 SHLES RECRDS BRKEN DWN BY UARTERS 2 Year Quarter Ne50l 999 i 1993 Spring 14 i Summer 54 i Autumn 162 i Winter EDIE i 1994 Spring 139 i Summer 59 1 Autumn 1T4 E Winter 19 l 1995 Spring 126 3 Summer 42 3 Autumn 162 M Winter 16 f e e e we IIIIII quotE quot J ti 1L Ihert Wizardl You can start with the Chart Icon as shown on the right There are 4 steps in using the Chart wizard as shown below Step 1 244 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Chart Wizard Step 1 nf 4 Chart Type Standard Types Custem Types Chart type Chart subtype EIIIIZIlIJ rI39IFI E Eiar lg Line 3 Pie W Scatter I Ftrea Deughnut J Ftadar SurFace 3 Bubble Em Stcuclt j Clustered Celumn Ccumpares values acress categcuries Press and Held tcu ew Sample I Cancel Heste I Einish I Chart llo39J39izatrd Step 2 nf 4 Chart Snurce Beta Data Range 1 Series l Te create a chartJ click in the Data range hex ThenJ en the IllcurlsheetJ select the cells that centain the data and labels yeu want in the chart Qata range Series in C aws P CcILIn39ns l Cancel si atk eet I Einish 245 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ehart Wizard Step 3 f 4 Ehart ptine I hew legencl Placement F Etcuttcum r Cgrner lquot Inn f ight f LeFt lCancel ci aclt eet Einish Ehart Wizard Step 4 f 4 Ehart Leatin F39lace chart III P like new gheet lCl39IEII lZl ll 397 n5 glaject in Eitleetii IIIIIIIIIIIIIIIIIIIIIIIIIIIIIE i Cancel c ack x I I Einieh I The dialog boxes are selfexplanatory Let us look at the example above and see how Chart wizard was used First the data was selected on the worksheet Next the Chart Wizard was selected We chose Column Graph as the option as you can see in the slide below We clicked Next 246 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 ak am r We metal a a a m2 its net Ej U D a fr tin rmed liltlll lrirli nibartth M u lma 3 I ail Ed39sE i ange1amp2 n yme ezereame aI i You can see the selection of Column graph in the slide below Ehart Wizard Step 1 pf 4 Ehart Type Standard Types I Custprn Types I ghart type Chart subtype I III I U m Fl E Eiar IE Line 5 F39ie 31quotquot Scatter I Furea a Deughnut r Ftadar g SurFace 239 Bubble II t Stuck TI Clustered Celumn Iernpares yalues acress categeries Press and Held te gleny Sample I ICancel I ext I Einish I Under Step 2 the Chart Title Category X axis and value Y were entered as shown in the slide Then the button Next was clicked 247 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ehart Wizard Step 3 f 4 Ehart ptine Titles l Fixes l lErilzllines l Legend l Data Labels Data Table l ll39iarttitle l DDWN Bil QUpIR39IERS SALES HEEUHD EHDKEH DEVquot EquotIII HUAFITEFIE gategnrif it axis lQuarters H El Sari51 Eauemaxig g 15 H H ill Ill IEzrist 5 1quot ll ll EIEeriesS lsalesl 5 II II II II II II II II II mm 3 lllllllllllllllllllllll aeaEFMaEFWa EF ca 1333 1334 1335 l utters lCancel ack eet Einish Under the 4lll step the default values Chart1 and Sheet1 were selected Then the button Finish was clicked Ehart Wizard Step 4 f 4 hart Lcatin Plate thart lIlilSI39IEWEl39IEElI lCl39Iartl l3 Fi abject in Cancel Eat nish The result is shown below as a column graph 248 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 200 QUARTE 133 39 5 TEi ll i ll 1993 1994 uarters eelee I D Ser iesl I Series2 D Series D Series Spring Summer Winter Spring Summer Winter Wi rite r Quarter Spring eutumri eutumri Summer m m quotI re D U I I I Chart Wizard was used again to draw a Side by Side chart using the same data The result is shown below I I I SALES BRKEN BY UARTER D Seriesrl D SeriesS I SeriesZ D Seriesl LP 3 1 e 9 SALES I I I A line graph of the data was also prepared as shown below This graph shows the seasonal variations in the values of sales 249 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 539 m cu c1 TEJEJal tElEl i Quarter EXAMINATION OF GRAPH TREND The graph shows that there is a general upward or downward steady behaviour of figures There are Seasonal Variations also These are variations which repeat themselves regularly over short term less than a year There is also a random effect that is variations due to unpredictable situations There are cyclical variations which appear as alternation of upward and downward movement EXTRACTING THE TREND FROM DATA Look at the following data 170 140 230 176 152 233 182 161 242 There is no explanation regarding time periods What to do First step Plot figures on graph Horizontal as period 1 Vertical as period 2 Conclusion There is a marked pattern that repeats itself There is a well established method to extract trend with strong repeating pattern DATE 3m 252 r 3 2m 3 I 15E ii nw u w p E 1m I1 I i l I I I I 1 4 E 3 3 Funnels Copyright Virtual University of Pakistan 250 Business Mathematics amp Statistics MTH 302 VU MOVING AVERAGES Look at the data in the slide below There is sales data for morning afternoon and evening for day 1 2 and 3 We can calculate averages for each day as shown These are simple averages for each da Microsoft Excel Lecture32 Elle Edit view insert Fgrmal 112215 gate window Help 7 a neeeaa em ez aaarlevueee 812 c 1 E F L H I J a L 1111 122 AVERAGES 122 Data Meving Average Trend 121 Day 1 lll39lerning 17121 Afterneen 1412 122 Evening 23121 122 Average 1311 AVERAGEF124F123 122 Day 2 I ll39lerning 173 122 Afterneen 132 1212 Evening 233 121 Average 13 AVERAGEF123F1311 122 Day 3 Warning 132 122 Afterneen 131 121 Evening 242 122 Average 133 AVERAGEF132F134 122 Now let us look at the idea of moving averages First Avergqe DaJLl 170 140 2303 5403 180 Next Avergqe Morninq 140 230 1763 5463 182 Next AverageAfternoon 230 176 1523 186 Another method Drop 170 Add 176 l76l703 63 2 Last average 2 180 2 182 m You may make a mistake You saw how it is possible to start with the rst 3 values 170 140 and 230 for the rst day and work out the average 180 Next we dropped 170 and added 152 the morning value from day 2 This gave us an average of 182 Similarly the next value was calculated Look at the worksheet below for the complete calculation These averages are called moving averages You could have used the alternative method but you may make a mistake in mental arithmetic So let us only use EXCEL worksheets 251 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 Micreeeft Excel Lecture Eile Edit view insert Fgrmat 1414 gate window elp 553g39Li ii v e x DEi cli 139 E39El i nmeuEEE 1 NH C D E F G H l J K L lvl 7 114 MUING AVEReGES 1351 Feried Date ll39leving Average Trend 1411 M511 llllerning 170 141 Afterneen 14H 39 13 AVERAGElF14lIlF142l 142 Evening 231 r 132 AVERAGElF1411F143l 143 Eley2 llllerning 17 39 AVERAGElF1412F1441l 144 Afterneen 152 r 13 AVERAGElF1413F145l 145 Evening r AVERAGElF1441F145l 1413 D3513 llllerning 132 r 192 AVERAGElF1415F14l 141 Afterneen 1E1 39 195 AVERAGElF145F143l 14e Evening 2412 1419 15D The moving averages were plotted as shown below You can see that the seasonal variation has disappeared Instead you see a clear trend of increase in sales This plot shows that moving averages can be used for forecasting purposes lu39l vilhliii AVE Ell g Eeriee2 339 15 quot Eerie53 39 1111i Eerieei Elli Ee1iee5 i r a eriee 1 e 4 e e 1139 11111 nEvieef Fe ed 14 ANALYSING SEASONAI VARIATIONS Let us find out how much each period differs from trend Calculate Actual trend for each period Day 1I Afternoon Actual 180 Trend 140 Actual Trend 140 180 40 Here 40 is the seasonal variation Similarly other seasonal variations can be worked out Copyright Virtual University of Pakistan 252 Business Mathematics amp Statistics MTH 302 VU LECTURE 33 TIME SERIES AND EXPONENTIAL SMOOTHING PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 32 0 Time Series and Exponential Smoothing TREND As discussed briefly in the handout for lecture 32 the trend is given by the moving average minus the actual data Look at the slide shown below The average of the morning afternoon and evening of the first day is 180 This value is written in cell I179 which is the middle value for first day The next moving average is written in cell I180 This means that the last moving average will be written in cell I185 as the moving average of the morning afternoon and evening of 3rd day will be written against the middle value in cell F185 Now that all the moving averages have been worked out we can calculate the trend as derence of moving average and actual value Micrusnft Excel Lecture Eile Edit view Insert Fgrmat eels gate window elp 139 E X De em eEvEIMEfIUvBueee Live w 4 e e D E F e H I J r L M 7 1e rCTUPIL MINUS TREND we lay F39eriezd Date Meving Average Trend we 1 Merning 1TH 1e Afterneen 143 133 43 F173l173 en Evening 233 132 43 131 2 Merning 173 133 1lIl 1e Afterneen 132 13 33 1e Evening 233 133 44 1e 3 Merning 132 132 1lIl es Afterneen 131 133 34 ea Evening 242 13 133 The actual trend figures are now written as shown in the slide below with M for morning A for afternoon and E for evening The titles Day 1 day 2 and Day 3 were written on the left hand side of the table Further Total for each column was calculated The total was divided by the nonzero values in the column For example in column M there are 2 nonzero values Hence the total 20 was divided by 2 to obtain the average 10 Similarly the averages in column A and E were calculated This data is the seasonal variation and can now be used for estimating trend and random variations 253 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icreee Excel Lecture le Edit Eiew insert Fgrmat IDEIIS gate window Help v a D ll fen Ev l f1 v 3 A202 1 5 A I E C D E F G H 191 ACTUALTREND FIGURES TGETHER 192 M A E 194 Day 2 10 35 M 195 Day 1U 34 U 19 Ttal 20 1 09 92 193 Average 10 435 45 199 EDI EXTRACTING RANDOM VARIATIONS Day 1 Afternoon trend 180 Afternoon seasonal variation 36 Trend variation 180 36 144 Actual value 140 Random variation 140 144 4 Conclusion Expected Trend Seasonal Random Actual expected 254 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Efflimeeft Excel Lecture Elle Edit flew insert Fgrmet eels gate window Help E D lgul gr Er l fze 3 IEH 1 it B C D E F G H J K l EDS 2m ACTUAL 140 176 152 182 151 rue Expected 2etrenilseasnel 144 228 176 151 182 159 zerR llfllTl 2naectuelerpected 1 2 0 1 2 l 2 EDS Forecast for day 4 t Trend for afternoon of day 4 Seasonal adjustment for afternoon period Trend 180 to 195 6 intervals 156 25 per period Figure for evening of day 3 195 25 1975 Morning of day 4 1975 25 200 Afternoon of day 4 200 25 2025 After adjustment of seasonal variation 36 2025 36 1665 or 166 SEASONABLE VARIATIONS Seasonal Variations are regarded as constant amount added to or subtracted from the trends This is a reasonable assumption as seasonal peaks and troughs are roughly of constant size In practice Seasonal variations will not be constant These will themselves vary as trend increases or decreases Peaks and troughs can become less pronounced Seasonal variations as well as the trend are shown in the graph below You can see that the trend clearly shows a downward slide in values 255 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I I I 3 EEAEDHAL H i Ti HE l f 39 l 3 I v I 331331 lt hi I 39 V 39 533332 g 3 I L 3 5 1 i 3 ii V i i i7 i7 i7 V V 123433233131112 1 1 333333 i ll In the following slide the actual values are for 4 quarters per year Here there is no middle value per year The moving averages were therefore summarised against the 3rel quarter As this does not reflect the correct position the average of the first two moving averages was calculated and written as centred moving average in column H The first centred moving average is the average of 141 and 138 or 1395 This is used as the trend and the value ActualTrend is the difference of Actual Centred Moving Average Here also the last row does not have a value as the moving average was shifted one position upwards Microsoft Excel Lecture33 Eile Edit ew lnsert Fgrmat 1333 gate indow Help v 3 D t lt3 E131 13BH E Eta3313 MES v r E C D E F G H J I42 L M 39 5E1 39 53 TREND AND SEASNAL VARIATINS 33 uarter Actual Mving Centred Actual 31 Average MAverage trend 32 391 142 33 2 54 34 3 132 141 1333 223 a 4 233 133 1333 333 I 3 1 133 13 1333 33 3 2 33 143 1333 333 3 3 124 133 1333 333 3 4 133 13 1333 323 m 1 123 133 1333 33 31 2 42 132 1333 333 32 3 132 123 The data from the previous slide was summarised as in the following slide using the approach described earlier It may be seen that the average seasonal variation for Spring Summer Autumn and Winter is 8 888 295 and 653 respectively 256 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnenft Excel Lecturej Elle Edit Eiew insert Fgrmat eels gata window elp v e x 05353 533 Er l fl v uEEEEHE 3 Lane 3 it D E F e H l J K 4 51 7 2395 5 uarterly Veriatins 1 Spring Summer Autumn Winter re 155539 0 0 225 55 re 1554 5 555 52 5 1995 me 5 e 0 H 51 52 Tetal 450 4725 550 1505 ea Average 0 255 553 54 as Rounded 30 55 BE The expected value now is the sum of centred movin averae and random Microsoft Excel Lecturejii Elle Edit eiew insert Fgrmat Iools gate window elp ifCarrelinTerkel v 5 X D2355 53957 Evelmemee E fe e mm v e V e o E F e H if at CMPLETE TABLE 55 tr Aetual Meeing Centred Aetual Expeeted Randem 89 Average MAeerage trend en 1 142 91 2 54 E52H35 C02G02 92 3 152 141 225 1505 5 C02G0 ea 4 205 133 135 2035 25 54 1 130 13 5 1305 05 95 2 50 140 1350 50 500 00 95 3 154 133 135 355 1555 55 at 4 105 13 1350 520 2020 40 98 1 125 135 1335 5 1255 05 99 2 42 132 1305 5 43 10 we 3 152 120 151 4 135 variation The random variation is the difference between the Actual and Expected value This gives us a complete table with all the values The values in this table were plotted using the EXCEL Chart Wizard as shown below You can see that the different components can now be seen clearly 257 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU I I I COMPLETE TALE FOR VARIATIONS 250 200 A 150 IIA r 39I39 CHI W Actual IUD 1quot ii E V V V Moving 50 3 Centred U II I I I I II I I I I 39 tual 1 3 5 r a 11 13 EEDEHEG 3950 n Random 100 1150 Pe ods I I I FORECASTING APPLE PIE SALES Forecast Sale steadily declined from 1390 to 1305 Over 4 quarters the sales declined by 1390 1305 85 Trend in Spring 1995 was 1335 We can assume annual decrease as on the basis of decline over the last 4 quarters 85 Trend in 1996 trend in 1995 less decline 1335 85 125 Seasonal variation as already worked out 8 Hence Final forecast 125 8 117 FORECASTING IN UNPRQICTABLE SITUATIONS Two methods were studied above Each one has certain features If there is steady increase in data and repeated seasonal variations there are many cases that do not conform to these patterns There may not be a trend There may not be a short term pattern Figures may hover around an average mark How to forecast under such conditions Data for sales over a period of 8 weeks is summarized and plotted in the slide below You may see that the values hover around an average value without any particular pattern This problem requires a different solution 258 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Microsoft Excel Lecture33 Elle Edit iew Insert Fgrmat Innls ghart window elp v El D tt rev flzv 01111110 11 1 31 0 0 0 E F 0 H 1110 1111 SALES 1111 Week N Sales LES 111 1 4500 5000 111 2 4000 HY H 4000 153 m 3000 Week ND 154 I g 2000 Sales I 111 5 4000 mg 111 E 4200 3 15 123450T8 vv Vteek 111 3 4100 I 150 100 FORECAST Let us assume that the forecast for week 2 is the same as the actual data for week 1 that is 4500 Week no Actual sales Forecast 1 4500 2 4000 4500 The Actual sale was 4000 Thus the Forecast is 500 too high Another approach would be to incorporate the proportion of error in the estimate as follows new forecast old forecast proportion of error on Or new forecast old forecast or x old actual old forecast This method is called Exponential Smoothing We shall learn more about this method in lecture 34 259 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 34 FACTORIALS PERMUTATIONS AND COMBINATIONS OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 33 o Factorials o Permutations and Combinations Module 7 Module 7 covers the following Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 3536 ChiSquare Lectures 37 Binomial Distribution Lectures 38 FORECAST Please refer to the Example discussed in Handout 33 Let OL 03 Then Forecast week 3 week 2 forecast 0c x week 2 actual sale week 2 forecast 4500 03 x 500 4350 Conclusion Overestimate is reduced by 30 of the error margin 500 The slide below shows the calculation for normal error as well as alpha x error You can s that the error is considerably reduced using this approach Microsoft Excel Lecture33 Elle Edit iew insert Formal Iools gate window elp viii iii39viii yiiquot 1 v 539 X D2355 om Ev l 1 v 2 33 3 5155 v 5 E III D E F G H T 15 Week No Sales Forecast Error alpha 1 Error 153 1 4500 151 2 4000 4500 500 150 155 300 4350 550 1 05 155 4 4000 4135 415 1245 15 5 4000 43005 2005 72 155 0 4200 43907 1 007 155 7 3000 43137 37 221 110 4100 41104 404 451 151 52 Forecast E104H104 ErrorD104E104 15 Error lpha 013 260 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The forecast is now calculated by adding alpha x Error to the actual sales The error is the difference between the actual sales and the forecast The first value is the same as the sale last week Use of alpha 03 is considered very common This method is called Exponential Smoothing and alpha is Smoothing Constant Rule for obtaining a forecait Let A Actual and F Forecast Then F t F t 1 0LA t1 F t 1 ocA t 1 1 on F t1 F t 1 0L A t2 1 on F t 2 Substituting F3 on A t1 1 0c ocA t 2 1 on F t 2 on A t1 1 on A t 2 1 0c quot2F t 2 Replacing F t 2 by a t3 1 on F t 3 F t OtAt1 1 on A t 2 1 0c quot2A t3 1 0LF t 3 WHERE TO APPLY EXPONENTIAL SMOOTHING What kinds of situations require the application of Exponential Smoothing What are good values Diet The accepted Criterion is Mean Square Error MSE You can find MSE for by squaring all and including the present one and dividing by the number of periods included Sign of good forecast is when MSE stabilizes Generally alpha between 01 and 03 performs best Example The slide below shows the calculation of MSE Detailed formulas can be seen in the Worksheet for Lecture 34 Micrnsnft Excel Lecturejil Eile Edit Eiew insert Fgrmat nulls Qatar window Help DEi i lt31 Evtl imvnu et g g EiEiE 1r 5 B C D F G H 122 Frecast E151H154 Er39rrD154E154 m ErrrAlpha 121 15 Week Actual FurecastErrnr 232Erranrrer12 MSE 135 22 22011 2221 l l l l 1 23 2400 2221 2GB 41110131 200m we 24 261111 2260 340 102 115601 5135 19 25 281211 2362 438 1314 191341 86361 13m 25 3000 24934 1521 2513544 122313 131 Forecast D1YQF1YQ WISE UM1H1FEH13W5 133 EXCEL EXPONENTIAL SMOOTHING TOOL 26 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU It is possible to use the Exponential Smoothing Tool included in the EXCEL Tools Beta Analysis analysis Tools limos39a TwoFactor Without Replication lCorrelation Cancel I j39 39 3939 392 quotI I l FTest TwoSample For 39u39ariances J Fourier Finalssis Histogram irio39ring Fwerage Ranclom Number Generation j Input UK input Range I Qamping Factor I Cal39IEEl l Labels Help I lElutput options gutput Range l Qhart lCilutput l tanclarcl Errors Different items in the Dialog Box are described below Input Range Enter the cell reference for the range of data you want to analyze The range must contain a single column or row with four or more cells of data Damping factor Enter the damping factor you want to use as the exponential smoothing constant The damping factor is a corrective factor that minimizes the instability of data collected across a population The default damping factor is 03 Note Values of 02 to 03 are reasonable smoothing constants These values indicate that the current forecast should be adjusted 20 to 30 percent for error in the prior forecast Larger constants yield a faster response but can produce erratic projections Smaller constants can result in long lags for forecast values Labels Select if the rst row and column of your input range contain labels Clear this check box if your input range has no labels Microsoft Excel generates appropriate data labels for the output table Output Range Enter the reference for the upperleft cell of the output table If you select the Standard Errors check box Excel generates a twocolumn output table with 262 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU standard error values in the right column If there are insuf cient historical values to project a forecast or calculate a standard error Excel returns the NA error value Note The output range must be on the same worksheet as the data used in the input range For this reason the New Worksheet Ply and New Workbook options are unavailable Chart Output Select to generate an embedded chart for the actual and forecast values in the output table Standard Errors Select if you want to include a column that contains standard error values in the output table Clear if you want a singlecolumn output table without standard error values Example Use of the Exponential Smoothing Tool is shown in the following slides First the Exponential Tool was selected Eiimmiiii first listiim iii quot31311 39 Eile Edit Eiew insert Fgrmet eels gate winders Help v E X DE i ll 5 il i Teii39ii 1 EEEIE 139 r C i E F G H J 7 es EKPUNENTIAL SMOOTHING USING EXCEL WIZARD 13E 220i NlA 1e 240i 22m eteinelieis 1ee 250i 234i nite iineee5ingeFetter x 139 iineee TweFecter With Fieplitetien am E entire TweFecterWitheutReplicetien 39 39 Cerreetien 19D 1 I Ceeeriente HE 191 Destritiee Statistics iEiiiEIIIIrIEFItlEil Sri39IIIIIIItl39IlnIJ 192 FTest TeeSample fer iierientes 193 Feurier Finelysis 194 Histegrem j 195 1 19 Next the Input and Output Range were specified Labels Chart Output and Standard Errors were ticked as options in check boxes 263 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E 111191msigisii i FTTTE i Qll a H i Qi Eli l ll lm Eile Edit Eiew lnsert Format Iools gate window elp Weee D gi v i 1de E209 1quot 13931 e e E F e H 195 EKFGNENTIAL smooTHiNG USING EXCEL s r 1e13 2200 7 13 Expo ne ntiel Smoothing Input v 11313 2000 I I lnputFianoe 7C196C191 1 v GK V 139 Eampingfacmr V Cancel l 1939 1 Labels V elp 1511 3100 output options 192 EUtF39UtRme isoslaesosm E39s 39 i l i i 1913 F shart Output 19E tsnsstssrtsrsi 197r 199 199 The output alon with standard raphs is shown on the followin slide Microsoft Excel Lecture33 Eile Edit ew insert Format eels gate window elp 1 islftTerkel 51 x Diasgit rev Evsi i lnvnueeesiiii C2135 v is C o E F G H I J 7 1135 EXF ONENTIAL SMOOTHING USING EXCEL WIZARD 1313 2200 11111 11111 131 2100 21m 1111A 11313 2000 251a 911A 1139 200 2122 11111 F 1511 3000 25111313 siesisem 191 3400 1512 193 Exponential Smoothing 191 195 ions 1 oi 3mm 19r 1 Actual E 2min 199 2 mm Ferecast 199 see 393 39 39 39 2m 1 2 3 4 5 292 EintnPoiIrt 203 2131 264 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FACTORIAL Let us look at natural numbers Natural Numbers 1 2 3 Let us now define a factorial of natural numbers say factorial of 5 Five Factorial 5 54321 or 12345 Similarly factorial of 10 is Ten Factorial 12345678910 10987654321 In general n nn1n2321 or n nn1n2 nn1 FACTORIAL EXAMPLES 10 109876543213628800 85 8765 5 876 336 129 1211109l9 121110 1320 10895 109876595 10876 3360 WAYS lf operation A can be performed in m ways and B in n ways then the two operations can be performed together in mn ways Example A coin can be tossed in 2 ways A die can be thrown in 6 ways A coin and a die together can be thrown in 26 12 ways PERMUTATIONS An arrangement of all or some of a set of objects in a definite order is called permutation Example 1 There are 4 objects A B C and D Permutations of 2 object A amp B AB BA Permutations in three objects A B and C ABC ACB BCA BAC CAB CBA Example 2 Number of permutations of 3 objects taken 2 at a time 3P2 332 32 6 AB BA AC CA BC CB Number of permutations of n objects taken r at a time nPr nnr Example 3 Let39s say you and a friend love going to movies and you get a Saturday afternoon free so you can indulge yourselves You go to a multiplex that is showing 6 movies simultaneously each starting at 200 pm 400 pm and 600 pm after which you have to get back home How many different ways can you watch the most different movies Answer You have a choice of 6 movies so this is your set You can watch one movie at 265 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Answer You have a choice of 6 movies so this is your set You can watch one movie at 200 one at 400 and one at 600 therefore you can watch 3 movies and you39re looking for the number of 3permutations We have then P6 3 6 63 6 3 6543 3 654 120 There are 120 different ways for you to watch 3 of the 6 movies that Saturday afternoon Example Suppose there are 100 numbers to choose from 00 to 99 you must choose 5 numbers in a specific order and you can only choose a number once What are your chances of winning the grand prize There are 100 choices and we39re only picking 5 of those so we have P1005 100 100 5 9034502400 ways to pick 5 numbers in a specific order Since only one of those is the winning sequence your chances of winning are l in 9034502400 Example A government keeps some confidential information in a heavily guarded room that is locked with a 5card mechanism Eight different off1cials each carry a card and to get access the cards must be inserted in a speci c order The order changes daily and 3 of the 8 cards will not be used on any given day A novice spy needs to acquire some documents in this room He manages to acquire all eight cards and slip past the guards but doesn39t realize until he gets to the door that only five cards are used and they must be inserted in the correct order A wrong entry brings with it a mass of large mean guys with big guns What are his chances of getting the right sequence Solution The spy has a set of 8 cards to choose 5 from therefore n8 and F5 and P85 83 6720 Only one of those 6720 possibilities is correct so he has a l in 6720 chance or 16720 000015 0015 266 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PERMUTATIONS OF n OBJECTS Number of n permutations of n different objects taken n at a time is n nPn nnn n 0 n 1 n nn1n2321 Number of permutations of n objects of which n1 are alike of one kind n2 are alike of one kind and nk are alike Nin1in2ink1 Examgle 3 How many possible permutations can be formed from the word STATISTICS S3A1 T3 l 2 C1 m nPr nn1n2nk 103132110987654333l2l 50400 PERMUT EXCEL function PERMUT can be used to calculate number of permutations Returns the number of permutations for a given number of objects that can be selected from number objects A permutation is any set or subset of objects or events where internal order is significant Permutations are different from combinations for which the internal order is not significant Use this function for lotterystyle probability calculations Syntax PERMUTnumbernumberchosen Number is an integer that describes the number of objects Numberchosen is an integer that describes the number of objects in each permutation Remarks Both arguments are truncated to integers lf number or numberchosen is nonnumeric PERMUT returns the VALUE error value If number S 0 or if numberchosen lt 0 PERMUT returns the NUM error value If number lt numberchosen PERMUT returns the NUM error value The equation for thelnumber of permutations is 2 Fit a 3 Example Suppose you want to calculate the odds of selecting a winning lottery number Each lottery number contains three numbers each of which can be between 0 zero and 99 inclusive The following function calculates the number of possible permutations 267 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E3 Micrnsnft Excel Baum Eile Edit iew insert Fgrmat nulls ata window Help DEQJL r Ev i fl v BED v 5 A E r D E F 3 1 PERMUTU IUH IIJE UWbELChDSEI I 3 ata escripti n 4 Number if nbjects 5 Number if mbjects in each permutatien a PERMUTMA5 268 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 35 COMBINATIONS ELEMENTARY PROBABILITY PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 34 o Combinations 0 Elementary Probability COMBINATIONS Arrangements of objects without caring for the order in which they are arranged are called Combinations Number of n objects taken r at a time denoted by nCr or n given by r nCr nrnr Difference between Combination and Permutation Suppose we have to form a number of consisting of three digits using the digits 1234 To form this number the digits have to be arranged Different numbers will get formed depending upon the order in which we arrange the digits This is an example of Permutation Now suppose that we have to make a team of 11 players out of 20 players This is an example of combination because the order of players in the team will not result in a change in the team No matter in which order we list out the players the team will remain the same For a different team to be formed at least one player will have to be changed Example Number of combinations of 3 different objects A B C taken two at a time 3232 62 3 These combinations are AB AC and BC COMBINATIONS EXAMPLES Here are a few examples of combinations which are based on the above formula Example 3 ln how many ways a team of 11 players be chosen from a total of 15 players n15r11 15 1 141 1211 1 141 12 C 15 5 3 5 3 1365Ways 11 1115 11 114 4321 Example4 There are 5 white balls and 4 black balls ln how many ways can we select 3 white and 2 black balls 5 4 3X 2 35 3X24 2 Example 5 269 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU If a committee of 3 people is to be selected from among 5 married couples so that the committee does not include two people Who are married to each other how many such committees are possible Solution Total number of ways of picking 3 out 5 married couples or 10 people 10C3 120 A set can have any of the ve married couples gt 5 The third person can be any one of the remaining eight gt 8 one married couple is already part of the chosen set so total Number of ways in Which the set of three can have a married couple 58 40 number of combinations Which don39t have any of the married couples 10C3 58 80 RESULTS OF SOME COMBINATIONS Here are some important combinations that can simplify the process of calculations for Binomial Expansion nCO nCn 1 eg 400 404 1 nC1 nCn1 n eg 401 403 4 nCr nCnr eg 5C2 503 270 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU BINOMIAL EXPANSION An expression consisting of two terms joined by or sign is called a Binomial Expression Expressions such as ab ab xyquot2 are examples of Binomial Expressions We can verify that Xyquot1 X y xyquot2 xquot2 2xy yquot2 xyquot3 xquot3 3xquot2y 3xyquot2 yquot3 xyquot4 xquot4 4xquot3y 6xquot2yquot2 4xyquot3 yquot4 Expressions on the right hand side are called Binomial Expansions COEFFICIENTS OF BINOMIAL EXPANSION The coefficients of the binomial expansion for any binomial expression can be written in combinatorial notation xyquot5 5C0xquot5 5Clxquot4y 5CZxquot3yquot2 5C3xquot2yquot3 5C4xyquot4 5C5yquot5 Solving xquot5 5xquot4y 10xquot3yquot2 10xquot2yquot3 5xyquot4 yquot5 CALCULATION OF BINOMIAEXPANSION COEFFICIENTS Coefficient of first and last term is always 1 Coefficient of any other term coefficient of previous termpower of x from previous termnumber of that term Example First term xquot5 Last term yquot5 Second coefficient 51 8 Third coefficient 542 10 Fourth coefficient 1033 10 Fifth coefficient 1024 5 PROJECT DEVELOPMENT MANAGER S PROBLEM A toys manufacturer intends to start development of new product lines A new toy is to be developed Development of this toy is tied with a new TV series with the same name There is 40 chance of TV series The production in such a case is estimated at 12000 units The Profit per toy would be Rs 2 Without TV seriessale there may be demand for 2000 units Already 500000 Rs has been invested A rival may bring to the market a similar toy If so the sale may be 8000 units The chance of rival bringing this toy to the market is 50 M The company has two choices Abandon new product Risk new development How should the company tie it all to nancial results Sample space Event The set of collection of all possible outcomes of an experiment is called the sample space Each possible outcome of an experiment is called event Thus an event is a subset of the sample space For example all six faces of a die make a sample space By rolling the dice occurrence of number 1 is an event Probability 271 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Probability is the numerical measure of the chance that an uncertain event will occur The probability that the event A will occur is usually denoted by pA The probability of any event must be between zero and one inclusive For any event A 0 S pA S 1 pA 1 means certain pA 0 means impossible PROBABILITY EXAMPLE 1 How can we make assessment of chances Look at a simple example A worker out of 600 gets a prize by lottery What is the chance of any one individual say Rashid being selected m Chance of any one individual say Rashid being selected 1600 The probability of the event quotRashid is selectedquot is the probability of an event occurringpRashid 1600 This is a priori method of finding probability as we can assess the probability before the event occurred 272 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU PROBABILITY EXAMPLE 2 When all outcomes are equally likely a priori probability is defined as pevent Number of ways that event can occurTotal number of possible outcomes If out of 600 persons 250 are women then the chance of a women being selected pwoman 250600 PROBABILITY EMPIRICAL APPROACH In many situations there is no prior knowledge to calculate probabilities What is the probability of a machine being defective Method 1 Monitor the machine over a period of time 2 Find out how many times it becomes defective This experimental or empirical approach EXPERIMENTAI AND THEORETICAL PROBABILITY pevent Number of times event occursTotal number of experiments Larger the number of experiments more accurate the estimate Experimental probability approaches theoretical probability as the number of experiments becomes very large m Consider two events A and B What is the probability of eitherA or B happening What is the probability ofA and B happening What is the number of possibilities Probability of A or B happening Number of ways A or B can happen Total number of possibilities Number of ways A can happen number of ways B can happen Total number of possibilities Or Number of ways A can happen Total number of possibilities Number of ways B can happen Total number of possibilities Probability ofA happening Probability of B happening Condition for Or Rule A and B must be mutually exclusive When A and B are mutually exclusive MA or B MN pB OR RULE EXAMPLE If a dice is thrown what is the chance of getting an even number or a number divisible by three peven 36 pdiv by 3 26 peven or div by 3 36 26 56 The number 6 is not mutually exclusive Hence Correct answer 46 AND RULE Probability ofA and B happening Probability ofA x Probability of B Example 273 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU In a factory 40 workforce are women Twenty five percent females are in management grade Thirty percent males are in management grade What is the probability that a worker selected is a women from management grade m pwoman chosen 25 25 females management grade 30 of males management grade pwoman amp Management grade pwoman x pmanagement Assume that the total workforce 100 pwoman 04 p management 025 pwoman x p management 04 x 025 01 or 10 SET OF MUTUALLY EXCLUSIVE EVENTS To cover all possibilities between mutually exclusive events add up all the probabilities Probabilities of all these events together add up to 1 PW I003 I0C I0N 1 EXHAUSTIVE EVENTS A happens orA does not happen then A and B are Exhaustive Events pA happens A does not happen 1 The sum of the probabilities of all mutually exclusive and collectively exhaustive events is always equal to 1 That is pA pB pC 1 if A B C are mutually exclusive and collectively exhaustive events 274 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 1 pyou pass 09 pyou fail 1 09 01 EXAMPLE1 EXHAUSTIVE EVENTS A production line uses 3 machines The Chance that 1st machine breaks down in any week is 110 The Chance for 2nd machine is 120 Chance of 3rd machine is 140What is the chance that at least one machine breaks down in any week w pat least one not working pall three working 1 pat least one not working 1 pall three working pall three working p1st working x p2nd working x p3rd working p1st working 1 p1st not working 1110 910 p2nd working 1920 p3rd working 3940 pall working 910 x 1920 x 3940 66698000 pat least 1 working 1 66698000 13318000 APPLICATION OF RULES A firm has the following rules When a worker comes late there is A chance that he is caught First time he is given a warning Second time he is dismissed What is the probability that a worker is late three times is not dismissed m Let us use the denominations 1C Probability of being Caught first time 1NC Probability of being Not Caught first time 2C Probability of being Caught 2nd time 2NC Probability of being Not Caught 2nd time 3C Probability of being Caught 3rd time 3NC Probability of being Not Caught 3rel time Probabilities of different events can be calculated by applying the AND Rule 1C14 amp 2C14 Dismissed 1 116 464 1C14 amp 2NC34 amp 3C14Dismissed 2364 1C14 amp 2NC34 amp 3NC34Not dismissed 1964 1NC34 amp 2C14 amp 3C14Dismissed 3364 1NC34 amp 2C14 amp 3NC34Not dismissed 2964 1NC34 amp 2NC34 amp 3C14Not dismissed 3964 1NC34 amp 2NC34 amp 3NC34Not dismissed 42764 pcaught first time but not the second or third time A x x 964 pcaught only on second occasion x A x 964 plate three times but not dismissed pnot dismissed 1 pnot dismissed 2 pnot dismissed 3 pnot dismissed 4 964 964 964 2764 5464 guqht using OR Rule pcaught pdismissed 1 pdismissed 2 pdismissed 3 464 364 364 1064 guqht and pnot caught using rule about Exhaustive events pnot caught 1pnot caught 1 1064 5464 275 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 LECTURE 36 ELEMENTARY PROBABILITY PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 35 0 Elementary Probability PROBABILITY CONCEPTS REVIEW Most of the material on probability theory along with examples was included in the handout for lecture 35 You are advised to refer to handout 35 Some of the concepts and examples have been further elaborated in this handout Probability means making assessment of chances The simplest example was the probability of Rashid getting the lottery when he was one of 600 The probability of the event was 1600 PERMUT EXAMPLE ln handout for lecture 35 we looked at the function PERMUT that can be used for calculations of permutations An example is shown in the slide kE Micrueuft Excel HunkI Elle Edit iew Insert Fgrmat eels Data indew Help DEQJE rm Evil iH vHHEEE BED 139 i5 e e r e E r e 1 PERMUTI1LIITIIJEI39I lLll l lel hUSEl l 3 Date Description 4 l Number bf bbjeete e 13 Number bf bbjeete in each permutetien e r PERMUTMA5 below 9 OR RULE REVIEW When two events are mutually exclusive the probability of either one of those occurring is the sum of individual probabilities This is the OR Rule This is a very extensively used rule A and B must be mutually exclusive The formula for the OR rule is as under MA or B MN pB Copyright Virtual University of Pakistan 276 Business Mathematics amp Statistics MTH 302 Example If a dice is thrown what is the chance of getting an odd number or a number divisble by three Podd 36 pdiv by 3 26 podd or div by 3 36 26 56 The number 6 is not mutually exclusive Hence correct answer 56 AND RULE REVIEW The AND Rule requires that the events occur simultaneously Example 60 workforce are men pman chosen 35 25 females management grade 30 of males management grade What is the probability that a worker selected is a man from management grade Example pman amp management grade pman x pmanagement Total workforce 100 pman 06 p management 03 pman x p management 06 x 03 018 or 18 SET OF MUTUALLY EXCLUSIVE EVENTS REVIEW Between them they cover all possibilitiesProbabilities of all these events together add up to 1 Exhaustive Events are events that happen or do not happen pit rains 09 pit does not rain 1 09 01 Example In Handout for lecture 35 we studied the problem of the three machines A production line uses 3 machines Chance that 1St machine breaks down in any week was 110 Chance for 2nd machine was 120 Chance of 3rd machine was 140 What is the chance that at least one machine breaks down in any week What are the probabilities Probability that one or two or three machines are not working in other words at least one not working and that all three areworking add up to 1 as exhaustive events Pat least one not working paII three working 1 From the above the probability that at least one is not working is worked out Pat least one not working 1 paII three working Now to work out the probability that all three are working we need to think in terms of machine 1 and machine 2 and machine 3 working This means application of the AND Rule paII three working p1st working x p2nd working x p3rd working Now the probability of machine 1 working is not known The probability that machine 1 is not working is given These two events working and not working are exhaustive events and add up to 1 Thus the event that machine 1 is working p1st working can be calculated as 1 p1st not working 1 110 910 The calculations for the other machines are p2nd working 1120 1920 p3rd working 1 140 3940 277 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Now the combined probability of pall working is a product of their individual probabilities using the AND Rule 910 X 1920 X 3940 66698000 Finally Pat least 1 working or 1 66698000 13318000 278 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 37 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 36 0 Patterns of Probability Binomial Poisson and Normal Distributions MODULE 7 Module 7 covers the following Factorials Permutations and Combinations Lecture 34 Elementary Probability Lectures 35 36 Patterns of probability Binomial Poisson and Normal Distributions Pan14 Lectures 37 40 MODULE 8 Module 8 covers the following Estimating from Samples Inference Lectures 41 42 Hypothesis Testing ChiSquare Distribution Lectures 43 44 Planning Production Levels Linear Programming Lecture 45 Assignment Module 7 8 EndTerm Examination EXAMPLE 1 We covered in the past two lectures Elementary Probability Most of the material was included in Handout 35 Some questions were discussed in detail in handout 36 In lecture 37 the example where the employee was warned on coming late and dismissed if late twice will be discussed The material for this example is given in handout 35 Here we shall cover the main points and the method A firm has the following rules When a worker comes late there is A chance that he is caught First time he is given a warning Second time he is dismissed What is the probability that a worker is late three times is not dismissed m How do we solve a problem of this nature The answer is to develop the different options first Let us see how it can be done First time There are two options Caught 1C Not Caught 1NC 2nd time Caught 2C Not Caught 2NC 3ml time Caught 3C Not Caught 3NC 279 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Look at combinations up to 239quot1 stage 1CampZC 1Camp 2NC 1NCamp 2C 1NCamp 2NC Look at combinations unto 3ml stage 1Camp20amp3C 1C amp 2NC amp 3C 1C amp 2NC amp 3NC 1NC amp 2C amp 3C 1NC amp 2C amp 3NC 1NC amp 2NC amp 3C 1NC amp 2NC amp 3NC You saw that the first case is 1C amp 2C Here the employee was caught twice and was dismissed He can not continue Hence this case was closed here In other cases the combinations were as given above Now the probability of being caught was A As an exhaustive event the probability of not being caught was 1 A Now the probabilities can be calculated as follows 1C amp 2C 14X14 116 1C amp 2NC amp 3C 14X 34X14 364 1C amp 2NC amp 3NC 14X34X34 964 1NC amp 2C amp 3C 34x14x14 364 1NC amp 2C amp 3NC 34x14X34 964 1NC amp 2NC amp 3C 34x34x14 964 1NC amp 2NC amp 3NC 34x34x34 2764 The probabilities for each combination of events are now summarized below First Caught Second Caught Dismissed 10 14 amp 20 14 Dismissed 1 116 464 First caught Second Not Caught 3rel Caught Dismissed 10 14 amp 2NC 34 amp 30 14 Dismissed 2 364 First caught Second Not Caught 3rel Not Caught Not Dismissed 10 14 amp 2NC 34 amp 3NC 34 Not dismissed 1 964 First Not Caught Second Caught 3rd Caught Dismissed 1NC 34 amp 20 14 amp 30 14 Dismissed 3 364 First Not caught Second Caught 3rd Not Caught Not Dismissed 1NC 34 amp 20 14 amp3NC 34 Not dismissed 2 964 First caught Second Not Caught 3rel Caught Not Dismissed 1NC 34 amp 2NC 34 amp 30 14 Not dismissed 3 964 First caught Second Not Caught 3rd Not Caught Not Dismissed 1NC 34 amp 2NC 34 amp 3NC 34 Not dismissed 4 2764 Probabilities pcaught The probability of being caught can be calculated by thinking that these are mutually events All situations where there was a dismissal can be considered Probabilitycaught pdismissed 1 pdismissed 2 pdismissed 3 464 364 364 1064 280 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU pnot caught Once we have the probability of being caught we can find out the probability of not being caught as an exhaustive event Thus pnot caught 1 pcaught 1 1064 5464 EXAMPLE 2 Two firms compete for contracts A has probability of 3A of obtaining one contract B has probability of A What is the probability that when they bid for two contracts firm A will obtain either the first or second contract m PA gets first or A gets second 64 Wrong Probability greater than 1 We ignored the restriction events must be mutually exclusive We are looking for probability that A gains the first or second or both We are not interested in B getting both the contracts pB gets first x pB gets both A x A 116 pA gets one or both 1 116 1516 Alternative Method Split quotA gets first or the second or bothquot into 3 parts A gets first but not second 3A x A 316 A does not get first but gets second A x 316 A gets both x 916 PA gets first or second or both 316 316 916 1516 EXAMPLE 3 In a factory 40 workforce is female 25 females belong to the management cadre 30 males are from management cadre lf management grade worker is selected what is the probability that it is a female Draw up a table first 281 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Male Female Total Management NonManagement Total 40 100 Calculate Total male 100 40 60 Management female 025 x 40 10 NonManagement female 40 10 30 Management male 03 x 60 18 NonManagement male 60 18 42 Management total 18 10 28 NonManagement total 42 30 72 m Male Female Total Management 18 10 28 NonManagement 42 30 72 Total 60 40 100 pmanagement grade worker is female 1028 EXAMPLE 4 A pie vendor has collected data over sale of pies This data is organized as follows No Pies sold lncomeX Daysf fX Rs 40 x 35 1400 20 28000 50 1750 20 35000 60 2100 30 63000 70 2450 20 49000 80 2800 10 28000 Total 100 203000 Meanday 203000100 2030 The selling price per pie was Rs 35 What was the mean sale per day Such a question can be solved by calculating the sale in each slab and then dividing the total sale by number of pies days is the probability lf multiplied with the income from each pie the expected sale from all pies can be calculated The overall expected value was 203000 When divided by the number of days 100 an average of 2030 Rs Per day was obtained as average sale per day EXPECTED VALUE EMV Z probability of outcome x financial result of outcome Example In an insurance company 80 of the policies have no claim In 15 cases the Claim is 5000 Rs For the remaining 5 the Claim is 50000 Rs What is the Expected value of claim per policy Applying the formula above EMV 08 x 0 015 x 5000 005 x 50000 0 750 2500 3250 Rs 282 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU TYPICAL PRODUCTION PROBEM In a factory producing biscuits the packing machine breaks 1 biscuit out of twenty p 120 005 What proportion of boxes will contain more than 3 broken biscuits This is a typical Binomial probability situation The individual biscuit is broken or not two possible outcomes Conditions for Binomial Situation 1 Either or situation 2 Number of trials n known and fixed 3 Probability for success on each trial p is known and fixed CUMULATIVE BINOMIAL PROBABILITIES The Cumulative Probability table gives the probability of r or more successes in n trials with the probability p of success in one trial In the table The total number of trials n 1 to 10 The number of successes r 1 to 10 The probability p 005 01 02 025 03 035 04 045 05 283 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 38 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 37 0 Patterns of Probability Binomial Poisson and Normal Distributions CUMULATIVE BINOMIAL PROBABILITIES Probability of r or more successes in n trials with the probability of success in each trial Look in column for n Look in column for r Look at column for value of p005 to 05 In other words a cumulative binomial probability refers to the probability that the binomial random variable falls within a specified range eg is greater than or equal to a stated lower limit and less than or equal to a stated upper limit For example we might be interested in the cumulative binomial probability of obtaining 45 or fewer heads in 100 tosses of a coin see Example 1 below This would be the sum of all these individual binomial probabilities bxg45 100 05 bx 0 100 05 bx 1 100 05 bx 44 100 05 bx 45 100 05 x n You can calculate these values by using the formula Px S c ij 1 P H 620 x Or directly from the table Example The probability that a student is accepted to a prestigious college is 03 If 5 students from the same school apply what is the probability that at most 2 are accepted m To solve this problem we compute 3 individual probabilities Using the binomial formula The sum of all these probabilities is the answer we seek Thus 284 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU bx 5 2 5 03 0837 Table Z t tttil ti Eirmltiial 111 0l111bliiii 55 17 139 H P 1139 1 11 E ii Lain 1x 13quot E 111121 111I1 171111 113131 11411 1511 111111 1111 113131 111211 H115 nl 11 12335112 11912121 11311112 LTtil LEEDS ISLEIIIIII ISL ltIZIII 1111quot ELEVII 111IIIII1 11115131 1 11312111 1111E 11211112 1111111 11211111 11111 1111111 1tIIIIIIII 1111111 lIIIIIIII1 1111111E 313 11 111111 1131121 111411 H410 13611 011111 III l III 11115 IEll 1111111 1111113 1 1311 119 tit1 DELI ISLE111 LEI ELIEll 11110 IIL1ISIII 111911 111193 1 11113111 111E 11213111 1112110 1111111 1111111 HitII 1tIIIIIIII 111110 lIIII111 11111 E 313 11 LEST 1511 H143 H116 H115 IltiISl 111111 ISLE2115 11IIII11 1311111131 1 113211 131911 1131 LTE4 III lE ISLEIZIIZI H151 11116 01114 111113 13111111 1 1121111 11932 111211 I 73 01115 0315 0134 1155 ISL135 11111 131143 3 11312111 1111E 11211112 1111111 11211111 11111 1111111 112IIIIII 1111111 lIIIIIIII1 1111111E i4 11 111127 111555 14111 Illll 11111 ISLE163 111115 111105 ISLE2111 1111011 1311111131 1 115136 131941 113111 01551 I143 H111 Il39lTE 1111134 ISLE1 111111 13111 1 1121111 1319 11111 01116 IILEEfl USES 17112 11143 IIL lE1 1111112 1311114 3 11313111 111E 03111 DEFl 03115 LETIII 11ISIII IILEE39III 11144 171135 4 11312111 1111E 11211112 1111111 11211111 11111 1111111 1tIIIIIIII 1111111 lIIIIIIII1 11111111 11 1114 1315 1313 H165 113113 011111 01110 111101 Eli2110 11III111 13111E 1 1531quot 131311quot 113 0515 I131quot 1155 ISLEIE 1111111 Eli211 1111011 1311111131 P 1133 11991 1941 ELSEE ISLEIIIIII I111 111161 ISLE155 11IIII15 11111111 3 11213111 111E 1111 01165 1911 LEE 0661 11411 01151 111151 111123 BINOMDIST Returns the individual term binomial distribution probability Use BINOMDIST in problems with a fixed number of tests or trials when the outcomes of any trial are only success or failure when trials are independent and when the probability of success is constant throughout the experiment For example BINOMDIST can calculate the probability that two of the next three babies born are male Syntax BlNOMDISTnumberstrialsprobabilityscumulative Numbers is the number of successes in trials Trials is the number of independent trials Probabilitys is the probability of success on each trial Cumulative is a logical value that determines the form of the function If cumulative is TRUE then BINOMDIST returns the cumulative distribution function which is the probability that there are at most numbers successes if FALSE it returns the probability mass function which is the probability that there are numbers successes Remarks Numbers and trials are truncated to integers lf numbers trials or probabilitys is nonnumeric BINOMDIST returns the VALUE error value If numbers lt 0 or numbers gt trials BINOMDIST returns the NUM error value If probabilitys lt 0 or probabilitys gt 1 BINOMDIST returns the NUM error value 285 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The binomial probability mass function is I I sass a s where as Js39 a The cumulativehbinomial distribution is Sines Zbis s 4 Iii39339 Micrusnft Excel Lecturej Eile Edit Eiew insert Fgrmat eels gate induw elp 3 Er 7 E Q i E sun v x a s slscuulsrrsssasssasa A E 0 II E 1 BINMDISTnumberstrialsprbabilityscumulatiue 2 Data Descriptin s BlNumber f successes in trials 4 10 Number f independent trials 5 0539Prbability f success n each trial BINMDISTlA3A4A5FALSE a Prbability f exactly 6 f 10 trials 9 being successful 0205078 1 Example In the above example the BINOMDIST function was used to calculate the probability of exact 6 out of 10 trials being successful Here the value of Cumulative was set as False The following example also shows a similar calculation 286 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE Micrusuft Excel Lecture33 Eile Edit ew insert Fgrmat Innls gate indnw Help Eggs itav ma fluv uggss i ass 2 s a E c 12 EXAMPLE 22 Five sins are tssed simultaneusly 21 What is the chance f btaining heads 22 22 Data Descriptin 22 Number f successes in trials 22 5 Number f independent trials 22 05 Prbability f success n each trial 2 22 03125 22 Prlsalsility f exactly f 5 trials 22 being successful 03125 31 EXAMPLE 1 The probability of wet days is 60 Note that the figure 06 is beyond the maximum value 05 as given in the tables Let us first convert our problem to pdry 1 06 04 Now p5 or more wet days can be restated as p2 or less dry days The BINOMDIST function is for pr or more Let us convert p2 or less dry days to 1 p3 or more days Now the value of n 7 r 3 and p 04 Using BINOMDIST the answer is 04199 Note that the value of cumulative was TRUE 287 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E P icrusuft Excel Lecture3 Eile Edit Eiew insert Fgrmat eels gate indnw Help De ie g t e v v ezveememmw Ariel vIDvBIHEEEEE E gfdgfg D419 139 fa A El I3 35 EXAMPLE 344 Prebability ef wet deye in current menth 34 Prebebility ef er mere wet deye next week 34 p The teblee are fer p upte 39 Turn the queetien Pljdry 1 04 4D pf er mere wet days 7 er leee dry days 44 p er leee dry days 1 pj 3 er mere dry days 42 Date Deeeriptien 43 2 Number ef eueeeeeee in triele 44 739 Number 11quot independent trials 45 04 Prebebility ef eueeeee en each trial 45 BNOMDISTfA43A44A45TRUE 44 At meet eueeeeeee 1 EXAMPLE 2 In a transmission where 8 bit message is transmitted electronically there is 10 probability of one bit being transmitted erroneously What is the chance that entire message is transmitted correctly We can state that the probability required is for 0 successes errors in 8 trials bits pone bit transmitted erroneously 01 PX0 8 p01 0430 For exact binomial distribution n x n x 8 0 8 0 Pxn P1 P 0 011 01 0430 x 288 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU T0010 30111111011120 011101111101 15101301511003 Emmimed c 005 000 030 040 050 000 000 000 000 005 0003 0100 0050 0010 0004 0001 0000 0000 0000 0000 0043 0503 0255 0100 0035 0000 0001 0000 0000 0000 0004 0003 0000 0552 0315 0145 0050 0011 0001 0000 0000 1000 0005 0044 0000 0504 0503 01004 0050 0010 0000 0000 1000 1000 0000 0042 0020 0030 0400 0104 0050 0005 0000 5 1000 1000 0000 0000 0050 0055 0005 0440 0303 0030 0000 0 1000 1000 1000 0000 0001 0005 0004 0045 0400 0100 0050 0 1000 1000 1000 1000 0000 0000 0003 0043 0031 0500 0530 0 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 n 0 0 0030 03300 0134 0040 0010 0002 0000 0000 0000 0000 0000 1 0020 0005 0430 0100 0001 0000 0004 0000 0000 0000 0000 2 0002 0041 0030 0403 0233 0000 0025 0004 0000 0000 0000 5 0000 0002 0014 0030 0403 0254 0000 0035 0003 0000 0000 4 1000 0000 0000 0001 0033 0500 0200 0000 0000 0001 0000 5 1000 1000 0000 0005 0001 0040 0510 0000 0000 0000 0001 0 1000 1000 1000 0000 0005 0010 0000 0530 0302 0053 0000 0 1000 1000 1000 1000 0000 0000 0000 0004 0504 0205 0001 r7 050 1100 in i N a 1 20 ii 01 51 Using BINOMDIST The data was for 0 or more successes BINOMDIST function gives the value for at most r successes Hence the answer was obtained directly 289 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejlicrnsnft Excel Lecturej Elle Edit Eiew insert Fgrrnat Innis gate window Help assesses rm gEv lii i jmnaT Mal 1D39BIHEEE T63E3 E ElEri v 5 A 51 EXHMPLE 52 Prbability f1 errneus bit 01 53 Prbability f 8 crrect bits l errneus hits a Data Descriptin 55 1 Number f successes in trials as 8 Number f independent trials 5 01 Prbability f success n each trial as 041305 BINMDISTBS B EB 7TRUE as M mst U successes 04305 El a E El EXAMPLE 3 A surgery is successful for 75 patients What is the probability of its success in at least 7 cases out of randomly selected 9 patients psuccess in at least 7 cases in randomly selected 9 patients Here n 9 psuccess 075 pat lease 7 cases p 075 is outside the table Let us invert the problem pfailure 1 075 025 Success at least 7 Failure 2 or less Pxgt7 n9p075 1 pxlt7 n9p075 1 pxlt6n9p075 1 03995 06005 60 290 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 i 003 0003 1 0943 3 0994 3 1000 4 1000 5 1000 0 1000 3 1000 3 1000 0030 1 0939 3 0992 3 0999 4 1000 5 1000 1000 3 1000 3 1000 010 0430 0313 0903 0993 1000 1000 1000 1000 1000 033 03 094 0993 0999 1000 1000 1000 1000 030 0103 0503 039 0944 0990 0999 1000 1000 1000 0134 0430 0333 0914 0930 099 1000 1000 1000 Qlculgtion usinq BINOMDIST Here the question was inverted We had to find 7 successes out of 9 The probability was 75 for success It becomes 030 003 3 0130 0532 0300 0942 0939 0999 1000 1000 0040 0190 0403 030 0901 093 0990 1000 1000 040 001 0100 0313 0394 0330 0950 0991 0999 1000 0010 001 0233 0433 033 0901 093 0990 1000 P 050 0004 0030 0140 0303 003 035quot 0903 0990 1000 0002 0030 0090 0354 0500 0340 0910 0930 0993 000 0001 0009 0050 0134 0400 0035 0394 0933 1000 25 for failure Now let us restate the problem in terms of failure We are interested in 7 or more successes It means 2 or less failures Now the BINOMDIST function gives us at most r successes In other words 2 or less Hence if we specify r 2 we get the answer 06007 directly 0000 0001 0011 0053 0194 04743 0343 0943 1000 0000 0000 0004 0033 0099 030 0000 0000 0001 0010 0030 0303 049 0333 1000 0300 090 0000 0000 0000 0000 0003 0033 013 0330 1000 0000 0000 0000 0000 0001 0003 01 31935 093 0000 0000 0000 0000 0000 0000 0105 033 1000 0000 0000 0000 0000 0000 0001 0003 001 030 291 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EJhicrusuft Excel Lecturej Elle Edit Eiew insert Fgrmat Innis Qatar induw Help 139 51 near sea a a m a sltl aamnro a Fll39ial vl v f ggg ga3 3 amp 234 v 5 A E C D E as EXAMPLE rr Prbability f success 075 Failure 025 ra Prbalaility f 7 successes ut f 9 r 2 r less failures F9 an Data Descriptin m 2 Number f successes in trials as 9 Number f independent trials as 025 Prbability f success n each trial a 16007 BINMDISTBB1BB2383TRUE as At mst 2 successes 06007 SE 292 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NEGATIVE BINOMIAL DISTRIBUTION A negative binomial experiment is a statistical experiment that has the following properties The experiment consists of X repeated trials Each trial can result in just two possible outcomes We call one of these outcomes a success and the other a failure The probability of success denoted by P is the same on every trial The trials are independent that is the outcome on one trial does not affect the outcome on other trials The experiment continues until rsuccesses are observed where r is specified in advance Consider the following statistical experiment You flip a coin repeatedly and count the number of times the coin lands on heads You continue flipping the coin until it has landed 5 times on heads This is a negative binomial experiment because The experiment consists of repeated trials We flip a coin repeatedly until it has landed 5 times on heads Each trial can result in just two possible outcomes heads or tails The probability of success is constant 05 on every trial The trials are independent that is getting heads on one trial does not affect whether we get heads on other trials The experiment continues until a fixed number of successes have occurred in this case 5 heads Negative Binomial Formula Suppose a negative binomial experiment consists of xtrials and results in r successes If the probability of success on an individual trial is P then the negative binomial probability is W6 r F x1cr1 Pr 1 P Example Bob is a high school basketball player He is a 70 free throw shooter That means his probability of making a free throw is 070 During the season What is the probability that Bob makes his first free throw on his fifth shot 30111 170quot This is an example of a geometric distribution which is a special case of a negative binomial distribution Therefore this problem can be solved using the negative binomial formula or the geometric formula We demonstrate each approach below beginning with the negative binomial formula The probability of success P is 070 the number of trials X is 5 and the number of successes r is 1 We enter these values into the negative binomial formula 293 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU bX r X1Cr1 Pr QXr b5 1 07 400 071 034 b5 3 07 000567 NEGBINOMDIST Returns the negative binomial distribution NEGBINOMDIST returns the probability that there will be numberf failures before the numbersth success when the constant probability of a success is probabilitys This function is similar to the binomial distribution except that the number of successes is fixed and the number of trials is variable Like the binomial trials are assumed to be independent For example you need to find 10 people with excellent reflexes and you know the probability that a candidate has these qualifications is 03 NEGBINOMDIST calculates the probability that you will interview a certain number of unqualified candidates before finding all 10 qualified candidates Syntax NEGBINOMDISTnumberfnumbersprobabilitys Numberf is the number of failures Numbers is the threshold number of successes Probabilitys is the probability of a success Remarks 0 Numberf and numbers are truncated to integers o If any argument is nonnumeric NEGBINOMDIST returns the VALUE error value o If probabilitys lt 0 or if probability gt 1 NEGBINOMDIST returns the NUM error value o If numberf numbers 1 S 0 NEGBINOMDIST returns the NUM error value 0 The equation for the negative binomial distribution is xr1 H I with 1 P 1 P where x is numberf r is numbers and p is probabilitys 294 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NEGBLNOMDIST EXAMPLE You need to find 10 people with excellent reflexes and you know the probability that a candidate has these qualifications is 03 NEGBINOMDIST calculates the probability that you will interview a certain number of unqualified candidates before finding all 10 qualified candidates lilicreeeft Excel Lecture Elle Edit iew insert Fgrmet eels gate window Help T D g rg r Evsl mmv ueeetset BEE 1r f e s c u E 11 NEGBINMDISTlnumberj numbersprbabilitys 12 Date Descriptin 3 10 Number f failures 4 5 Threshld number f successes 5 025 Prbebility f success a 1r Negative binmiel distributin fr the 1E terms shire 0055040 19 m CRITBINOM Returns the smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value Use this function for quality assurance applications For example use CRITBINOM to determine the greatest number of defective parts that are allowed to come off an assembly line run without rejecting the entire lot Syntax CRITBINOMtriaIsprobabilitysalpha Trials is the number of Bernoulli trials Probabilitys is the probability of a success on each trial Alpha is the criterion value Remarks If any argument is nonnumeric CRITBINOM returns the VALUE error value If trials is not an integer it is truncated lf trials lt 0 CRITBINOM returns the NUM error value If probabilitys is lt 0 or probabilitys gt 1 CRITBINOM returns the NUM error value If alpha lt 0 or alpha gt 1 CRITBINOM returns the NUM error value i i i i Example A B 1 Data Description 2 6 Number of Bernoulli trials 3 05 Probability of a 295 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 4 075 Formula CRITBNOMA2A3A4 success on each trial Criterion value Description Result Smallest value for which the cumulative binomial distribution is greater than or equal to a criterion value 4 296 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 39 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 3 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 38 0 Patterns of Probability Binomial Poisson and Normal Distributions CRITBINOM EXAMPLE The example shown under CRITBINOM in Handout 38 is shown below Microsoft Excel Lecture3 Eile Edit Eiew losert Format Iools gate window Help v infcig 39 739 3 1 E SUM v X of f3 CRITEiNDM1 EB EEtAFDj A I El es CRITBINMtrialsprbabilitysalphaj EE 5 Data Description 53 Number of Bernoulli trials 05 Prbability f a success h each 5E trial a U75Criterin value Smallest value fr which the cumulative e CRITBIN binomial distribution is greater than a WAE or equal to a criterion value 4 e A69ATU as EXPECTED VALUE EXAMPLE A lottery has 100 Rs Payout on average 20 turns Is it worthwhile to buy the lottery if the ticket price is 10 Rs Expected win per turn pwinning x gain per win plosing x loss if you loose 120x 100 10 1920x 10 Rs 9020 19020 Rs 45955Rs So on an average you stand to loose 5 Rs DECISION TABLES Look at the data in the table below No of Pies demanded Occasions 25 10 30 20 35 25 40 20 45 15 50 10 297 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Price per pie Rs 15 Refund on return Rs 5 Sale price Rs 25 Profit per pie Rs 25 15 Rs 10 Loss on each return Rs 15 5 Rs 10 How many pies should be bought for best profit To solve such a problem a decision table is set up as shown below The values in the first column are number of pies to be purchased Figures in columns are the sale with share of sale within brackets If the number of pies bought is less than the number that can be sold the number of pies sold remains constant If the number of pies bought exceeds the number of pies sold then the remaining are returned This means a loss For every value the sum of profit for sale and loss for pies returned is calculated The average sale for each row is calculated by multiplying the profit for each sale with sale in the column An example calculation is given as a guide for 30 pies DECISION TABLES 2501 3002 35025 4002 45015 5001 EMV 25 250 250 250 250 250 250 250 30 200 300 300 300 300 300 290 35 150 250 350 350 350 350 310 Buy 40 100 200 300 400 400 400 305 45 50 150 250 350 450 450 280 0 100 200 300 400 500 240 Expected profit 30 Dies 01x 200 02 x 300 025 x 300 02 x 300 015 x 300 01x 300 206075604530 290 Rs Best Profit It may be noted that the best profit is for 35 Pies Rs 310 DECISION TREE TOY MANUFACTURING CASE The problem of the manufacturer intending to start manufacturing a new toy under the conditions that the TV series may or may not appear that the rival may or may not sell a similar toy is now solved below Here a Decision tree has been developed with the possible branches as shown below Each sequence represents an application of the AND rule 1A Abandon 18 Go ahead gt2A Series appears 60 gt28 No series 40 gt2Agt3A Rival markets 50 gt2Agt3B No Rival 50 Production Series no rival 12000 units Series rival 8000 units No series 2000 units Investment Rs 500000 Profit per unit Rs 200 Loss if abandon Rs 500000 What is the best course of action 298 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Decision Tree Profit if rival markets series appears 8000 x 200 500000 1600000 500000 1100000 Rs Profit if no rivals 12000 x 200 500000 2400000 500000 1900000 Rs ProfitLoss if no series 2000 x 200 500000 400000 500000 100000 Rs No series EMV Rival markets and no rivals 05 x 1100000 05 x 1900000 1500000 Series EMV 06 x 1500000 04 x 100000 900000 40000 860000 Rs Conclusion It is clear that in spite of the uncertainty there is a likelihood of a reasonable profit Hence the conclusion is Go ahead THE POISSON DISTRIBUTION The Poisson distribution is most commonly used to model the number of random occurrences of some phenomenon in a speci ed unit of space or time For example 0 The number of phone calls received by a telephone operator in a 10minute period 0 The number of aws in a bolt of fabric 0 The number of typos per page made by a secretary It has the following characteristics Either or situation No data on trials No data on successes Average or mean value of successes or failures This is a typical Poisson Situation Characteristics Eitheror situation Mean number of successes per unit m known and fixed p chance unknown but small event is unusual For a Poisson random variable the probability that X is some value X is given by the formula x PXx ue39 x0l X where J is the average number of occurrences in the speci ed interval For the Poisson distribution EX VarX Example 299 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The number of false re alarms in a suburb of Houston averages 21 per day Assuming that a Poisson distribution is appropriate the probability that 4 false alarms will occur on a given day is given by 2 146 21 PX 4 T 00992 THE POISSON TABLES OF PROBABILITIES Gives cumulative probability of r or more successes Knowledge of m is required Table gives the probability of that r or more random events are contained in an interval when the average number of events per interval is m Example 2 Attendance in a factory shows 7 absences What is the probability that on a given day there will be more than 8 people absent m Method 1 PX gt81 PX 8l PxlPx2Px3Px4Px5 Px6Px7Px8 7161 7262 7363 7464 7565 7666 7767 7868 1 l 2 3 4 5 l 7 7 l 00064002230052l009 l20 l2770 1490 1490 1304 02709 Method 2 PX gt 8 l PX S 8 l0729l 02709 3 00 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU p9 or more successes 02709 Iif3333333i3ti3te Pnieenn Dietrihntinn T3333 Table 31333333 ennntletit39e 13r3h3l3iiit fnnetiene 3f P3333333 Dietrihentien with 33333313 339 E33333 13qie t3 nd the 13333133hiiit33 if where here 3 Pnieeon Distribution with 339 2 13333 in r3w 3 3333 311333333 3 t3 nd if 33085 1 where i3 P33333332 3 33 3 33 3 3 33 3 33 3 33 3 33 b 33 3 3 33 33 3 33333 33333 33333 33333 33333 33333 33333 33333 33333 33333 3 33333 33333 333 33 1 33333 33333 33333 333 33333 33333 33333 33333 33333 33333 53 33333 33333 33333 33 33333 33333 33333 33333 33333 33333 3 33333 33333 33333 33133 33333 33333 33333 33333 33333 3333 Example 3 An automatic production line breaks down every 2 hours Special production requires uninterrupted operation for 8 hours What is the probability that this can be achieved m a 82 4 x 0 no breakdown 0 0 p x0 00183 1 83 0 From Table 301 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Chitiiulatite Peleebn Dietributibh Table Table ahawe cumulative parbbabilit le tjti I lE bf Paiaaen Dietributieh with variants rt Exam ple tb nd the prebabilitgr PEX if where X haa a Paiaaen Dietributibh with rt 2 lablt in raw 4 and mlumn 4 tb nd if 3U85T1 where ia Pbaieabaii j Q at i la a a 35 4L at a Elu l II 3 1337 II 23 1 II 1 35 3 13213132 1 EIUIHQE II393III392 iEi1l 3 3 ll l l l 1 El IJ EIIE T EII E IZIIQE393 InT3rir3 II SHE 414113161 In33T3 b1991 411359 illfligll lufli ll ll lanai b9353 LB 1939 b3383 Ill T T In3433 b4333 Ill33EI398 233 l 1321733 El 12439 b9933 3931i 9344 Illt CiTll LT5713 IEIh LTE II 531313 b4335 1313433 Ell 2135i b9993 1319963 Kilt31314 419433 1313912 1313 153 II T354 1162233 b5331 creative 1El39llllililit II Q SELL II 9955 ID 983 4 II 958i lug lift 1 IIJEETE i 735 l 117132 9 DUB l i ll U39EIIIEIEIJ II 9 991339 II 139939 ll ID 93995 5 II 985 3 lug E363 5 3934 II i 3 3 l l El T Ilfi 3 3 ll IEIIIEIIIEIEIJ Il IIJIIJIIJII i 9998 ID 9 93 9 II 995 3 lug 38 ll il39ElT33 i Q43 9 LE1 Il U39 3 131313 a HL lJt a 39lll39ll7irimtt39 LLHR t mmm a mmn a FLA t nan t FEE45 Example 4 An automatic packing machine produces on an average one in 100 underweight bags What is the probability that 500 bags contain less than three underweight bags Solution m 1x500100 5 pxlt3 0 5 pxlt2 0 5 012471247 Chitiiulatitre Pbieebh Dietributibh Table Table ahbwa euhiulatiwe prbbabilit funetibha emf PtJIlSEDIH Dietribtitiah with rarieua rt Exam ple tb nd the probability if where haa a Peiaabrt Dietributibh with be 2 lbbl in raw 4 and ebluirih 4 ta nd if 3 85 l where ia Peieabh CI ml at i 15 a a 35 a 45 C 5 aetiaa aaara aaaai aiaaa aaaai amaa abate ataaa aaiii timer agree araaa aaara aaaaa aaara b1991 aiaaa abate aaaii errata agate b9113 aeriaa aerer aaaaa aaaaa aaaaa aaaai airaa aaaaa again b9344 man were were aaaee aaaaa aaaaa aaaaa aaaaa aaaea t19814 aaara aaaia aaiaa araaa aaaaa aaaai Haiti retire 4119an aaaaa aaaaa traaaa aaiei aaare arear artea aeiea retire aaaaa aaaai agate aaaaa aaaea aaaar aaaaa aaaii areaa inane their aaaaa aaeaa aaaaa b9381 aaraa aaaaa 1191134 aaaea were that iiith aaaaa aaaaa aaaaa again aaraa uaaar aaaia retire termini were three aaaar 399813 aaaar aaaia tiaaaa aaaaa 39E i i iim ri b gl aiei 302 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 40 PATTERNS OF PROBABILITY BINOMIAL POISSON AND NORMAL DISTRIBUTIONS PART 4 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 39 0 Patterns of Probability Binomial Poisson and Normal Distributions Part 4 POISSON WORKSHET FUNCTION Returns the Poisson distribution A common application of the Poisson distribution is predicting the number of events over a specific time such as the number of cars arriving at a toll plaza in 1 minute Syntax POISSONxmeancumulative X is the number of events Mean is the expected numeric value Cumulative is a logical value that determines the form of the probability distribution returned If cumulative is TRUE POISSON returns the cumulative Poisson probability that the number of random events occurring will be between zero and x inclusive if FALSE it returns the Poisson probability mass function that the number of events occurring will be exactly x Remarks 0 Ifx is not an integer it is truncated Ifx or mean is nonnumeric POISSON returns the VALUEI error value If x S 0 POISSON returns the NUM error value If mean S 0 POISSON returns the NUM error value POISSON is calculated as follows For cumulative FALSE 1 xii PGLS SGN I For cumulative FJIALSE 1 E A I CUMPGISSGN Z irI If Example An application of the POISSON function is shown below In this slide the value of Cumulative was TRUE It means that the probability is for at the most case 3 03 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Ejl icreseft Excel Lecture39 Eile Edit iew Lnsert Fgrmat eels gate indew Help if cit IE v r v s v 41 e SLIM v x a s F39DISSDHEA3MTRLJEJ A El 3 D E 1 PISSNxmeeneumuletiue 2 Date Deserintin 3 2 Number f events 4 5 Expected mean 5 s PIS Cumulative Pissn pirbeleility with Sm the terms slave 012d652 3 e3 1 M 12 In the slide below the Cumulative is FALSE which means that the probability is for ectl 2 events Ejiliiicrueeft Excel Lecture E Eile Edit iew insert Fgrmat eels gate window elp 29 it s m 2 v e SLIM 1 X J 52 PDISSDNWFA IEFALSEJ A e t D E 15 PISSNmmean umuletive 15 Date Descriptin 1 2 Number f events W SHExpected mean 19 SPISSM Cumulative Pissn prbebility with 21 MEMB the terms alive 0384224337 3 FALSE 24 3 04 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE PATTERN In Binomial and Poisson the situations are eitheror Number of times could be counted In the Candy problem with underweight boxes there is measurement of weight Binomial and Poisson are discrete probability distributions Candy problem is a Continuous probability distribution Such problems need a different treatment FREQUENCY BY W GHT Look at the frequency distribution of weight of sample bags Microsoft Excel Lecture Elle Edit iew lnsert Fgrrnal Iools Qatar window elp De neaavt lt3 azv irinaluuorove trial vIDvBIH tutitt F3r v 5 A E C D E F G H J 23 24 FREQUENCY BY WEIGHT No of bags 25 5013 but under 505 2 25 505 but under 507 12 2 507 but under 509 21 23 509 but under 511 29 29 511 but under 5113 an 513 but under 515 11 31 515 but under 517 2 32 Frequency distribution graph of the sample is shown below You may see a distinct shape in the graph It appears to be symmetrical I I I FEEUENC V DIETEIEUTIN El Series l I Series a El Series I I El Seriesdl E l E E E l SeriesE I 1 39U 1 E g g H E r g Ln EISenesE g E E E E E E lSeriesf L 3 E El Seriesa E m E E m Weight I I I 3 05 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU The shape of the distribution is that of a Normal Distribution as shown as New distribution in the slide below On this slide you also see a Standard Normal Distribution with 0 mean and standard deviations 1 2 3 4 etc Standard distribution New distributidn 43543212D 12345l3 3 06 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NORMAL DISTRIBUTION The blue Curve is a typical Normal Distribution A standard normal distribution is a distribution with mean 0 and standard deviation 1 The Yaxis gives the probability values The Xaxis gives the 2 measurement values Each point on the curve corresponds to the probability p that a measurement will yield a particular 2 value value on the xaxis Probability is a number from 0 to 1 Percentage probabilities multiply p by 100 Area under the curve must be one Note how the probability is essentially zero for any value 2 that is greater than 3 standard deviations away from the mean on either side Mean gives the peak of the curve Standard deviation gives the spread Weiqht distribution Gag Mean 510 g StDev 25 g What proportion of bags weighs more than 515 g Proportion of area under the curve to the right of 515 g gives this probability 3 07 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EA UNER THE STANDARD NOFIMAI CURVE The normal distribution table gives the area under one tail only zvalue Ranges between 0 and 4 in first column Ranges between 0 and 009 in other columns Example Find area under one tail for zvalue of 205 Look in column 1 Find 20 Look in column 005 and go to intersection of 20 and 005 32233 2323 222 33302 Ell033 31313 ELI Els 313329 33333 33333 33333 33333 333333333 3333 33333 32 ll 303323 32512 0552 3063935 2323 0325 2234 2253 332 32322 223 0932 2222 223 1033 E Hilfi 342 til3 322 1222 1255 3223 2332 2353 4326 l342 3423 314 1554 152 3322 l r W 39 2236 11222 322 2244 1229 35 1915 3250 3225 239 2053 2032 2 22 2 3 52 2 323339 2223l 33 2252 2221 2323 2352 2322 2322 2354 2423 25 2 2542 122 2523 2331 l 2332 2323 2 2233 2223 2234 2233 2223 2352 322 2221 2912 2922 2932 2225 33222 225 33222 2223 3 353 2 2122 2136 2212 32 22 3254 2222 3215 2235 232939 L3 3333 3333 3333 3333 3333 3313 lllJ39 3553 4564 4523 3522 342 3522 4203 1133 3 4625 3333 333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 12 3213 4219 3226 4222 4232 4243 42 533 3253 3263 3262 3333 333 3333 3333 3333 3333 3333 3333 333 393 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3911 quot339 l39ill 32332 E d 39 39l l 333339 33121 31E di dl d You can nd the probability 04798 at the intersection of 20 and 005 ie 25 which is corresponding zvalue As the probability at right from the center of the curve is 05 and also at left is 05 E33333 3333333 EH333333 3333333333 333quot3333333333 3333 3331333 33333333 3333333 33333 373333 3 3 33 2 3 0 8 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Now if we want to nd the probability corresponding to 2 value greater than equal to 25 then we have to subtract the value of probability correspoing to given 2 value from 05 EMMU LawFE HEREAIL FREQUENCY quotEETREUTlDH area unde standard normal curve from E the Z 31 0504798 00202 202 309 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU CALCULATING Z VALUES 2 Value x MeanStDev Process of calculating z from x is called Standardization 2 indicates how many standard deviations the point is from the mean W Find proportion of bags which have weight in excess of 515 g Mean 510 StDev 25 g Solution 2 515 51025 2 From tables probability corresponding to 2 value is 04772 J hrLE 555 552 552 525 255 5525 22 252 522 2 55255 52555 5522 55552 55155 55225 55222 5555 52552 51 5555 5555 5555 5555 5555 5555 5555 5555 5555 5555 52 5555 55 52 555 5 55 55 5555 5555 5525 139555 5 555 5 155 53 1555 1215 5255 5253 1355 5255 5555 1453 5555 555 55 1555 l 555 5 525 5555 5 53955 5 555 5 552 5555 5 555 5555 5 2555 525 1525 2252 2555 2525 2525 2552 2552 2225 55 2255 2255 2525 2555 2555 2522 2555 2555 2515 2555 55 2555 255 5 2552 2555 2555 2555 2555 2555 2525 2552 55 2555 255 2555 2555 2555 5525 5555 5555 5555 5555 55 5555 5555 5252 5255 5255 5255 5555 5555 5555 5555 15 5515 3435 3555 5555 5555 555 3555 5555 255 3625 55 5555 5555 5555 5555 5525 5555 5555 5555 5555 5555 52 5555 5555 5 55 5555 5525 5555 5552 5555 5555 5555 5 5552 5555 5552 5555 55 5 5 5555 5 55 5 552 55 55 15 5152 5252 5225 5252 5255 5255 5252 5555 5222 5 5 5552 5555 5555 5555 5552 5555 5555 5555 5525 5555 5 5 5552 5555 5555 5555 5555 4555 55 5 5525 5535 5555 5 5 55 55 5555 5555 5552 5555 5555 555 555 5 5525 5555 5 5 5555 5555 5555 5555 555 5 5555 5555 5555 5555 5555 55 5555 55 5 5525 5552 5555 5555 5555 5555 5555 5555 5222 5255 5222 5252 5255 5552 5252 522 5555 39l 5525 5525 5535 5535 5555 5552 555 5555 5555 5555 22 5555 5554 5555 5555 5555 5 5555 55 5 5555 5555 5555 25 5 5225 5555 55 5555 5555 55 551 5 555 5 5555 l w l 5 l 391 05 04772 00228 3 10 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 2 What percentage of bags filled by the machine will weigh less than 5075 9 Mean 510 g StDev 25 9 Solution 2 5075 51025 1 Look at value of 2 1 2111111111111119111111 91111111111111111 2111111111111 1111 11111111 111111111 111111111 111111 111m 9 11 E 11 By table 05 03413 01587 1111quot 111 111 111 111 1111 11111 111 111 99 999199 111 991099 111111211 11111111 911199 99219 99219 911119 119359 I11 9399 9499 9419 115 11 9551 99 9929 9915 91 14 9159 2 9191 11992 9911 119 19 9949 9991 1929 1964 1 1111 1 141 113 1119 1211 1255 1293 1311 1193 1999 1443 1491 151 114 1554 1591 1129 1114 1199 1199 1112 1199 1944 1219 1 1 1111 1111 1111 1111 1151 1111 1111 1111 1111 1111 I1 1 2251 2291 2324 2151 2199 2422 2494 2491 2511 2549 11 1 2599 2511 2942 2911 213994 2194 2114 2194 2923 2952 11 2 2991 2919l 2999 2991 2995 9923 3951 3919 3191 9131 11 9 3159 3191 2212 1213 3219 1229 3315 3141 1165 1399 1411 1411 1111 1111 1114 1111 1199 3111 1i 1626 1199 9129 3149 21111 11911 219 111 3939 39 I 3999 3991 3925 3944 1962 1999 9991 4915 13 4992 9949 4919 4992 4999 41 1 1 4131 4141 4112 4111 14 4192 4291 4222 4231 4251 4295 4219 4292 4196 4319 15 4332 4349 43 51 4219 4392 4394 4499 441 9 4429 4441 11 4151 1111 4111 1114 1415 4511 4115 4111 4131 1145 111 45 13 i 3 quot 22111 1 4625 4633 1 9 4941 4149 4159 4114 4911 4119 4991 4692 4699 4199 19 4112 4119 4129 4122 4129 4144 4159 4159 4161 4111 1 11 1111 1111 1113 4111 1111 4191 1111 4111 4111 111 139 2 1 4221 4929 4939 4234 9999 9242 4941 4959 4994 4951 1 1 1111 1114 1111 4111 1111 1111 411 1114 1111 4191 2 9 4993 4991 4999 4991 4994 4991 4999 4911 4913 4919 2 4 4919 4929 4922 4925 4921 4929 4931 139 4932 49 94 4916 3 1 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU W What is the probability that a bag filled by the machine weighs less than 512 g z 512 51025 08 m 22222 2243 22224 2135 3336 22222 2133 2223 2222 23232232 21W 42202322 22222222 2222622 423124 321234 62223 2222322 22352 121 223 6433 4423 215 2 2 4552 4536 23636 1262 5 42 24 4233 0233 23 32 2232 2 223 223 3342 4332 22226 1664 2 2223 2 142 23 1223 1212 2253 2293 1332 2363 2446 2443 2432 2522 44 2554 2542 2622 2664 2244 2236 2222 2443 2244 2323 2925 2254 1425 2222 2254 2443 2223 2252 2244 2224 2252 222 2 2324 2 352 2344 2422 2454 2426 2512 2 544 2524 262 2 2642 2623 2244 2234 2264 2 23934 2223 2252 7 222 i 29 222 2339 2362 2445 34223 365 2 3623 3 2 236 3233 3253 3236 3222 3233 3264 32562 3325 3346 3365 3333 22 3413 3432 3462 3425 3543 3532 3354 3522 3339 3622 2 2 3643 3665 3636 3243 3222 3243 32212 32222 33 222 3632 t 2 3443 3363 3334 3362 342 5 3344 3962 3226 3292 4222 5 2 3 4232 4443 4232 42233 42 2 5 4232 4 42 4262 42 2393 24 4232 4242 4222 4236 4252 4265 4222 4232 4346 4323 2 5 4332 4345 43 32 4324 4332 4394 4446 442 3 4423 4442 26 4452 4463 4434 4434 4493 4305 39 39 4335 4343 22 45544564 4544 4542 4542 4544 4625 4633 2 2 4642 4642 4656 4664 4622 4622 4629 4326 EMMULATWENURMM 2666621464 DETREUTMN 426a M d fi 42464464 6642662 411646 2mm 23 66 E 0502881 07881 3 12 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Example 4 What percentage of bags weigh between 512 and 515 21 512 51025 08 Solution Area 1 02119 22 515 51025 2 Area 2 002275 pbags weighs between 512 and 515 Area 1 Area 2 02119 002275 018915 189 Required Area 39 iumu LATEWE Nammt FREQUENCY quotESTREMTtDH area underi Standard i t t curve ff m it m E 11 Areal 050288102119 CUMULATWE NURMAL FREQUENCY l ETREMTtnw area funded standard nurmat curve firam E be 3 2413 I2 l 115 US39 Area 2 05i 47720022s ampd 39 5 Ea ain manr 39CNJMUMTWE NDRMAL FREQUENCY 3quotESTREUTtDN area Handed standard nmmal curve from El he E zna i2 l I15 05quot 3 13 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0211900228 01891 1891 So we obtain CMMLII LATWE HEREEL FEEQUEHC quotESTREUTJIDH area runderi standard marina Eu t tram in E F quot quot215 22 n5 05quot 314 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 41 ESTIMATING FROM SAMPLES INFERENCE PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 40 0 Estimating from Samples Inference NORMDIST Returns the normal distribution for the specified mean and standard deviation This function has a very wide range of applications in statistics including hypothesis testing Syntax NORMDISTxmeanstandarddevcumulative X is the value for which you want the distribution Mean is the arithmetic mean of the distribution Standarddev is the standard deviation of the distribution Cumulative is a logical value that determines the form of the function If cumulative is TRUE NORMDIST returns the cumulative distribution function if FALSE it returns the probability mass function Remarks o If mean or standarddev is nonnumeric NORMDIST returns the VALUE error value o If standarddev S 0 NORMDIST returns the NUM error value o If mean 0 standarddev 1 and cumulative TRUE NORMDIST returns the standard normal distribution NORMSDIST o The equation for the normal density function cumulative FALSE is ix ixf fEItquot 435 1 e I t 0 When cumulative TRUE the formula is the integral from negative infinity to x of the given formula Example In the slide the x value is 42 Arithmetic mean is 40 Standard deviation is 15 The cumulative distribution is 09 32 315 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU E icrnsn Excel Lecture41 Elle Edit Eiew insert Fgrmat Innis gate window elp D al v Evai fl v DE v a a E t i NRMDISTWmeanstandartLdevcumulative 2 Data Descriptin 3 42 Value fr which yu want the distributin 1 4i arithmetic mean f the distributin 5 15 Standard deaiatin f the distributin j r Cumulative diatributian function for 39I39I39I lll til a the terms above 03079 9 W Returns the standard normal cumulative distribution function The distribution has a mean of 0 zero and a standard deviation of one Use this function in place of a table of standard normal curve areas Syntax NORMSDISTz z is the value for which you want the distribution Remarks lf 2 is nonnumeric NORMSDIST returns the VALUE error value The equation for the standard normal density function is Example The input to the NORMSDIST function is the zvalue The output is the cumulative probability distribution In the example 2 1333333 The normal cumulative probability function is 0908789 3 16 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Micrnsnft Excel Lecturejl Elle Edit iew insert Fgrmat Innls Qatar indnw Help H t a v n v E v E SLIM r X J f3 NDRMSDISTHEEEEHE E A 11 NURMSDISTCZ 12 A B 13 Frmula Descriptin Result NORMINV Returns the inverse of the normal cumulative distribution for the specified mean and standard deviation Syntax NORMINVprobabilitymeanstandarddev Probability is a probability corresponding to the normal distribution Mean is the arithmetic mean of the distribution Standarddev is the standard deviation of the distribution Remarks If any argument is nonnumeric NORMINV returns the VALUE error value If probability lt 0 or if probability gt 1 NORMINV returns the NUM error value If standarddev S 0 NORMINV returns the NUM error value If mean 0 and standarddev 1 NORMINV uses the standard normal distribution see NORMSINV NORMINV uses an iterative technique for calculating the function Given a probability value NORMINV iterates until the result is accurate to within 1 3x10quot7 f NORMINV does not converge after 100 iterations the function returns the NA error value Example Here the probability value arithmetic mean and standard deviation are given The answer is the xvalue 3 17 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE ilicrusuft Excel Lecture41 Eile Edit ew lnaert Fgrmat IDDIS Qatar window elp cit E v E E IE 3 SLIM r K inquot f NDRMINW i23 24 25j 4 E 2i NRMINUfprhahility meanatandarddeu 21 H B 22 Data Descriptin Prhahility crrea nding 22 t the nrmal distributin 10 i irithmetic mean f the 22 distribution 15 Standard deviatin f the 22 distr39ihutin 25 iNRMI NVA23 Z l 2 lnuerae f the nrmal cumulative 25 diatr39ihutin ferquot the terms ahdue 12 W Returns the inverse of the standard normal cumulative distribution The distribution has a mean of zero and a standard deviation of one Syntax NORMSINVprobability Probability is a probability corresponding to the normal distribution Remarks 0 If probability is nonnumeric NORMSINV returns the VALUE error value 0 If probability lt 0 or if probability gt 1 NORMSINV returns the NUM error value NORMSINV uses an iterative technique for calculating the function Given a probability value NORMSINV iterates until the result is accurate to within 1 3x10quot7 If NORMSINV does not converge after 100 iterations the function returns the NA error value Example In this case the input is the zvalue The corresponding cumulative distribution is calculated 3 18 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU lilicrnsnft Excel Lecturele Elle Edit Eleni Insert Fgrmat Innis Qatar intlnnl Help ll Ev W E r E sun r X J 8 NDRMSDIST1333333 A El 11 NURMSDISUZ 12 A B 13 Frmula Descriptin Result 14 N1RMSDIST1 15 Nrmal cumulative distributin functin lEi at 0908789 17 SAMPHNG VARIATIONS Electronic components are despatched by a manufacturer in boxes of 500 A small number of faulty components are unavoidable Customers have agreed to a defect rate of 2 One customer recently found 25 faulty components 5 in a box Was this box representative of production as a whole The box represents a sample from the whole output In such a case sampling variations are expected lf overall proportion of defective items has not increased just how likely is it that a box of 500 with 25 defective components will occur SAMPLING VARIATIONS EXAMPLE 1 In a section of a residential colony there are 6 households say Household A B C D E and F A survey is to be carried out to determine of households who use corn flakes of in breakfast Survey data exists and the following information is available Households A B C and D Use corn flakes Households E and F Do not It was decided to take random samples of 3 households The first task is to list all possible samples and find of each sample using corn flakes Possible Samples mple of users mple of users ABC 100 BCD 100 ABD 100 BCE 67 ABE 67 BCF 67 ABF 67 BDE 67 ACD 100 BDF 67 ACE 67 BEF 33 ACF 67 CDE 67 ADE 67 CDF 67 ADF 67 CEF 33 AEF 33 DEF 33 Percentage In Sample 3 19 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Out of 20 samples 4 contain 100 of users 12 contain 67 of users 4 contain 33 of users with required characteristic If the samples are selected randomly then each sample is likely to arise The probability of getting a sample with 100 of users is 420 or 02 with 67 1220 or 06 with 33 420 or 02 This is a Sampling Distribution SAMPLING DISTRIBUTION The sampling distribution of percentages is the distribution obtained by taking all possible samples of fixed size n from a population noting the percentage in each sample with a certain characteristic and classifying these into percentages Mean of the Sampling Distribution Using the above data Mean 100 x 02 67 x 06 33 x 02 67 Mean of the sampling distribution is the true percentage for the population as a whole You must make allowance for variability in samples Conditions For Sample Selection Number of items in the sample n is fixed and known in advance Each item either has or has not the desired characteristic The probability of selecting an item with the characteristic remains constant and is known to be P percent If n is large gt30 then the distribution can be approximated to a normal distribution STANDARD ERROR OF PERCENTAGES Standard deviation of the sampling distribution tells us how the sample values differ from the mean P It gives us an idea of error we might make if we were to use a sample value instead of the population value For this reason it is called STandard Error of Percentages or STEP STEP The sampling distribution of percentages in samples of n items ngt30 taken at random from an infinite population in which P percent of items have characteristic X will be A Normal Distribution with mean P and standard deviation STEP P100Pnquot12 The mean and StDev of the sampling distribution of percentages will also be percentages 320 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 42 Estimating from Samples Inference Part 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 41 0 Estimating from Samples Inference EXAMPLE 1 In a factory 25 workforce is women How likely is it that a random sample of 80 workers contains 25 or more women m standard deviation STEP P100Pnquot12 Mean P 25 N 80 STEP 25100 2580quot12 25 x 7580quot12 484 women in sample 2580 x 100 3125 2 3125 25484 129 Look for p against 2 129 in the table you will get 04015 39 z 333 331 333 333 333 333 33 333 00 000 030040 00000 01 I 30 001013 010100 00330 00030 003 39339 001350 0 1 0300 0430 0410 115 1 1 055 139 30595 0435 053 5 00 14 0053 01 0093 0033 Ii011 0310 0040 0031 1033 1054 1 103 1 141 03 1113 1211 1355 1303 1331 1300 1405 1443 1400 151Ir 04 1554 1 501 1333 1354 1100 39 333 1113 1003 1044 130311 05 1015 105I 1005 2010 2054 3000 3123 0153 3100 2234 015 3351 3201 3334 2 3511 3333 2433 2454 3403 3511 3 543 01 2500 351 1 3543 350 3 3104 3134 3104 3134 3023 2052 030 3301 39 0 3030 3331 3305 3033 3051 3035 3 1 03 31 3 3 00 31 5039 3133 3213 331 33 3304 3300 331 5 3340 3 335 3303 3 3313 3333 3331 3333 3333 3331 3333 3313 1 3333 3333 3333 3333 3333 3133 3313 3133 3333 3333 3333 3331 3333 3333 3333 3333 39 3 3333 3333 3333 3333 3333 311 3 3131 3133 1 3133 3333 3333 3333 3331 3333 3333 3333 15 4332 4345 4351 4330 4300 4304 441 3 4430 4441 10 4453 4453 4474 4404 4495 4505 3951 5 4535 4535 4545 11 4 JEEJ Atrial JEEP Ii l39 JEH39I mm Il39lf Ji39 Al l Al 39 Since we want to nd psample contains 25 or more women so 3 2 1 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU CMMULATIWE Notmm FREQUENCY 3quotESTRIEJIJTllDN area untied Standard normal curve from I lie 2 11 balms w 39 z1EE I I15 LEM psample contains 25 or more women 05 04015 00985 or about 10 APPLICATIONS OF STEP Some important issues are 0 What is the probability that such a sample will arise o How to estimate the percentage P from information obtained from a single sample 0 How large a sample will be required in order to estimate a population percentage with a given degree of accuracy 0 To obtain answers to these questions let us solve some typical problems CONFIDENCE LIMITS A market researcher wishes to conduct a survey to determine consumers buying the company s products He selects a sample of 400 consumers at random He finds that 280 of these 70 are purchasers of the product What can he conclude about of all consumers buying the product First let us decide some limits It is common to use 95 confidence limits These will be symmetrically placed around the 70 buyers In a normal sampling distribution 25 corresponds to a zvalue of 196 on either side of 70 With 95 con dence limit we have 5 chance of errors level of signi cance That mean 25 on each side of curve 25 0025 050025 0475 We have subtracted 0025 from 05 because we nd out the probability of acceptance region as the total probability under the curve is always 1 that mean 05 on right side of mean and 05 on left side of mean And 025 is rejection region so 0475 is acceptance region So 0475 is the probability of acceptance region 322 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Corresponding 2 value is196 z Ml lint an tam m are ill um acute tiltmm u 39 uizlleg 39 EEQix39 J ELM39939 Lil35939 1 i AME Illle i39 39 9135 ll 7 1355 l 0596 IDEISE 126 5 JDquot 14 JETS EISLE WQ39E39 332 IEliE39Tlt it til 0943 1393 lll l Jill64 li m3 it Hit 13 1111quot 121 llfl fi JEQS 133l 1563 1 1443 MEI ll l 04 1554 1591 3625 1664 H tt l 315 l39it39itl l l t l JET ti l 939 5 WEI 1935 33 t9 2354 3133 El 23 2 t 5 2 ll Eel 2224 LE 2257 229 2324 235 2339 2122 2154 2436 251 2549 a E i l 39 E ll l 2642 2623 ET EM 2234 all Ewell 2323 2352 ELIE z t E l 2939 295 2995 123 3tt5l Ellis jll 3l33 19 3159 EIEIE SZIZ 3233 3264 EaEEtt 331155 33 3365 3339 LG Jude13a 3433 3435 35 3531 355 35 3599 362l Lll 35 3665 E l EliTZEI39 EM NW BTQHU 33 Ml HESEl LE 3349 3369 3910 3925 3944 3962 3930 399 0115 3 4332 4349 4133 All itl 39l 3 All EM 414 4 ME tl I TI a 4192 AM 4236 4251 4265 AZ A292 4336 4319 4332 4345 43 5 ALETl ABE2 4394 sl ll 4429 Ala le MSE H l quot 4525 535 4545 A d a u 1211 quot AME Al 4633 AME 39 4 I 46 4699 was ATM 4229 4126 4 13932 4333 s 4 l A316 41 Now the sample percentage of 70 can be used as an approximation for population percentage P Hence STEP 70100 70400quot12 229 Confidence gmits Estimate for population percentage 70 196 x STEP Or 70 196 x 229 65515 and 7449 as the two limits for 95 confidence interval We can round off 196 to 2 323 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Then with 95 confidence we estimate the population percentage with that characteristic as lying in the interval P 2 x STEP EXAMPLE 2 A sample of 60 students contains 12 20 who are left handed Find the range with 95 confidence in which the entire left handed students fall Range 20 2 x STEP 20 2 x 20 x 10020l60quot12 967 and 3033 ESTIMATING PROCESS SUMMARY 1 Identify n and P the sample size and percentage in the sample 2 Calculate STEP using these values The 95 confidence interval is approximately P 2 STEP 99 confidence For 99 confidence limits zvalue 258 With 99 con dence limit we have 1 chance of errors level of signi cance That mean 05 on each side of curve 05 0005 050005 0495 We have subtracted 0005 from 05 because we nd out the probability of acceptance region as the total probability under the curve is always 1 that mean 05 on right side of mean and 05 on left side of mean And 005 is rejection region so 0495 is acceptance region So 0495 is the probability of acceptance region Corresponding 2 value is 258 from table as we did in example stated above 324 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU FINDING A SAMPLE SE To satisfy 95 confidence 2 x STEP 5 STEP 25 Pilot survey value of P 30 STEP 30 x 70nquot12 25 Mg n 336 We must interview 336 persons to be 95 confident that our estimate is within 5 of the true answer DISTRIBUTION OF SAMPLE MEANS The standard deviation of the Sampling Distribution of means is called STandard Error of the Mean STEM STEM E J sd denotes standard deviation of the population n is the size of the sample EXAMPLE 3 What is the probability that if we take a random sample of 64 children from a population whose mean IQ is 100 with a StDev of 15 the mean IQ of the sample will be below 95 m s 15 n 64 population mean 100 STEM 1564quot12 1564quot12 158 1875 2 100 95 STEM 51875 267 This gives a probability of 00038 So the chance that the average IQ of the sample is below 95 is very small 325 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 43 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION PART 1 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 42 o Hypothesis testing ChiSquare Distribution EXAMPLE 1 An inspector took a sample of 100 tins of beans The sample weight is 225 g Standard deviation is 5 g Calculate with 95 confidence the range of the population mean m STEM E J sd is not known Use sd of sample as an approximation STEM i 05 100 95 confidence interval 225 2 x 05 or From 224 to 226 g PROBLEM OF FAULTY COMPONENTS REVISITED Box of 500 components may have 25 or 5 faulty components Overall faulty items 2 P 2 n 500 STEP 2 x 98500quot12 0626 To find the probability that the sample percentage is 5 or over 2 5 2STEP 30626 479 Area against 2 479 is negligible Chance of such a sample is very small FINITE POPULATION CORRECTION FACTOR lf population is very large compared to the sample then multiply STEM and STEP by the Finite Population Correction Factor 1 nNA12 Where N Size of the population n Size of the sample n less than 01 N TRAINING MANAGER S PROBLEM New refresher course for training of workers was completed The Training Manager would like to assess the effect of retraining if any Particular questions ls quality of product better than produced before retraining Has the speed of machines increased 39Do some classes of workers respond better to retraining than others Training Manager hopes to 0 Compare the new position with established 326 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU 0 Test a theory or hypothesis about the course Case Study Before the course Worker X produced 4 rejects After the course Out of 400 items 14 were defective 35 An improvement The 35 figure may not demonstrate overall improvement It does not follow that every single sample of 400 items contains exactly 4 rejects To draw a sound conclusion Sampling variations must be taken into account We do not begin by assuming what we are trying to prove NULL HYPOTHESIS We must begin with the assumption that there is no change at all This initial assumption is called NULL HYPOTHESIS Implication of Null Hypothesis That the sample of 400 items taken after the course was drawn from a population in which the percentage of reject items is still 4 NULL HYPOTHESIS EXAMPLE Data P 4 n 400 STEP P100 Pnquot12 4100 4400quot12 098 At 95 confidence limit Range 4 2 x 098 204 to 596 Conclusion Sample with 35 rejects is not inconsistent No ground to assume that rejects has changed at all On the strength of sample there were no grounds for rejecting Null Hypothesis ANOTHER EXAMPLE Before the course 5 rejects After the course 25 rejects 10 out of 400 P 5 STEP 5100 5400quot12 5 x 95400quot12 109 Range at 95 Confidence Limits 5 2 x 109 282 to 718 Conclusion Doubt about Null Hypothesis most of the time Null hypothesis to be rejected PROCEDURE FOR CARRYING OUT HYPOTHESIS TE 1 Formulate null hypothesis 2 Calculate STEP amp P 2 x STEP 3 Compare the sample with this interval to see whether it is inside or outside If the sample falls outside the interval reject the null hypothesis sample differs significantly from the population If the sample falls inside the interval 327 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU do not reject the null hypothesis sample does not differ significantly from the population at 5 level HOW THE RUE WORKS Bigger the difference between the sample and population percentages less likely it is that the population percentages will be applicable 0 When the difference is so big that the sample falls outside the 95 interval then the population percentages cannot be applied Null Hypothesis must be rejected o If sample belongs to majority and it falls within 95 interval then there are no grounds for doubting the Null Hypothesis FURTHER POINTS ABOUT HYPOTHESIS TESTING 99 interval requires 258 x STEP Interval becomes wider It is less likely to conclude that something is significant A We might conclude there is a significant difference when there is none Chance of error 5 type 1 Error B We might decide that there is no significant difference when there is one Type 2 Error 328 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 44 HYPOTHESIS TESTING CHISQUARE DISTRIBUTION PART 2 OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 43 o Hypothesis Testing ChiSquare Distribution FURTHER POINTS ABOUT HYPOTHESIS TESTING This is a continuation of the points covered under Handout 43 3 We cannot draw any conclusion regarding the direction the difference is in A Possible to do 1tailed test Null Hypotheis P gt 4 against the alternative Pgt 4 z 164 for 5 significance level Range P 164 x STEP 098 Example Range 4 164 x 098 239 New figure 35 Hence There is no reason to conclude that things have improved 4 We cannot draw any conclusion regarding the direction the difference is in B Possible to do 2tailed test Null Hypothesis P gt 4 against the alternative Pgt 4 z 196 for 5 significance level Range P 196 x STEP 098 Example Range 4 2 X196 x 098 208 to 592 New figure 35 There is no reason to conclude that things have improved HYPOTHESES ABOUT MEANS Let us go back to the problem of retraining course discussed earlier Before the course Worker X took 25 minutes to produce 1 item StDev 05 min After the course Foe a sample of 64 items mean time 258 min Null hypothesis No change after the course STEM sdnquot12 0564 12 00625 Range 25 2 x 00625 2375 to 2625 min Conclusion No grounds for rejecting the Null Hypothesis There is no change significant at 5 level 329 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU ALTERNATIVE HYPOTHESIS TESTING USING ZVALUE z sample percentage population meanSTEP 35 4098 051 Compare it with zvalue which would be needed to ensure that our sample falls in the 5 tails of distribution 196 or about 2 z is much less than 2 We conclude that the probability of getting by random chance a sample which differs from the mean of 4 or more is quite high Certainly it is greater than the 5 significance level Sample is quite consistent with null hypothesis Null hypothesis should not be rejected PROCESS SUMMARY 1 State Null Hypothesis 1tailed or 2tailed 2 Decide on a significance level and find corresponding critical value ofz 3 Calculate sample zsample value population value divided by STEP or STEM as appropriate 4 Compare sample 2 with critical value of z 5 If sample 2 is smaller do not reject the Null Hypothesis 6 If sample 2 is greater than critical value of 2 sample provides ground for rejecting the Null Hypothesis TESTING HYPOTHESES ABOUT SMALL SAMPLES Whatever the form of the underlying distribution the means of large samples will be normally distributed This does not apply to small samples We can carry out hypothesis testing using the methods discussed only if the underlying distribution is normal If we only know the Standard Deviation of sample and have to approximate population Standard Deviation then we use Student s tdistribution STUDENT S tDISTRIBUTION Student s TDistribution is very much like normal distribution In fact it is a whole family of tdistributions As n gets bigger tdistribution approximates to normal distribution tdistribution is wider than normal distribution 95 confidence interval reflects greater degree of uncertainty in having to approximate the population Standard Deviation by that of the sample EXAMPLE Mean training time for population 10 days Sample mean for 8 women 9 days Sample Standard Deviation 2 days To approximate population Standard Deviation by a sample divide the sum of squares by n 1 STEM 28quot12 071 Null Hypothesis There is no difference in overall training time between men and women tvalue sample mean population meanSTEM 9 10071 141 Forn8v8 1 7 For 505 significance level looking at 0025 2tailed t 2365 Calculated table value m Do not reject the Null Hypothesis 3 30 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SUMMARY I If underlying population is normal and we know the Standard Deviation Then Distribution of sample means is normal with Standard Deviation STEM population sdn3912 and we can use a ztest SUMMARY II If underlying population is unknown but the sample is large Then Distribution of sample means is approximately normal With StDev STEM population sdn3912 and again we can use a ztest SUMMARY Ill lf underlying population is normal but we do not know its StDev and the sample is small Then We can use the sample sd to approximate that of the population with n 1 divisor in the calculation of sd Distribution of sample means is a tdistribution with n 1 degrees of freedom With Standard Deviation STEM sample sdn 12 And we can use a ttest SUMMARY IV If underlying population is not normal and we have a small sample Then none of the hypothesis testing procedures can be safely used TESTING DIFFERENCE BETWEEN TWO SAMPLE MEANS A group of 30 from production has a mean wage of 120 Rs per day with Standard Deviation Rs 10 50 Workers from Maintenance had a mean of Rs 130 with Standard Deviation 12 Is there a difference in wages between workers Difference of two sample means s1n1 1n23912 s n1s1quot2 n2s2quot2 n1 n2quot12 N1 30 n2 50 s1 10 s2 12 s 30 x 100 50 x14430 50 12 1129 Standard Error of Difference in Sample Means STEDM 1129130 150quot12 260 z difference in sample means 0STEDM 120 130260 385 This is well outside the critical 2 for 5 significance There are grounds for rejecting Null Hypothesis There is difference in the two samples PROCEDURE SUMMARY 1 State Null Hypothesis and decide significance level 2 Identify information no of samples large or small mean or proportion and decide what standard error and what distribution are required 3 Calculate standard error 4 Calculate z or t as difference between sample and population values divided by standard error 5 Compare your 2 or t with critical value from tables for the selected significance level if z ort is greater than critical value reject the Null Hypothesis 331 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU MORE THAN ONE PROPORTION Look at a problem where after the course some in different age groups shows improvement while others did not Let us assume that the expected improvement was uniform An improvement of 40 if applied to 21 24 and 15 would give 14 16 and 10 respectively who improved Let us write these values within brackets Subtracting 14 16 and 10 from the totals 21 24 and 15 gives us 7 8 and 5 respectively who did not improve This is the estimate if every person was affected in a uniform manner Let us write the observations as O in one line 17 17 6 4 7 9 Let us write down the expected as E in the next line as 14 16 10 7 8 8 Calculate OE Next calculate OEquot2 Now standardize OEquot2 by dividing by E Calculate the total and call it x2 Age Improved Did not improve Total Under 35 1714 47 21 35 50 1716 78 24 Over 50 610 95 15 Total 40 20 60 O 17 17 6 4 7 9 E 14 16 10 7 8 8 OE 3 1 4 3 1 4 OEquot2 9 1 16 9 1 16 OEquot2E 0643 00625 16 1286 0125 32 692 Measurement of disagreement Sum OEquot2E is known as Chisquared x2 Degrees of freedom v r1xc1 3121 2 There are tables that give Critical value of chisquared at different confidence limits and degrees of freedom v columns1 x rows1 In the above case v 21 x 31 2 In the present case the Critical value of chisquared at 5 and v 2 5991 The value 692 is greater than 5991 This means that the Sample falls outside of 95 interval Null hypothesis should be rejected CHISQUARED SUMMARY Formulate null hypothesis no association form Calculate expected frequencies Calculate x2 Calculate degrees of freedom rows minus 1 x columns minus 1 look up the critical x2 under the selected significance level 5 Compare the calculated value of x2 from the sample with value from the table if the sample x2 is smaller within the interval don t reject the null hypothesis if it is bigger outside reject the null hypothesis Example Look at the data in the slide below bF Nf 332 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EE IHIicrosoft Excel Ernp Elle Edit iew Lnsert Fgrmat Iools gate window Help D a vs 39 v EvsIMi fmv ae F1 1 a 111 B I D E F I I3 39 1 STATUS SEE AGE 2 I I 3 I a I 1 3 39Jquot 4 I 1 3 I 5 I 1 4 I a 1 I 3 IS a I 1 3 I a I 1 4 39Jquot a 1 I 3 3 1o 1 I 3 I 11 I 1 4 S 12 1 I 4 4 13 I 1 3 1 It is possible to carry out ttests using EXCEL Data Analysis tool Data Analysis analysis Tools Histogram Moving Fwerage I Ftantlom Number Generation 3 Flank and Percentile Regression Hall I Sampling tTest Paired Two Sample For Means itIastI39SamolaassnmionquotLreonal39ss39at39iaolas eTest Two Sample For Means h quotquot When you select the tool and press OK the ttest dialog box is opened as below 333 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU tTest TwinSample Assuming Unequal Variances Input 39u39arialzule 1 Range 39u39arialzule a Range Cal39IEEl I Hypethesisecl Mean Difference I ElIII I l Labels alpha CLUE Clutput eptiens F Qutput Range I F New Werksheet Ely I F New erkheelt The ranges for the two variables labels and output options are specified For the above data the output was as follows Micrnsuft Excel Emp Eile gait Eiew insert Fgrmat leels gata window elp v a x DEH 5 v Ev m WBH E s 2 D15 139 f3 A e c D t 1 t Teet Ten Sample Assuming Unequal ariane 2 3 33 J 4 Mean 3913 351 5 Eariance 3356 5252 s Uheervatinne 23 35 Hypntheeized Mean Di 0 3 df SE a t Stat 1TEE m PTt ne tail H43 5 n t Critical ne tail 1625 n Pthtl ten tail n3511 u t Critical ten tail 2 32 1 11 CHITEST Returns the test for independence CHITEST returns the value from the chisquared y2 distribution for the statistic and the appropriate degrees of freedom You can use v2 tests to determine whether hypothesized results are verified by an experiment Syntax CHlTESTactualrangeexpectedrange Actualrange is the range of data that contains observations to test against expected values Expectedrange is the range of data that contains the ratio of the product of row totals and column totals to the grand total Remarks 3 3 4 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU o If actuarange and expectedrange have a different number of data points CHITEST returns the NA error value 0 The v2 test first calculates a v2 statistic and then sums the differences of actual values from the expected values The equation for this function is CHITESTp Xgty2 where and where Aij actual frequency in the ith row jth column Eij expected frequency in the ith row jth column r number or rows c number of columns CHITEST returns the probability for a v2 statistic and degrees of freedom df where df r 1c 1 Exam le Microsoft Excel Lecture44 Elle Edit Eiew insert Fgrmat Tools Data window elp v E X a as v n v 2 v E e a SUM v X J f3 CHITESTA3EEEABQ E C T M Women Actual Description 2 Actual 3 58 35 Agree 1 11 25 Neutral 5 10 23 Disagree E Exgeeged Women Expected Description a 39 4535 4765 Agree 8 1756 1344 Neutral 9 1609 1691 Disagree a 35M The V2 statistic for the data above is 1515957 11 as with 2 degrees of freedom 0000303 The above example shows two different groups The calculation shows that the probability for chisquared 1616957 with 2 degrees of freedom was 0000308 which is negligible 3 3 5 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU LECTURE 45 Planning Production Levels Linear Programming OBJECTIVES The objectives of the lecture are to learn about 0 Review Lecture 44 0 Planning Production Levels Linear Programming INTRODUCTION TO LINEAR PROGRAMMING A Linear Programming model seeks to maximize or minimize a linear function subject to a set of linear constraints The linear model consists of the following components A set of decision variables X An objective function ch X A set of constraints 2 ailx 5 bi THE FORMAT FOR AN LP MODEL Maximize or minimize chxj c1x1 c2x2 cnxn Subject to ainjE bi 1m Nonnegativity conditions all x 0j 1 n Here n is the number of decision variables Here m is the number of constraints There is no relation between n and m THE METHODOLOGY OF LINEAR PROGRAMMING Define decision variables Handwrite objective Formulate math model of objective function Handwrite each constraint Formulate math model for each constraint Add nonnegativity conditions THE IMPORTANCE OF LINEAR PROGRAMMING Many real world problems lend themselves to linear programming modeling Many real world problems can be approximated by linear models There are wellknown successful applications in 0 Operations 0 Marketing 0 Finance investment 0 Advertising 0 Agriculture There are efficient solution techniques that solve linear programming models The output generated from linear programming packages provides useful what ifquot analysis ASSUMPTIONS OF THE LINEAR PROGRAMMING MODEL 1 The parameter values are known with certainty 2 The objective function and constraints exhibit constant returns to scale 3 There are no interactions between the decision variables the additivity assumption The Continuity assumption Variables can take on any value within a given feasible range A PRODUCTION PRO M A PROTOTYPEXAMPE A company manufactures two toy doll models Doll A mmewwe 336 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Doll B Resources are limited to 1000 kg of special plastic 40 hours of production time per week mirketinq requirement Total production cannot exceed 700 dozens Number of dozens of Model A cannot exceed number of dozens of Model B by more than 350 The current production plan calls for o Producing as much as possible of the more profitable product Model A Rs 800 profit per dozen 0 Use resources left over to produce Model B Rs 500 profit per dozen while remaining within the marketing guidelines Manaqement is seekinq a production schedule that will increase the company s profit A linear programming model can provide an insight and an intelligent solution to this problem Decisions variables X1 Weekly production level of Model A in dozens X2 Weekly production level of Model B in dozens Obiective Function Weekly profit to be maximized Maximize 800X1 500X2 Weekly profit subject to 2X1 1X2 1000 lt Plastic 3X1 4X2 2400 lt Production Time X1 X2 700 5 Total production X1 X2 350 5 Mix Xjgt 0 j 12 Nonnegativity ANOTHER EXAMPLE A dentist is faced with deciding how best to split his practice between the two services he offers general dentistry and pedodontics children s dental care Given his resources how much of each service should he provide to maximize his profits The dentist employs three assistants and uses two operatories Each pedodontic service requires 75 hours of operatory time 15 hours of an assistant s time and 25 hours of the dentist s time A general dentistry service requires 75 hours of an operatory 1 hour of an assistant s time and 5 hours of the dentist s time Net profit for each service is Rs 1000 for each pedodontic service and Rs 750 for each general dental service Time each day is eight hours of dentist s 16 hours of operatory time and 24 hours of assistants time 337 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU THE GRAPHICAL ANALYSIS OF LINEAR PROGRAMMING Using a graphical presentation we can represent all the constraints the objective function and the three types of feasible points GRAPHICAL ANALYSIS THE FEASIBLE REGION The slide shows how a feasible region is defined with nonnegativity constraints GRAPHIEAL ANAL I EIS THE FEASIBLE 3393 The nunnegativity constraints I 1quot THE SEARCH FOR AN OPTIMAL SOLUTION The figure shows how different constraints can be represented by straight lines to define a feasible region There is an area outside the feasible region that is infeasible THE FEASIBLE REEIDN The Plaitin mr tmi rt f Emug5 mm 1131quot Itdint ion Iu rat mirt angina redundant Inteaaihle F39rczluctizu n Tima 3x4 3mm fiIJIII lm It may be seen that each of the constraints is a straight line The constraints intersect to form a point that represents the optimal solution This is the point that results in maximum profit of 436000 Rs As shown in the slide below The procedure is to start with a point that is the starting point say 200000 Rs Then move the line upwards till the last point on the feasible region is reached This region is bounded by the lines representing the constraints 338 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 THE SDLUTIDN 35131315 arhinrym say un t HEM IIIIIIIIII A Then immaee te unlit tipnestle and cumint ur i hum eas e m Pro t FL5 lllll 5M fw n it SUMMARY OF THE OPTIMAL SOLUTION Model A 320 dozen Model B 360 dozen Profit Rs 436000 This solution utilizes all the plastic and all the production hours Total production is only 680 not 700 Model a production does not exceed Model B production at all EXTREME POINTS AND OPTIMAL SOLUTIONS If a linear programming problem has an optimal solution an extreme point is optimal EXT R EM E PIN s PTIMAL TIMItlgyi If a quotnew nrtr rnmming prelrleln has an liTil39l39l l E lllti ll an e reme mint i5 nptimnl Copyright Virtual University of Pakistan VU 339 Business Mathematics amp Statistics MTH 302 MULTIPLE OPTIMAL SOLUTIONS There may be more than one optimal solutions However the condition is that the objective function must be parallel to one of the constraints If a weightage average of different optimal solutions is obtained it is also an optimal solution MULTIPLE in w if QLUTI 1N3 For multiple optimal solutions to exist the olijeetiue funetion must he pamllelto one oftite eo nstmints i Inn IHeighten auemge of optimal solutions is also an optimal solution Jr Copyright Virtual University of Pakistan VU 340 Business Mathematics amp Statistics MTH 302 VU AMORDEGRC AMORLINC AVERAGE AVE RAG EA BINOMDIST CHITEST COMBIN CORREL COUNT COUNTA COUNTBLANK COUNTIF COVAR CRITBINOM CUMIPMT CUMPRINC DAVE RAG E DB DDB DPRODUCT DSUM Some useful functions of Excel Returns the depreciation of an asset for each accounting period by using depreciation coefficient French accounting system Returns the depreciation of an asset for each accounting period French accounting system Returns the average of its arguments Returns the average of its arguments including numbers text and logical values Returns the Binomial Distribution Probability Returns the test for independence CHITEST returns the value from the chisquared x2 distribution for the statistic and the appropriate degrees of freedom Returns Number of Combinations for a Given Number of Items Returns the correlation coefficient between two data sets Counts how many numbers are in the list of arguments Counts the number of cells that are not empty and the values within the list of arguments Counts the number of blank cells within a range Counts the number of nonblank cells within a range that meet the given criteria Returns covariance Returns smallest value for which the Cumulative Binomial Distribution is less than or equal to a criterion value Returns cumulative interest paid between two periods Returns cumulative principal paid on a loan between two periods Averages the values that match specified conditions Returns depreciation of an Asset for a specified period using fixed declining balance method Returns the depreciation of an asset for a specified period by using the doubledeclining balance method or some other method that you specify Multiplies the values in a list that match the specified condition Adds the numbers in a list that match specified conditions 341 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU EFFECT EVEN EXP FACT FORECAST FREQUENCY FV FVSCHEDULE GEOMEAN HARMEAN INT INTERCEPT IPMT IRR ISPMT LN LOG LOG10 MEDIAN MDETERM MINVERSE MIRR MMULT MODE NEGBINOMDIST NOMINAL NORM DIST NORMSINV Returns effective annual interest rate Rounds Up to the Nearest Even Integer e Raised to the Power of a Given Number Returns factorial of a Number Prediction by Trend Returns a frequency distribution as a vertical array Returns future Value of an Investment Returns Future value of an initial principal with variable interest rate Returns Geometric Mean of Positive Numeric Data Returns Harmonic Mean of Positive Numbers Rounds to the Nearest Integer Calculates Point Where Line Will Intersect Y Axis Returns the interest payment for an investment for a given period Returns the internal rate of return for a series of cash flows Calculates the interest paid during a specific period of an investment Returns Natural Logarithm of a number Returns Logarithm of a Number to a Specified Base Returns Base 10 Logarithm of a number Gives Median or Number in Middle Matrix Determinant of an Array Gives Inverse Matrix for the Matrix Stored in an Array Returns the internal rate of return where positive and negative cash flows are financed at different rates Matrix Product of Two Arrays Returns Most Frequent Value of an Array Returns Negative Binomial Distribution Returns annual nominal interest rate Returns Normal Cumulative Distribution Returns Inverse of Normal Cumulative Distribution 342 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU NPER NPV ODD PERCENTILE PERM UT PMT POISSON POWER PPMT PRODUCT PV QUARTILE RATE ROUND ROUNDDOWN ROUNDUP RSQ SLN SLOPE SQRT STDEV STDEVA STDEVP SUBTOTAL SUM SUMIF SUMPRODUCT Returns Number of Periods for an Investment Returns the net present value of an investment based on a series of periodic cash flows and a discount rate Rounds Number Up to the Nearest Odd Integer Returns K th Percentile of Values in a Range Returns Number of Permutations for a Given Number of Objects Returns the periodic payment for an annuity Returns Poisson Distribution Returns the result of a Number Raised to a Power Returns the payment on the principal for an investment for a given period Multiplies All Numbers Returns Present Value of an Investment Returns Specified Quartile of Data Set Returns Interest Rate of a Loan or Annuity Rounds Number to a Specific Number of Digits Rounds Number Down Towards Zero Rounds Number Up Away From Zero Returns Square of Pearson Product Moment Correlation Coefficient Returns the straightline depreciation of an asset for one period Returns Slope of a Linear Regression Line Returns Square Root of a Number Estimates standard deviation based on a sample Estimates standard deviation based on a sample including numbers text and logical values Calculates standard deviation based on the entire population Returns a Subtotal in a List Add all numbers in a range of cells Adds Specified Cells Multiplies Array and Gives Sum 343 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU SUMSQ SYD TREND TRUNC VAR VARP VDB XIRR XNPV Square Numbers and Then Add Returns the sumof years39 digits depreciation of an asset for a specified penod Return Values Along Linear Trend Truncates to an Integer by Removing Fractional Part Estimate Variance from Sample Calculates Variance from entire Population Returns the depreciation of an asset for a specified or partial period by using a declining balance method VDB stands for variable declining balance Returns internal rate of return for a schedule of cash flows that is not necessarily periodic Returns net present value for a schedule of cash flows that is not necessarily periodic 344 Copyright Virtual University of Pakistan Business Mathematics amp Statistics MTH 302 VU Area under Standard Normal Curve from 0 to Z 000 001 002 003 004 005 006 007 008 009 00 00000 00040 00080 00120 00159 00199 00239 00279 00319 00359 01 00398 00438 00478 00517 00557 00596 00636 00675 00714 00753 02 00793 00832 00871 00910 00948 00987 01026 01064 01103 01141 03 01179 01217 01255 01293 01331 01368 01406 01443 01480 01517 04 01554 01591 01628 01664 01700 01736 01772 01808 01844 01879 05 01915 01950 01985 02019 02054 02083 02123 02157 02190 02224 06 02257 02291 02324 02357 02380 02422 02454 02486 02518 02549 07 02580 02611 02642 02673 02704 02734 02764 02794 02823 02852 08 02881 02910 02939 02967 02995 03023 03051 03078 03106 03133 09 03159 03186 03212 03238 03264 03289 03315 03340 03365 03389 10 03413 03438 03461 03485 03508 03531 03554 03577 03599 03621 11 03643 03665 03686 03708 03729 03749 03770 03790 03810 03880 12 03849 03869 03888 03907 03925 03944 03962 03990 03997 04015 13 04032 04049 04066 04082 04099 04115 04131 04147 04162 04177 14 04192 04207 04222 04236 04251 04265 04279 04292 04306 04319 15 04332 04345 04357 04370 04382 04394 04406 04418 04430 04441 16 04452 04463 04474 04485 04495 04505 04515 04525 04535 04545 17 04554 04564 04573 04582 04591 04599 04608 04616 04625 04633 18 04641 04649 04656 04664 04671 04678 04686 04693 04690 04706 19 04713 04719 04726 04732 04738 04744 04750 04758 04762 04767 20 04772 04778 04783 04788 04793 04798 04803 04808 04812 04817 21 04821 04826 04830 04834 04838 04842 04846 04850 04854 04857 22 04861 04865 04868 04871 04875 04878 04881 04884 04887 04890 23 04893 04896 04898 04901 04904 04906 04909 04911 04913 04916 24 04918 04920 04922 04925 04927 04929 04931 04932 04934 04936 25 04938 04940 04941 04943 04945 04946 04948 04949 04951 04952 26 04953 04955 04956 04957 04959 04960 04961 04962 04963 04964 27 04965 04966 04967 04968 04969 04970 04971 04972 04973 04974 28 04974 04975 04976 04977 04977 04978 04979 04980 04980 04981 29 04981 04982 04983 04983 04984 04984 04985 04985 04986 04986 30 049865 04987 04987 04988 04988 04989 04989 04989 04990 04990 31 049903 04991 04991 04991 04992 04992 04992 04992 04993 04993 345 Copyright Virtual University of Pakistan Introduction to Cultural Anthropology SOC401 VU Lesson 01 WHAT IS ANTHROPOLOGY Anthropology can be best de ned as the study of the various facets of what it means to be human Anthropology is a multidimensional subject in which various components are studied individually and as a whole to develop a better understanding of human eXistence In this lecture we will not only be developing an understanding of the definition of anthropology we will also be looking at what an anthropologist does In addition to this we will also be looking at the various branches of anthropology with a focus on cultural anthropology Definition of Anthropology Anthropology is derived from the Greek words am bmpos for human and logos for study so if we take its literal meaning it would mean the study of humans In one sense this is an accurate description to the extent that it raises a wide variety of questions about the human eXistence However this literal definition isn t as accurate as it should be since a number of other disciplines such as sociology history psychology economics and many others also study human beings What sets anthropology apart from all these other subjects Anthropology is the study of people their origins their development and variations wherever and whenever they have been found on the face of the earth Of all the subjects that deal with the study of humans anthropology is by far the broadest in its scope In short anthropology aims to describe in the broadest sense what it means to be human Activities of an Anthropologist As we already know anthropology is the study of what it means to be human So the study of the in uences that make us human is the focus of anthropologists Anthropologists study the various components of what its means to be human Branches of Anthropology A Physical Anthropology Is the study of humans from a biological perspective Essentially this involves two broad areas of investigation a Human paleontology this sub branch deals with re constructing the evolutionary record of the human eXistence and how humans evolved up to the present times b Human variation The second area deals with how why the physical traits of contemporary human populations vary across the world B Archeology study of lives of people from the past by examining the material culture they have left behind C Anthropological Linguistics the study of human speech and language D Cultural Anthropology the study of cultural differences and similarities around the world Now that we have brie y defined the various branches of anthropology lets us now take an in depth view of cultural anthropology Copyright Virtual University of Pakistan 1 Introduction to Cultural Anthropology SOC401 VU Cultural Anthropology As we have discerned above cultural anthropology concerns itself with the study of cultural differences as well as the similarities around the world On a deeper level the branch of anthropology that deals with the study of specific contemporary cultures elbmgmpJy and the more general underlying patterns of human culture derived through cultural comparisons elbmogy is called cultural anthropology Before cultural anthropologists can examine cultural differences and similarities throughout the world they must first describe the features of specific cultures in as much detail as possible These detailed descriptions elbmgmpbz39es are the result of extensive field studies in which the anthropologists observes talks to and lives with the people under study On the other hand ethnology is the comparative study of contemporary cultures wherever they are found The primary objective of ethnology is to uncover general cultural principals rules that govern human behavior Areas of Specialization in Cultural Anthropology I Urban Anthropology studies impact of urbanization on rural societies and the dynamics of life within cities H Medical Anthropology studies biological and socio cultural factors that effect health or prevalence of illness or disease in human societies HI Educational Anthropology studies processes of learning of both formal education institutions and informal systems which can use story telling or experiential learning IV Economic Anthropology studies how goods and services are produced distributed and consumed within different cultural contexts V Psychological Anthropology studies relationship between cultures and the psychological makeup of individuals belonging to them Holistic and Integrative Approach Cultural anthropologists consider in uences of nature and nurture across all locations and across different periods of time When various specialties of the discipline are viewed together they provide a comprehensive view of the human condition Common Responses to Cultural Difference A Ethnocentrism a belief that one s own culture is not only the most desirable but also superior to that of others B Cultural relativism looks at the inherent logic behind different cultures and practices in the attempt to understand them Relevance of Cultural Anthropology Cultural anthropology enhances understanding of differences and prevents oversimplified generalizations It increases self knowledge about our own thinking values and behavior and helps develop cognitive complexity through integration interconnectedness and differentiation different aspects of a singular entity Cultural anthropology is also useful in facilitating meaningful interaction with other cultures and sub cultures Copyright Virtual University of Pakistan 2 Introduction to Cultural Anthropology SOC401 VU Useful Terms Components parts Paleontology specialized branch of physical anthropology that analyses the emergence and subsequent evolution of human physiology Variation degree of difference Archeology sub field of anthropology that focuses on the study of pre historic and historic cultures through the excavation of material remains Contemporary current Urban city based Ethnocentrism the practice of viewing the customs of other societies in terms of one s own Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 1 in Cultural Ablbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 73 23971 blbropology by Ember cmd Peggrz39be Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture Which provide useful and interesting information Copyright Virtual University of Pakistan 3 Introduction to Cultural Anthropology SOC401 VU Lesson 02 THE CONCEPT OF CULTURE AND THE APPLICATION OF CULTURAL ANTHROPOLOGY Examining Culture We began this course by defining anthropology and its various branches We also looked at the chief duties of an anthropologist In this session we will be taking a more detailed look at cultural anthropology and its application We will also be dissecting the phenomena of culture and looking at the special functions of applied anthropology Last but not least as we all know all human occupations have their own set on ethical implications in this lecture we will be analyzing what an anthropologist owes to their profession and to society at large Before we take a more in depth look into cultural anthropology we must take a moment to first define what exactly is meant by culture In a non scientific way culture refers to such personal refinements as classical music the fine arts cuisine and philosophy So an example of this theory a person is considered more cultured if he listens to Bach rather than Ricky Martin or to make this example more nationalistic a person is said to be cultured if he listens to Nusrat Fateh Ali rather than Abrar ul Haq However anthropologists use this term in a much broader term than the average man Anthropologists don t differentiate between the cultured people and un cultured people All people have culture according to the anthropological definition We will define culture as every thing people have think and do as members of a society This definition can be most useful since the three verbs correspond to the three major components of culture That is everything people have refers to material possessions everything people think refers to the things they carry around in their heads such as ideas values and attitudes and everything people do refers too behavior patterns Thus all cultures compromise material objects ideas values and attitudes and patterned ways of behaving Just to give you better understanding of culture let us look at some of its main attributes 0 Culture includes everything that people have think and do as members of a society 0 All people have a culture 0 Culture comprises material objects ideas values and attitudes and patterned ways of behaving 0 Culture is a shared phenomenon For a thing behavior or idea to be classified as being cultural its must have a meaning shared by most people in a society Because people share a common culture they are able to predict with in limits how others will think and behave Cultural influences are reinterpreted and thus do not yield uniform effects Culture is learned One very important factor to remember about culture is that it s learned If we stop to think about it a loot of what we do during our waking hours is learned Brushing our teeth eating three times a day attending school tying our show laces these are all actions that we had to learn and yet they are an integral part of our culture While humans do have instincts culture is not transmitted genetically The process of learning culture is called enculturation which is similar in process but differs in terms of content Culture is necessary for our survival and effects how we think and act People from the same culture can predict how others will react due to cultural conditioning Copyright Virtual University of Pakistan 4 Introduction to Cultural Anthropology SOC401 VU Cultural Universals Cultural universals include economic systems systems of marriage and family education systems social control systems and systems of communication Some cultural systems are seemingly invisible such as insurance in the form of family based social safety nets many people in the developing world do not have insurance instead they rely on their families for support While it seems that these people have no one to help them in times of need they in fact do have social safety nets in the form of family support The versatility of cultural systems illustrates how exible and adaptable humans are Adaptive and Maladaptive Features of Culture Human beings rely more on cultural than biological adaptation to adjust to different types of environments including deserts and very cold areas The clothing habits of Eskimos in the North Pole allows them to live in a place which is naturally very inhospitable Biologically they are the same as us but they have learned to wear more appropriate clothing with lots of fur to keep the cold out These items of clothing have become a cultural trade mark with them Whenever we think of Eskimos we think of them laden with furs Humans can now even live in outer space or under water for limited periods of time Maladaptive or dysfunctional aspects of culture such as pollution can threaten or damage human environments The consumption of leaded petrol is bad for the environment yet given our reliance on automobiles it is difficult to do without them So what started of as an adaptive aspect allowing us to travel great distances has no become a maladaptive aspect of culture due to the sheer number of cars to be found around the world Integrative Aspects of Culture Cultures are logical and coherent systems shaped by particular contexts Various parts of culture are interconnected Yet culture is more than a sum of its parts Culture and the Individual Although culture in uences on the thoughts actions and behavior of individuals it does not determine them exclusively There is a diverse range of individuality to be found within one culture Most cultures are also comprised of subcultures for example artists in most societies have a slightly different way of dressing talking and thinking that mainstream people in their communities Applied versus Pure Anthropology Pure anthropology is concerned refining methods and theories to obtain increasingly accurate and valid anthropological data On the other hand applied anthropologists aims to understand and recommend changes in human behavior to alleviate contemporary problems ProblemOriented Research Anthropologists can apply anthropological data concepts and strategies to the solution of socio economic political problems facing different cultures Anthropologists can focus on development research or advocacy to help improve the human condition Specialized Functions for Applied Anthropologists a Policy Researcher provides cultural data to policy makers to facilitate informed decisions b Evaluator use research skills to determine how well a policy or program has succeeded in its objectives Copyright Virtual University of Pakistan 5 Introduction to Cultural Anthropology SOC401 VU C Impact Assessor measuring or assessing the effect of a particular project or policy 1 Needs Assessor use research skills to determine particular needs of a community of people e Trainer impart cultural knowledge about certain populations to different groups Ethical Implications Responsibility to the People Studied Anthropologists have an ethical responsibility to the people they are studying they need to present their finding in an unbiased way so that the true picture of their culture way of life can be presented Responsibility to the discipline The chief concern of all anthropologists should be to their discipline They must conduct their research in such a way that their findings play an integral part in consolidating their discipline Responsibility to Sponsors Most research that is done in the field is sponsored by one organization or another or in some cases some individuals are carrying out the burden of sponsorship the anthropologists must ensure that he carries out his duties with the utmost sense of responsibility Responsibility to Own and Host government Most researchers conduct research internationally where they have to respect the laws of their own country and that of the host country Useful Terms Implications results Dissection to take apart Enculturation the process by which human infants learn their culture Versatile different having a varying range Ethical moral Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapters 2 and 3 in Cuz umAm bropoogy A Applied Pmpecz z39ve 7y Fermrro cmdor Cbapz er 73 23971 H Zbropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Applied Anthropology htt wwwindianaedu wanthro a liedhtm Copyright Virtual University of Pakistan 6 Introduction to Cultural Anthropology SOC401 VU Lesson 03 MAJOR THEORIES IN CULTURAL ANTHROPOLOGY What is a theory A theory suggests a relationship between different phenomenons Theories allow us to reduce the complexity of reality into an abstract set of principles which serve as models to compare and contrasts different types of realities Theories are based on hypotheses which provide a proposition that needs to be tested through empirical investigations If what is found is consistent with what was expected the theory will be strengthened if not the theory will be either abandoned or some more time will be spent on it to revise it Anthropological theory changes constantly as new data comes forth Anthropological theories attempt to answer such questions as why do people behave the way they do And how do we account for human diversity These questions guided the early nineteenth attempts to theorize and continue to be relevant today We will explore the in chorological order the major theoretical schools of cultural anthropology that have developed since the mid nineteenth century Some of the earlier theoretical orientations such as diffusionism no longer attract much attention however others such as evolutionism have been modified and re worked into something new It is easy in hindsight to demonstrate the inherit flaws in some of the early theoretical orientations However we should keep in mind however that contempary anthropological theories that may appear plausible today were built on what we learnt from those older theories Cultural Evolutionism According to this theory all cultures undergo the same development stages in the same order To develop a better understanding of these various development stages it is important to brie y review these various stages and their sub stages Savagery barbarism and civilization were three classifications that classical anthropologists used to divide culture However in 1877 Lewis Henry Morgan wrote a book titled A dem 06769 in it the three stages of cultural anthropology were further classified into 7 stages which are as follows 0 Lower Savagery From the earliest forms of humanity subsisting on fruits and nuts 0 Middle Savagery Began with the discovery of fishing technology and the use of fire 0 Upper Savagery Began with the invention of bow and arrow 0 Lower Barbarism Began with the art of pottery making 0 Middle Barbarism Began with the domestication of plants and animals in the old world and irrigation cultivation in the new world 0 Upper Barbarism Began with the smelting of iron and the use of iron tools 0 Civilization Began with the invention of the phonetic alphabet and writing 1877212 Evolution is unidirectional and leads to higher levels of culture A deductive approach used to apply a general theory to specific cases Evolutionists were often ethnocentric as they put their own societies on top of the evolutionary ladder Yet it did explain human behavior by rational instead of supernatural causes Diffusionism Like evolutionism diffusionism was deductive and rather theoretical lacking evidence from the field It maintained that all societies change as a result of cultural borrowing from one another The theory highlighted the need to consider interaction between cultures but overemphasized the essentially valid idea of diffusion Copyright Virtual University of Pakistan 7 Introduction to Cultural Anthropology SOC401 VU Historicism Any culture is partially composed of traits diffused from other cultures but this does not explain the existing complexity of different cultures Collection of ethnographic facts must precede development of cultural theories inductive approach Direct eldwork is considered essential which has provided the approach a solid methodological base emphasizing the need for empirical evidence Each culture is to some degree unique So ethnographers should try to get the view of those being studies not only rely on their own views Historicists emphasized the need for training female anthropologists to gain access to information about female behavior in traditional societies Their anti theoretical stance is criticized for retarding growth of the anthropological discipline Psychological Anthropology Anthropologists need to explore the relationships between psychological and cultural variables according to this theory Personality is largely seen to be the result of learning culture Universal temperaments associated with males and females do not exist in practice based on research conducted by psychological anthropologists for example it was noticed that there are no universally consistent personality traits like being hard working on the basis of being a male or a female Functionalism Like historicism functionalism focused on understanding culture from the viewpoint of the native It stated that empirical fieldwork is absolutely essential Functionalists stressed that anthropologists should seek to understand how different parts of contemporary cultures work for the well being of the individual and the society instead of focusing on how these parts evolved Society was thought to be like a biological organism with all of the parts interconnected The theory argued that change in one part of the system brings a change in another part of the system as well Existing institutional structures of any society are thought to perform indispensable functions without which the society could not continue NeoEvolutionism Neo Evolution states that culture evolves in direct proportion to their capacity to harness energy The theory states that culture evolves as the amount of energy harnessed per capita per year increases or as the efficiency of the means of putting energy to work increases Leslie White1900 1975 Culture I Energy x Technology Culture is said to be shaped by environmental and technological conditions Therefore people facing similar environmental challenges are thought to develop similar technological solutions and parallel social and political institutions Cultures evolve when people are able to increase the amount of energy under their control according to this theory Given this emphasis on energy the role of values ideas and beliefs is de emphasized Useful Terms Theory a general statement about how two or more facts are related to one another Hypotheses an educated hunch as to the relationship among certain variables that guides a research project Evolutionism the 19th century school of cultural anthropology represented by Morgan and Tyler that attempted to explain variations in cultures by the single deductive theory that they all pass through a series of evolutionary stages Copyright Virtual University of Pakistan 8 Introduction to Cultural Anthropology SOC401 VU Savagery the rst amongst the three basic stages savagery barbarism and civilization of cultural evolution Barbarism the middle of the three basic stages of the 19th century theory developed by Lewis Morgan that all cultures evolve from simple to complex systems Civilization a term used by anthropologists to describe any society with cities Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 4 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermrro and or Chapter 14 in Am bropoogy 7y Ember cmd Peggrz39 e Internet Resources Anthropological Theories http wwwasuaeduantFacultymurphy436anthroshtm Use the hyperlinks on the above website to read up on the following theories for today s lecture Social Evolutionism Diffusionism and Acculturation Historicism Functionalism American Materialism Cultural Materialism Copyright Virtual University of Pakistan 9 Introduction to Cultural Anthropology SOC401 VU Lesson 04 GROWTH OF ANTHROPOLOGICAL THEORY continued French Structuralism French Structuralism focused on identifying the mental structures that underpin social behavior drawing heavily on the science of linguistics Structuralism thought that cognition based on inherent mental codes is responsible for culture Structuralism focused on underlying principles that supposedly generate behavior at the unconscious level rather than observable empirical behavior itself It focused more on repetitive structures rather than considering reasons for cultural change or variation Cultural alterations and variation are explained by reference to external environmental and historical influences Structuralism is criticized for being overly theoretical and not easily verifiable through empirical evidence EthnoScience Ethno science describes a culture using categories of the people under study emic approach rather than by imposing categories from the ethnographer s culture etic approach This theory tires to minimize bias and make ethnographic descriptions more accurate by focusing on underlying principles and rules of a given context Due to the time consuming nature of this methodology ethno science is confined to describing very small segments of given cultures It is difficult to compare native data collected by ethno scientists since there is no common basis for comparison Despite its impracticality the theory draws attention to the relativity of culture and its principles are useful for other theorists as well Cultural Materialism Cultural materialists rely on supposedly scienti c empirical and the etic approach of an anthropologist rather than relying on the viewpoints of the native informant Cultural materialists argue that material conditions and modes of production determine human thoughts and behavior Material constraints that arise from the need to meet basic needs are viewed as the primary reason for cultural variations For cultural materialist the importance of political activity ideology and ideas is considered secondary since it can only retard or accelerate change not be the cause for it Post Modernism Post modernism refutes the generalizing tendency in anthropology and does not believe that anthropologists can provide a grand theory of human behavior Instead it considers each culture as being unique Post modernism is in uenced by both cultural relativism and ethno science Post modernists want anthropology to stop making cultural generalizations and focus on description and interpretation of different cultures They consider cultural anthropology to be a humanistic not a scientific discipline Post modernists argue that ethnographies should be written collaboratively so that the voice of the anthropologist co exists alongside that of local people Interpretive Anthropology Emerging out of post modernism interpretive anthropology focuses on examining how local people themselves interpret their own values and behaviors Using an emic approach interpretive anthropologists focus on the complexities and living qualities of human nature Useful Terms Structural functionalism a school of cultural anthropology that examines how parts of a culture function for the well being of society Copyright Virtual University of Pakistan 10 Introduction to Cultural Anthropology SOC401 VU Confined limited Cultural materialism a contemporary orientation in anthropology that holds that cultural systems are most in uenced by such material things as natural resources and technology EtiC Relying on the views of the researcher or the cultural anthropologists Emic Relying on views of local people Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 4 in Cultural A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 74 23971 Lil z bropoogy by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following website for this lecture Anthropological Theories http wwwasuaeduantFacultymurphy436anthroshtm Use the hyperlinks on the above website to read up on the following theories for today s lecture Ecological Anthropology Cognitive Anthropology Structuralism Symbolic amp Interpretive Anthropologies Postmodernism amp Its Critics Copyright Virtual University of Pakistan 11 Introduction to Cultural Anthropology SOC401 VU Lesson 05 METHODS IN CULTURAL ANTHROPOLOGY Fieldwork A distinctive feature of Cultural Anthropology is its reliance on experiential eldwork as a primary way of conducting research Cultural Anthropologists collect cultural data and test their hypothesis by carrying out eldwork in different parts of the world The areas where this eldwork is conducted can include both urban and rural areas in highly industrialized rich countries or poor developing nations of the world Detailed anthropological studies have been undertaken to study the way in which people belonging to different cultures and sub cultures think and behave Comments on Fieldwork Since the credibility of ethnographic studies rests on their methods of research often termed the methodology so cultural anthropologists have begun focusing on how to conduct fieldwork While every fieldwork situation is unique there are a number of issues in common like the need to prepare for fieldwork or to obtain permission from the country s government where this research is to be conducted Even if a researcher is doing research within his her own country often permission from the concerned level of the local government is required particularly if thee research is considering how government structures institutions like schools or health clinics for example effect the lives and behavior of a particular group of people Stages of Fieldwork Selecting a research problem Formulating a research design Collecting the data Analyzing the data Interpreting the data Selecting a Research Problem 991 91 Cultural Anthropologists have moved away from general ethnographies to research that is focused specific and problem oriented The problem oriented approach involves formulation of a hypothesis which is then tested in a fieldwork setting Formulating a Research Design The independent variable is capable of effecting change in the dependent variable The dependent variable is the one that we wish to explain whereas the independent variable is the hypothesized explanation If we want to look at the effect of urbanization on family interactions the independent variable will be urbanization Defining Dependent Variables Dependent variables must be defined specifically so they can be measured quantitatively To ascertain family interaction the following issues deserve attention 0 Residence Patterns 0 Visitation Patterns 0 Mutual Assistance Copyright Virtual University of Pakistan 12 Introduction to Cultural Anthropology SOC401 VU 0 Formal Family Gatherings 0 Collecting and Analyzing Data Once the hypothesis is made concrete the data is collected through an appropriate data collection technique Once collected the data is coded to facilitate analysis For example if a questionnaire is being used to get views of 100 people in a given community all those people who say yes to a given question could be identified using a code to obtain a statistical number Then a similar questionnaire in another community could identify people responding positively to the same question In this way a researcher could compare how many people in both communities responded positively to the same question In addition to surveys other research techniques can also be coded even ethnographies can be coded to enable comparison of peoples attitudes and behavior in different communities Interpreting the Data Interpretation is the most difficult step in research which involves explaining the ndings to refute or accept the hypothesis A researcher could hypothesize that there is a link between urbanization and increasing poverty and then go into a community to see if increasing poverty is responsible for more people shifting into the city based on these findings the hypothesis could either be rejected or accepted Findings of a particular study can be compared to similar studies in other areas to get more extensive information about a particular problem or how different communities with different cultures deal with similar problems The problem of poverty and how different people react to this problem is a good example of a research problem that can be examined by different researchers and their findings compared to see how different cultures respond when they are faced by poverty Need for Flexibility A technique originally mentioned in the research proposal can prove to be impractical in the field Cultural anthropologists need some options and remain exible in choosing an appropriate technique given surrounding circumstances Difficulties in Fieldwork Research in remote locations carries risks such as exposure to diseases or different forms of social violence Researchers can encounter psychological disorientation commonly termed mz me Mac when they have to live and deal with circumstances completely alien to their own surrounds Researchers must also try to find a balance between subjectivity and objectivity if they want to assure the quality of their research and to prevent its criticism on the basis of being biased by the researcher s own viewpoints Many anthropological studies have been criticized for being biased or ethnocentric in their attempt to look at how other people live Useful Terms Ethnography detailed anthropological study of a culture undertaken by a researcher Ethnocentric the view that one s own cultural is superior Data collection of facts Biased prejudiced holding an unfair view Culture shock psychological disorientation brought on due to cultural difference Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Copyright Virtual University of Pakistan 13 14 Introduction to Cultural Anthropology SOC401 VU Lesson 06 METHODS IN CULTURAL ANTHROPOLOGY continued Participant Observation Anthropologists use this technique more extensively and frequently than other social scientists Participant Observation means becoming involved in the culture under study while making systematic observations about what goes on in it Guidelines for Participant Observation Fieldwork Before approaching the field it is advisable to obtain clearance from all appropriate levels of the political administrative hierarchy Local people at the grassroots level know their own culture better than anyone else and their views need to be given due respect Advantages of Participant Observation It allows distinguishing between what people say they do and what they actually do The greater the cultural immersion is the greater is the authenticity of cultural data It allows observation of non verbal behavior as well Disadvantages of Participant Observation There are problems of recording observations while using this technique The technique has an intrusive effect on subject of study Also a smaller sample size is obtained through this technique than through other techniques and the data obtained is hard to code or categorize making standardized comparisons difficult Interviewing Enables collection of information on what people think or feel allitudim data as well as what they do bebcwz39om dam Ethnographic interviews are often used alongside other data gathering techniques Structured and Unstructured Interviews In structured interviews interviewers ask respondents exactly the same set of questions in the same sequence Unstructured interviews involve a minimum of control with the subject answering open ended questions in their own words Guidelines for Researchers To minimize distortions in collected data researchers can check the validity of their findings by either asking cross check information given by respondents or repeat the same question at a later time It is important to frame the questions neutrally Instead of asking You don t smoke do you ask Do you smoke Census Taking Collecting basic demographic data at the initial stages of fieldwork is the least intrusive manner to begin investigating the state of a given community Copyright Virtual University of Pakistan 15 Introduction to Cultural Anthropology SOC401 VU Document Analysis Documentary analysis of administrative records newspapers and even popular culture like song lyrics or nursery rhymes is often surprisingly revealing about the circumstances aspirations and values of different people Genealogies Mapping relations of informants particularly in small scale societies is very revealing since they tend to interact more closely with their families than people in more complex societies which have a greater number of institutions and professionals Photography Cameras and video recorders allow researchers to see without fatigue without being selective and provide a lasting record of cultural events and physical surroundings Some local communities however can object to the use of cameras due to their conservative values or they consider it an intrusion on their privacy Choosing a Technique Choice of technique depends on the problem being studied Choice of a technique also depends on the receptiveness of the community in question to a particular technique For example if a given community does not allow the anthropologist undertaking research to use cameras the researcher will have to respect the wishes of the community in question and document descriptions of relevant events instead of being able to take a photograph by which this information could have been captured more easily Undertaking CrossCultural Comparisons For undertaking such comparisons particular with the help of statistics the following issues deserve attention 0 Quality of data being compared must be consistent and based on the same methodology information based on interviews conducted in one culture cannot be compared with information obtained from questionnaires in another culture 0 Units of analysis must be comparable it s not possible to compare different levels of social systems a village cannot be compared to a city for example 0 Contrasting cultural traits out of context from their remaining culture is problematic but useful in identifying similarities across different cultures which is an important objective for cultural anthropology Useful Terms Attitudinal based on how people think or feel about something Receptiveness response to a particular action Participation being a part of something Perspective point of view Cultural traits particular features of a culture Crosscultural comparison of differences between cultures Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Copyright Virtual University of Pakistan 16 17 Introduction to Cultural Anthropology SOC401 VU Lesson 07 COMPARATIVE STUDY OF PRODUCTION DISTRIBUTION AND CONSUMPTION IN DIFFERENT PARTS OF THE WORLD What is Economic Anthropology Economic Anthropology involves examining how different cultures and societies produce distribute and consume the things they need to survive All cultures need to be able to manage these processes in accordance with their given circumstances to ensure the survival of their people Differentiating Economics from Economic Anthropology While economists assume that people are preoccupied by the need to maximize profits and this is the basic impulse due to which they produce goods and services Economic anthropologists do not believe profit maximization is equally important for all cultures They point out that there are several other processes besides profit maximization which exist in different cultures of the world by the allocation distribution of resources need to produce goods and services and the distribution of the goods and services takes place For example these economic anthropologists look at how different cultures distribute land which is an important resource needed for production of agricultural goods and have noticed that different cultures have different ways in which this distribution takes place However economic anthropologists realize that like economists they too must answer some basic questions concerning basic economic needs of human beings which all cultures around the world face given that some human needs are universal and must be met no matter what type of culture people belong to Economic Universals Economic anthropologists have to consider the following economic universals which are of vital importance to human beings no matter what their cultural systems are like a Regulation of Resources How land water and other natural resources like minerals are controlled and allocated b Production How material resources sugarcane are converted into usable commodities sugar c Exchange How the commodities once produced are distributed among the people of a society Examining the Issue of Land Rights Free access to land is found in environments where water and pasturage is scarce Land rights are more rigidly controlled among horticulturalists and agriculturalists than among foragers and pastoralists Division of Labor Durkbez39m the famous sociologist responsible for establishing this branch of study in the early twentieth century had distinguished between two types of societies those based on mecbam39m solidarz and others based on organic solidarigl Societies with a minimum specialization of labor are held together by mechanical solidarity based on commonality of interest In these societies people are more self reliant therefore they need other people to a lesser degree than people in societies where people focus on production of a very specific good or service and then rely on others to provide them other necessities of life in exchange for their specialized product Copyright Virtual University of Pakistan 18 Introduction to Cultural Anthropology SOC401 VU Highly specialized societies are held together by organic solidarity based on mutual interdependence Such societies emphasize the need for specialization and people depend on other people in order to obtain the different things that they need Gender Roles and Age Specialization Generally many cultures allocate specific responsibilities on the basis of age and gender Ole people and those very young are given lighter tasks in most cases where circumstances permit cases of extreme poverty child labor can also take place Similarly women are usually allocated tasks which allow them to maintain exible timings so that they can look after their homes as well There are exceptions to this rule however since many educated women do work as long as men often leaving their children to the care of day centers In many countries around the world the process of urbanization has led men to move away to the cities in order to earn more cash often leaving women behind to undertake agricultural work which was previously done by men Circumstances also compel poor women to take on heavy work burdens like their men folk to ensure the survival of their families Moreover the same type of activity weaving may be associated with the opposite gender in different cultures the division of labor by gender is seen as being arbitrary Is Nepotism Always Bad In many societies people relate to each other based on the principle of particularism family and kinship ties rather than on universalistic terms using standardized exams interviews Nepotism is not necessarily a sign of corruption since consideration of ground realities like kinship ties can often help determine how people will adjust to specific work environments Useful Terms Allocation of resources the distribution of resources Barter the direct exchange of commodities between people that does not involve a standardized currency Division of Labor the set of rules found in all societies dictating how the day to day tasks are assigned to the various members of a society Reciprocity the practice of giving a gift with an expected return Globalization the world wide process dating back to the demise of the Berlin wall which involves a revolution in information technology opening of markets and the privatization of social services Labor specialization a form of having command over one activity Copyright Virtual University of Pakistan 19 Introduction to Cultural Anthropology SOC401 VU Internet Resources In addition to reading from the textbook please visit the following website for this lecture Economic Anthropology http enwikipediaorg wiki Economic anthropology Use the hyperlinks on the above website to read up on the following aspects of Economic Anthropology for today s lecture Anthropological theories of value The Anthropological view of Wealth Copyright Virtual University of Pakistan 20 Introduction to Cultural Anthropology SOC401 VU Lesson 08 ECONOMIC ANTHROPOLOGY continued THE DISTRIBUTION OF GOODS AND SERVICES Modes of Distribution Economic Anthropologists categorize the distribution of goods and services in three modes reciprocity redistribution and market exchange Based on these three forms of exchange cultures around the world distribute the goods and services produced by them in order to ensure the survival of the various people which belong to that particular culture 1 Reciprocity implies exchange of goods and services of almost equal value between two trading partners 2 Redistribution most common in societies with political bureaucracies is a form of exchange where goods and services are given by a central authority and then reallocated to create new patterns of distribution 3 Market exchange systems involve the use of standardized currencies to buy and sell goods and services Types of Reciprocity The idea of reciprocity can be divided into the following distinct types of practices evident in cultures around the world 1 Generalized reciprocity involves giving gifts without any expectation of immediate return For example the parents look after their children and these children when they grow older look after their aging parents This is an unsaid rule or obligation towards one family which people undertake willingly out of love and concern and without any external compulsion or the idea of getting something back in return for their caring attitudes 2 Balanced reciprocity involves the exchange of goods and services with the expectation that the equivalent value will be returned within a specific period of time For example if a neighbor s son or daughter is getting married the neighbors will take gifts to the wedding and then expect the same courtesy when their own child s wedding The notion of birthday gifts is even more time specific and thus serves as a good example of balanced reciprocity 3 Negative reciprocity involves the exchange of goods and services between equals in which the parties try to gain an advantage in order to maximize their own profit even if it requires hard bargaining or exploiting the other person Redistribution Whereas reciprocity is the exchange of goods and services between two parties redistribution involves a social centre from which goods are redistributed Often this redistribution takes place through a political or bureaucratic agency eg the revenue collection or tax department which is found in most countries or even the meal system in Pakistan based on a religious ideology which is meant to redistribute wealth to those who are destitute Market Exchange Market exchange is based on use of standardized currencies or through the barter exchange of goods and services This system of exchange is much less personal than either reciprocity or redistribution People Copyright Virtual University of Pakistan 21 Introduction to Cultural Anthropology SOC401 VU trade in a marketplace to maximize their pro ts The greater the specialization of labor that exists in a society the more complex is the system of market exchange to be found in that society Globalization Globalization involves the spread of the free market economies to all parts of the world based on the assumption that more growth will take place when free trade and competition becomes a universal phenomenon Globalization has begun to show visible impacts on the cultures and lives of people around the world There are people who favor globalization thinking it will help remove poverty across much of the world but they are also those who think that globalization will do the exact opposite Useful Terms Organic Solidarity a type of social integration based on mutual inter dependence Particularism the propensity to be able to deal with people according to one s particular relationship to them rather than according to a universal standard Production a process where by goods are taken from the natural environment and then altered to become consumable goods for society Property Rights western concept of individual ownership Standardized Currency a medium of exchange with well defined and an understood value Universalism the notion of awarding people on the basis of some universally applied set of standards Internet Resources In addition to reading from the textbook please visit the following website for this lecture Economic Anthropology http enwikipediaorg wiki Economic anthropology Use the hyperlinks on the above website to read up on the following aspects of Economic Anthropology for today s lecture Non market economics Copyright Virtual University of Pakistan 22 Introduction to Cultural Anthropology SOC401 VU Lesson 09 FOCUSING ON LANGUAGE An Anthropological Perspective Language is a unique phenomenon which allows human beings to communicate meaning to others and express our thoughts and feelings to other people Perhaps the most distinctive feature of being human is our capacity to create and use language Many anthropological linguists would agree that without language human culture could not exist beyond a very basic level The Nature of Language The meaning we give to language is arbitrary random It is due to this arbitrary nature of language there is such a diversity of languages Languages of the World Almost 95 percent of people speak fewer than 100 languages of the approximately 6000 languages that are currently found in the world Due to this many languages face the threat of extinction with an increasingly small number of people who know the language This evident dying out of rarely spoken languages is an issue of concern to cultural anthropologists since the extinction of a language also means the death of a way of thinking and expressing human thought Of the more widely spoken languages Mandarin Chinese dialect is spoken by almost 1 in 5 people in the world Hindi is also spoken by multitudes of people Yet English is the most popular second language spoken by people all around the world Communication Human versus Nonhuman Humans are not the only species that communicate Animals use calls to mate find food and signal danger Human communication amongst humans is however much more complex than that of animals We can combine words in unique ways to express our innermost feelings or even very complicated ideas which can be understood by others who can speak the same language Open and Closed Communication Systems Animal sounds are mutually exclusive Closed Communication systems they cannot be combined to express new meanings A warning sound of an animal is always the same and this sound is used to convey the same message always it cannot be combined with other sounds to convey different types of meaning Only humans can put different meanings together through using of an Open Communication system which is the language they speak This categorization of Open and Closed communication systems has been questioned by anthropological linguists based on research conducted using sign language A chimpanzee for example can in fact combine two words to create a third word Researchers have trained a chimpanzee to learn the sign language for water and for bird but not shown it how to say duck using sign language This chimpanzee has however been able to create the two known words water and bird to refer to a duck indicating that other species could also use open communication systems like humans However no linguist has yet made the claim that any animal species has evolved language to a degree which can express the complexities of meaning that human beings can Copyright Virtual University of Pakistan 23 Introduction to Cultural Anthropology SOC401 VU Displacement Humans can speak of purely hypothetical and abstract things of things which happened in the past or may happen in the future Whereas animals only communicate in the present about things concerning their immediate surroundings animals cannot express abstract thoughts Learning to Communicate Imitating adult speech is partially responsible for acquisition of language Linguists like Noam Chomsky at the Massachusetts Institute of Technology think that children are born with a universal grammatical blueprint which helps them pick up the rules of the language being spoken around them so quickly and that this is a biological gift that only the human species seems to possess since no other species has such compleX communication abilities Structure of Language All languages have logical structures or rules which are followed by all those who can speak read and write that particular language 0 Phonology provides the sound structure to a language so it can be commonly understood when spoken 0 Morphemes the smallest units of speech that convey meaning art ist s by standing alone or being bound to other words 0 Grammar provides the unique rules of a language which help give a logical structure to a language Grammar also provides rules by which words are arranged into sentences AgWax Consider the words Adam apples likes eating which make no sense since the verb eating and the adjective likes are not in their grammatically correct position Correcting the mistake will make the sentence clear Adam likes eating apples The underlying structure of sentences which enables us to correct such a mistake and speak in a clear manner is due to the grammatical rules of syntaX The fact that we can even say this sentence is due to phonology and morphemes help us create a sentence by providing us with different meanings in smaller words eat ing like s Useful Terms Displacement the ability that humans have to talk about things remote in time and space Free Morphemes morphemes that appear in a language without being attached to other morphemes Grammar the systematic way in which sounds are combined in any given language to send and receive meaning full utterances Phonology the study of language s sound system Copyright Virtual University of Pakistan 24 Introduction to Cultural Anthropology SOC401 VU Internet Resources In addition to reading from the textbook please Visit the following website for this lecture Copyright Virtual University of Pakistan 25 Introduction to Cultural Anthropology SOC401 VU Lesson 10 FOCUS ON LANGUAGE continued Changes in Language Language evolves over time Linguists can undertake a Jymbmm39c mag3239s to understand language structures and its underlying rules at a given point in time Undertaking a diacbmm39c mag5239s however means looking at how a given language changes over time Language Families Language families include languages derived from a proto language Linguists began clustering languages upon finding similarities between Sanskrit and classical Latin and Greek in the 18805 From the perspective of language families Germanic is mother tongue of English French and Spanish are its sister languages They all belong to the Indo European language family All languages have internal dialects as well as sharing features with other languages as well particularly with those belonging to the same language family as them Levels of Complexity Linguists have proven that languages of less technological societies are as capable of communicating abstract ideas as advanced societies For example the Navaho do not have singular and plural nouns like English does but their verbs contain much more information than English Instead of merely saying going the Navaho say how they are going if they are going on a horse they must further indicate how fast the horse is going which is a lot more information than a phrase in English which just mentions I am going Cultural Emphasis The vocabulary of languages emphasizes significant words in a given culture This is known as a cultural emphasis Technologically related words show emphasis on technology in highly industrialized countries There are numerous new words to describe computer technology in various languages which did not even exist a few decades ago Language and Culture Some ethno linguists suggest that language is more than a symbolic inventory of experiences from our physical world experiences According to them language even shapes our thoughts and provides a standardized way to react to experiences The SapirWhorf Hypothesis According to this hypothesis different cultures see the world differently due to their different languages Language in uences and channels our perceptions and thus shapes our resulting behavior as well The hypothesis has conducted several tests in the attempt to validate its claim Linking Language to Culture It is difficult to establish causation to prove either that language determines culture or that culture in uences language Language does mirror values of a culture consider for example the emphasis on self in individualistic societies On the other hand in more traditional societies like Japan the use of collective words like we is much more evident Copyright Virtual University of Pakistan 26 Introduction to Cultural Anthropology SOC401 VU SocioLinguistics Socio linguistics examines links between languages and social structures While earlier cultural linguists focused on language structures there is now greater focus on the situational use of language ie how the same language is used to speak in different manners depending on the context of the conversation Diglossia Often two varieties of the same language are spoken in different social situations High forms are associated with literacy and education and the elite whereas the lower forms for example Pidgin are considered to be less sophisticated Language and Nationalism Language has important implications for ethnic identities To forge national unity political leaders have often suppressed use of local languages in favor of standardized national languages to provide a sense of unity to the nation and to develop a common means of communication Useful Terms Evolve Develop Synchronic Analysis the analysis of cultural data at a single point in time rather than through time Diachronic Analysis the analysis of socio cultural data through time rather than at a single point in time Derived taken from Abstract Not clear or vague Emphasis To lay importance on Perceptions Viewpoints Suppressed concealed or covered up Dialect form of speech peculiar to particular region Internet Resources In addition to reading from the textbook please visit the following website for this lecture Copyright Virtual University of Pakistan 27 Introduction to Cultural Anthropology SOC401 VU Lesson 11 OBTAINING FOOD IN DIFFERENT CULTURES Strategies for Obtaining Food Food obtaining strategies vary from culture to culture Food obtaining strategies are developed in response to particular environments There are five major food obtaining strategies found in different cultures of the world These five forms of obtaining food are not mutually exclusive and within each category there are evident variations due to technological and environmental differences Therefore often one form of obtaining food predominates within a given culture While food obtaining strategies vary widely around the world none is necessarily superior then another Major Food Obtaining Strategies Food Collection collecting wild vegetation hunting and fishing Horticulture cultivation using simple tools and small and shifting plots of land Pastoralism keeping livestock and using its produce for food Agriculture cultivation using animals irrigation and mechanical implements Industrialization producing food using complex machinery Most developed countries and an increasing amount of developing countries rely on industrialized processes to obtain food Food Environment and Technology Some environments enable a number of modes of food acquisition while others permit limited number of adaptations Technology provides the advantage of adaptation to a given environment It can be said that specific food obtaining modes are in uenced by the interaction of a people s technological and environmental conditions The extent to which any society can procure food depends on sophistication of tools used and the abundance of plant and animal life in a given area Productivity of agriculturalists not only depends on technology but also availability of natural resources like water and fertile soil Anthropologists agree that while the environment does not set limits on food obtaining patterns it does place a limit on the adaptations possible and on the ultimate productivity of an area People with simple technologies also cope well with their environments and are intelligent given their circumstances and surroundings The environmental capacity of a given area is referred to as carrying capacity The natural consequence of exceeding carrying capacity is to harm the environment Optimal Foraging Many foraging societies spend extra time and effort to obtain a particular food Ethnographic studies of the Ache in Paraguay for example have revealed that this is not irrational behavior but due to caloric returns of these food sources despite the energy expended in killing collecting and preparing it This reveals that optimal foraging is a calculated strategy not a irrational whim Useful Terms Foraging collecting or gathering Copyright Virtual University of Pakistan 28 Introduction to Cultural Anthropology SOC401 VU Optimal best or most feasible Expended spent Environmental capacity carrying capacity of the environment ie the amount of productive pressure the air water and soil can take Without being damaged Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermrro cmdor Cbapz er 76 23971 Lil z bropology 7y 29 Introduction to Cultural Anthropology SOC401 VU Lesson 12 FOOD AND CULTURE continued Food Collection Food collection involves systematic exploration of natural plants and animals available in given natural environments People have been foragers for an overwhelming majority of time and have only developed other options to secure food in the last 10000 years or so Food Collectors Most societies prefer to produce food but half a million people in different cultures live by foraging even today There are considerable variations in the life patterns of current foragers but it is possible to make some generalizations about them Contemporary Food CollectorsForagers Food collecting societies have low population density They are usually nomadic or semi nomadic rather than sedentary since their prey often migrates The basic social unit amongst food collectors is a family or a band a loose federation of families Contemporary food collectors occupy remote and marginal habitats due to pressure from food processing people with their dominating technology and thirst for more land While food collectors hunt as well as collect wild plants vegetation provides almost 80 of their food intake Food collecting people live in a wide variety of environments including deserts tropical forests mountains and the polar regions of the Artic and Antarctic circles Unlike food producers food collectors possess inbuilt mechanisms low population and little use of technology which prevents it from becoming too efficient and completely destroying their own source of food Do Foragers Live Well Despite inhabiting the most unproductive parts of Earth foragers are well off and dubbed the original af uent society by anthropologists They enjoy leisure time have enough food and use remarkable intelligence and ingenuity in securing their food Most contemporary foraging societies remain small scale unspecialized egalitarian and non centralized The Khung in the Kalahari Desert in Namibia and the Inuit in the Artic region provide good examples of hunting and gathering peoples today Food Production About 10000 years ago humans made a transition from collecting to producing food by cultivating crops and keeping herds of animals The earliest cultivation occurred in the Fertile Crescent of the Middle East Archeologists think this transition was due to demographic and environmental pressures Early farmers paid a high price for this new food strategy They did not switch convinced by the superiority of agriculture which was more monotonous less secure and required more labor and time Evidence reveals early cultivators also experienced a decline in nutritional and health standards because they had to shift from collecting to growing food Changes Resulting From Neolithic Revolution Food production resulted in the first population explosion Fertility rates also increased since children could make an economic contribution People became sedentary and civilizations began to develop As farming became more efficient people had more free time and began making farm implements and pottery leading to the division of labor and specialization The egalitarianism of foraging societies was replaced by social inequalities and the thirst for private ownership Copyright Virtual University of Pakistan 30 Introduction to Cultural Anthropology SOC401 VU Useful Terms Population number of people in a given area Sedentary settlement or settled down in one place Strategy a thought out method to obtain some objective or goal Monotonous boring Nutritional value amount of energy Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um Ambropoogy A Applied Pmpm z39ve 7y Fermrro cmdor Cbapz er 76 23971 Lil z bmpoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Food http2 WWWarchaeolinkcom anthropology of food general reshtm Copyright Virtual University of Pakistan 31 Introduction to Cultural Anthropology SOC401 VU Lesson 13 OBTAINING FOOD IN DIFFERENT CULTURE continued Horticulture Horticulture is the simplest form of farming using basic tools no fertilizer or irrigation and relying on human power Horticulturists use shifting cultivation techniques also referred to as slash and burn cultivation Horticultural Crops Crops growth by horticulturists can be divided into three categories tree crops seed crops and root crops Common tree crops include bananas figs dates and coconuts Major seed crops are high in protein Wheat barley rice millet oats and sorghum are all seed crops Major root crops are high in starch and carbohydrates Yams sweet potatoes potatoes are all root crops The Lacondon Maya of Chiapas Mexico are more productive than mono crop agriculturalists They achieve three levels of production from the same land and do so by maintaining by imitating the dispersal patterns found within ecological systems of tropical rainforest rather than displacing them Slash and Burn Technique In unused areas of vast land slash and burn can be a reasonably efficient form of production Ash fertilized soil resulting from slashing and burning wild vegetation must lie fallow to restore fertility Under drought conditions of Al Nino during the 1990s horticulturists were severely criticized for destroying large tract of grasslands and forests in Madagascar Brazil and Indonesia since the fires they lit for clearing land often raged out of control Pastoralism Keeping domesticated livestock as a source of food is widely practiced in areas where cultivation is not possible Pastoralism involves a nomadic or semi nomadic lifestyle within small family based communities Pastoralists also maintain regular contact with cultivators to help supplement their diets Agriculture More recent than horticulture agriculture uses technologies like irrigation fertilizers and mechanical equipment to produce high yield and large populations Agriculture is associated with permanent settlements and high levels of labor specialization Intensive agriculture leads to even further specialization and use of technological inputs It also leads to social stratification political hierarchies and administrative structures Industrialization Since several centuries people have used industrialized food getting strategies There is increasing amounts of mechanical power available for the purpose of obtaining storing and processing food Industrialization also uses a mobile labor force and a complex system of markets which has led to the increasing commercialization of food Therefore food is grown not only for consumption but also for exporting to other countries of the world Copyright Virtual University of Pakistan 32 Introduction to Cultural Anthropology SOC401 VU Biotechnology provides a current example of industrialized food getting as does laser leveling or use of GPS transmitters on grain harvesters All these technological innovations have been incorporated into the food production process and helped to increase food output Yet there are environmental costs resulting from exceeding the carrying capacity of land and from overuse of technological innovations such as pesticides and fertilizers The use of biotechnology in food production is also a much debated topic Useful Terms Horticulture A form of small scale crop cultivation characterized by the use of simple technology and the absence of irrigation Carbohydrates energy source found in particular types of food group Tropical humid Drought lack of rainfall Criticize disapprove of Supplement add on Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 7 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro cmdor Cbapz er 76 23971 Lil z bropoogy by 33 Introduction to Cultural Anthropology SOC401 VU Lesson 14 RELEVANCE OF KINSHIP AND DESCENT Kinship Defined Kinship is the single most important social structure in all societies Kinship is based on both consanguineal blood and affinal marriage relations or even fictive ties adoption godparents Functions of Kinship Vertical Function Kinship systems provide social continuity by binding together a number of generations Horizontal Function Kinships provide social solidarity and continuity Within the same generation as well Cultural Rules Regarding Kinship Kinship systems group relatives into certain categories and call them by the same name and behave With them in a similar manner Yet how particular cultures categorize relatives varies according to different principles of classification Kinship Criteria Different societies use different rules in formulating kinship ties Some of these are Generation uncles are in one generation cousins in another Gender cousins do not occupy gender determined kin categories Lineality kin of a single line ie son father grandfather Consanguineality kin through a linking relative Wife s brother Relative Age one kinship term for father s older brother another for his younger brother eg m and chacha Gender of Connecting Relative using different kinship terms for the father s brother s daughter his sister s daughter Social Conditions different kinship terms for a married or an unmarried bother Side of the Family different kinship terms for father s and mother s sides of the family eg phupho and khala Rules of Descent Rules of descent enable the affiliation of people with different sets of kin for example Patrilineal descent affiliates a person With the kin of the father Matrilineal descent affiliates a person With the kin of the mother Ambilineal descent permits an individual to affiliate With either parent s kin group Consanguineal versus Affinal Kin Copyright Virtual University of Pakistan 34 Introduction to Cultural Anthropology SOC401 VU Some societies make a distinction in kinship categories based on whether people are related by blood mmcmgm39 ea km or through marriage 7161 km For example take the difference between a sister and a sister in law or a brother and a brother in law Comparing Descent Groups Patrilineal descent groups are most common around the world The relations between man and wife tend to be more fragile in matrilineal societies Useful Terms Unilineal descent tracing descent through a single line such as matrilineal or patrilineal as compared to both sides bilateral decent Bilateral able to accommodate two sides simultaneously Matrineally mother s side of the family Patrineally father s side of the family Prevalent common amongst many Kinship relationship Merging integration Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 10 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 27 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture which provide useful and interesting information Kinship Terms wwwmnsuedu emuseumz culturalz kinship 1 terms html Copyright Virtual University of Pakistan 35 Introduction to Cultural Anthropology SOC401 VU Lesson 15 KINSHIP AND DESCENT continued Tracing Descent In societies that trace their descent unilineally people recognize that they belong to a particular unilineal descent group or series of groups Sixty percent of cultures in the world are unilineal Unilineal groups are adaptive and clear cut social units based on birthright which in turn in uence inheritance marriage and prestige issues Kinship Organization Kinship is organized on the basis of different groups of varying sizes Lineages are based on a set of kin who can trace their ancestry back through known links Clans are unilineal groups which claim descent but they are unable to trace all their genealogical links Phratries are groups of related clan Moieties are two halves of a society related by descent Bilateral Descent A person is related equally to both sides of the family on the basis of bilateral descent This form of descent is prevalent in foraging and industrialized societies Bilateral systems are symmetrical and result in the formation of kindred which are loose kinship networks rather than being permanent corporate functioning groups Double Descent A double unilineal descent system is one where descendents are traced matrineally and patrineally As a result both sides of the family have a useful social function such as enabling inheritance Under this system it is possible for moveable property such as livestock or agricultural produce to be inherited from the mother s side whereas non moveable property land may be inherited from the father s side This system is found in only 5 of world cultures for eg the Yako in Nigeria Primary Kinship Systems There are siX basic types of kinship systems used to define how cultures distinguish between different categories of relatives Eskimo are found in one tenth of the world societies this system involves bilateral descent focusing on nuclear relations and lumping external relatives cousins uncles and aunts Hawaiian are found in a third of world societies this system uses the same term for all relatives of the same gender and generation so the term walker is used not only for the mother but also for her sisters and the father s sisters Cousins are termed brothers and sisters This is an ambivalent system which submerges the nuclear family into a larger kinship group Iroquois are a less prevalent system which emphasizes the importance of unilineal descent groups by distinguishing between members of one s own lineage and those belonging to other lineages Sudanese are the system is named after the country where the system was first identified It is the most pluralistic system since it makes the most terminological distinctions Copyright Virtual University of Pakistan 36 Introduction to Cultural Anthropology SOC401 VU Omaha emphasizes patrilineal descent the father and his brothers are referred to by the same term and the paternal cousins are called siblings but cross cousins are referred to by separate terms On the mother s side there is a merging of generations all her male relatives are called mother s brother Crow is the exact opposite of the Omaha system as it emphasizes maternal relations which are all important for determining the descent group of children Kinship Diagrams Cultural Anthropologists often use kinship diagrams to help explain family structures which use simple symbols for males and females and to indicate what their relationships are to each other The diagram below depicts a married couple and their two children a son and a daughter Useful Terms 0 j Descent established linage Ambivalent unclear In uence power Social function the particular purpose served for society A G Prestige Social honor or respect within a society Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 10 in Cuz um A lbropology A Applied Pmpecz z39be by Fermrro andor Cbapz er 27 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web pages for this lecture which provide useful and interesting information Descent Terms and Concepts http referenceallrefercom encyclopedia D descenthtml Copyright Virtual University of Pakistan 37 Introduction to Cultural Anthropology SOC401 VU Lesson 16 THE ROLE OF FAMILY AND MARRIAGE IN CULTURE Family and Marriage The family is a social unit in which its members cooperate economically manage reproduction and child rearing and most often live together Families can be based on lineage and marital ties Marriage the process by which families are formed is a socially approved union between male and female adults Marriage is based on the assumption that it is a permanent contract Yet there is a discrepancy between real and expected behavior within marriages given the high rates of divorce in many countries of the modern world Functions of Families Families reduce competition for spouses They also regulate the division of labor on the basis of gender Families also meet the material educational and emotional needs of children Marriage Restrictions Cultures restrict the choice of marriage patterns by exogamy which means marrying outside a given group E dogamy on the other hand implies marrying within a given group Conservative Hindus are mostly endogamous as are Rwandans in Central African It is important to note that endogamous groupings can be based on lineage or even ethnic or economic similarities Moreover it is possible to simultaneously have an endogamous marriage within an ethic group that is also exogamous outside one s lineage Types of Marriage Monogamy a marriage arrangement that implies having one spouse at one time Polygamy a marriage arrangement that implies a man marrying more than one woman at one time Polyandry a marriage arrangement that implies a woman marrying more than one husband at one time Economic Aspect of Marriage Marriages involve transfer of some type of economic consideration in exchange for rights of union legal rights over children and rights to each other s property There are many cultures in the world which consider marriage as more than a union of man and wife but instead an alliance between two families Types of Marriage Transactions Bridewealth transfer of wealth from a groom s family to that of the bride s approximately 47 Bride service labor in exchange for a wife common in small scale societies lacking material wealth approximately 17 Dowry transfer of wealth from a bride s family to that of the groom s This practice was popular in medieval Europe and may still prevail in several parts of Northern India approximately 3 Copyright Virtual University of Pakistan 38 Introduction to Cultural Anthropology SOC401 VU Woman Exchange two men exchanging sister s as wives This practice is limited to a small number of societies approximately 3 in Africa and the Paci c region Reciprocal Exchange a roughly equal exchange of gifts between bride and groom families Found amongst traditional Native Americans and islands in the Paci c region approximately 6 Note These above statistics are not very recent and should not be taken literally but rather as an indication of the popularity of the above types of transactions Useful Terms Discrepancy difference Reciprocal equal Groom husband Reproduction process of giving birth to children Transaction exchange pages for this lecture which provide useful and interesting information Family enwikipediaorg wiki Family Copyright Virtual University of Pakistan 39 Introduction to Cultural Anthropology SOC401 VU Lesson 1quot ROLE OF FAMILY AND MARRIAGE IN CULTURE continued Residence Patterns Residence patterns are in uenced by kinship systems For eg patrilocal residence is common in patrilineal cultures Residence patterns can be disrupted due to events such as droughts famines wars or even due to economic hardship The most common types of residence patterns evidenced around the world are Patrilocal the couple can live with or near the relatives of the husband s father most prevalent Matrilocal the couple can live with or near the relatives of the wife s father Avunculocal the couple can live with or near the husband s mother s brother Ambilocal or bilocal the couple can live with or near the relatives of either the wife or the husband Neolocal Where economic circumstances permit the couple can also establish a completely new residence of their own Residence patterns are not static The Great Depression in America during the 19305 for example compelled neolocal residents to shift back to living with one of their parents again due to economic reasons Similar circumstances keep recurring in different societies of the world and result in changing residence patterns In many traditional societies joint family systems are also very common The dynamics of a joint family system differ from widely from living independently implying a shared responsibility for household responsibilities often under the charge of the oldest member of the household Family Structures Cultural Anthropologists distinguish between two types of family structures the nuclear family and the extended family Nuclear families are based on marital ties whereas the extended family is a much larger social unit based on blood ties among three or more generations Nuclear Family A two generation family formed around the marital union While a part of bigger family structures nuclear families remain autonomous and independent Nuclear families are often found in societies with greatest amount of geographic mobility Nuclear family patterns were encouraged by industrialization and technology but also have remained evident in foraging societies Where resources are scare it makes sense for people to remain in nuclear families whereby retaining a certain level of mobility independence Nuclear families are therefore called the basic food collecting unit in addition to being the most dominant mode of family life in many modern day families around the world Extended Families Blood ties are more important than ties of marriage which form the basis of extended families Extended families can be matrilineal or patrilineal The Anthropological Atlas of 1967 noted 46 out of 862 societies Copyright Virtual University of Pakistan 40 Introduction to Cultural Anthropology SOC401 VU as having some form of extended family organization These numbers have no doubt increased over the past few decades given the increasing world population ModernDay Families Modernization and urbanization have seen progressive movement towards nuclear family structures In developing countries this correlation is not necessary The lack of employment security makes extended families serve as social safety nets Migrant families also hold onto traditional family structures even after having gone to live abroad In western societies even nuclear families are not so common given high divorce and separation rates Useful Terms Prevalent commonly found in different places Migrant refugee Correlation association between two entities Scarce in short supply Evident obvious site for this lecture which provide useful and interesting information Family enwikipediaorg wiki Family Copyright Virtual University of Pakistan 41 Introduction to Cultural Anthropology SOC401 VU Lesson 18 GENDER AND CULTURE Meaning of Gender Gender refers to the way members of the two sexes are perceived evaluated and expected to behave It is not possible to determine the extent to which culture or biology determines differences in behaviors or attitudes between males or females Although biology sets broad limits on gender de nitions there is a wide range of ideas about what it means to be feminine or masculine Margaret Mead demonstrated this gender based variation in her classical study of sex and temperament in New Guinea Gender Roles In some cultures gender roles are rigidly defined in other cultures they can overlap In general terms however there is considerable uniformity in gender roles found throughout the world Men engage in warfare clear land hunt and trap animals build houses fish and work with hard substances Women on the other hand tend crops prepare food collect firewood clean house launder clothes and carry water tasks compatible with child rearing Yet there are many exceptions to the rule For example in parts of Eastern Africa and in other parts of the developing world women carry enormous amounts of firewood on their backs For the foraging Agta of the Philippines hunting is not an exclusively male activity Status of Women The status of women is multidimensional involving such aspects as the division of labor the value placed on women s contributions economic autonomy social and political power legal rights levels of deference and the extent to which women control the everyday events of their lives The status of women varies around the world but it is unfortunate that in most cases it continues to remain below that of men Gender Stratification Gender stratification contrasts the status assigned by different cultures on the basis of gender It is important to release that status is itself a multidimensional notion involving issues of economic social and political empowerment Stratification on the basis of gender is a common phenomenon The relationships between men and women vary in both degree and in extent across different cultures of the world Many cultures in Asia for example are very stratified along gender lines On the other hand foraging societies like the Mbuti Pygmies of Central Africa possess a very egalitarian gender approach all their elders are called tata Gender stratification need not be static However in most critical areas women tend to be subordinate to men in most societies of the world It is difficult to measure the comparative status of men and women in different societies since there are various components of stratification which can vary independently of each other Useful Terms Status social ranking Copyright Virtual University of Pakistan 42 Introduction to Cultural Anthropology SOC401 VU Acquisition gaining Interaction communication Abundance profusion or great quantity Multidimensional many sided Stratification hierarchical division Irrational Without logic Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 11 in Cuz um A lbropology 43 Introduction to Cultural Anthropology SOC401 VU Lesson 19 GENDER ROLES IN CULTURE continued Gender Ideology Gender ideology is used in most societies to justify the universal male dominance Deeply rooted values about the superiority of men the ritual impurity of women and the preeminence of men s work are used to justify subjugation of women However it has been demonstrated in recent years that women do not perceive themselves in the same ways that they are portrayed in largely male gender ideologies Negative Impact of Biased Gender Ideologies In some societies gender ideologies become so extreme that females suffer serious negative consequences such as female infanticide female nutritional deprivation honor killings and domestic violence These atrocities are due to the negative impact of gender ideologies as well as due to the disempowerment of females which is another simultaneous consequence of these ideologies Women Employment Although the words breadwinner and housewife accurately described the middle class western household around the beginning of the twentieth century the separate gender roles implied by these two terms have become more myth than reality Over the past four decades the number of women in working outside the home has increased dramatically This is true for not only industrialized but also developing countries due to the ongoing phenomenon of globalization which has led more and more women into the workforce Occupational Segregation The economy of most countries is characterized by a high rate of occupational segregation along gender lines Not only are occupations gender segregated but women tend to earn considerably less than men Feminization of Poverty There has been a trend in recent decades toward the feminization of poverty Being disempowered women fall victims to poverty much more easily then men They also have less access to resources with which to fight against poverty Women often are responsible for looking after their children and their poverty results in declining health standards of both women and their children Useful Terms Segregation separation Resources means of production or more generally the nancial means required to do something Ideology an established way of thinking Decade a period of ten years Disempowered without any say or without any authority or power Suggested Readings Copyright Virtual University of Pakistan 44 Introduction to Cultural Anthropology SOC401 VU Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 11 in Cuz um Ambropoogy 45 Introduction to Cultural Anthropology SOC401 VU Lesson 20 STRATIFICATION AND CULTURE Stratification and its Manifestations Individuals in different cultures and societies have varying amounts of access to wealth power and prestige This evident inequality leads to stratification whereby groups or categories of people are ranked hierarchically relative to one another Social Ranking Social ranking is an important feature found to one degree or another in all societies The degree to which societies rank individuals however varies and results in varying amounts of inequality to be found in the world Dimensions of Inequality According to Max Weber stratification takes place on the basis of three reasons People are distinguished from each other on the basis of wealth or economic resources they posses Secondly stratification takes place on the basis of differing levels of power Power is the ability to achieve one s goals and objectives even against the will of others The amount of power often correlates to amount of wealth individuals possess Types of Societies Stratified societies which are associated with the rise of Civilization range from open Class societies which permit high social mobility to more rigid caste societies which allow for little or no social mobility Class societies are associated with achieved status the positions that the individual can Choose or at least have some control over Caste societies on the other hand are based on ascribed statuses into which one is born and cannot change The United States is often cited as a prime example of a class society with maximum mobility Although its national credo includes a belief in the possibility of going from rags to riches most people in the United States remain in the class into which they are born because social environment has an appreciable effect on a person s life Chances The mobility in less developed countries is even more restricted Hindu India is often Cited as the most extreme form of caste society found in the world Social boundaries among castes are strictly maintained by caste endogamy and strongly held notions of ritual purity and pollution Useful Terms Inequality unevenness Purity cleanliness Pollution environmental degradation or physical corruption deterioration Copyright Virtual University of Pakistan 46 Introduction to Cultural Anthropology SOC401 VU Social mobility ability to change one s status Ritual a social routine Suggested Readings pages for this lecture Which provide useful and interesting information Stratification WWWsocicanterburyacnZ resources glossary socialstshtml Copyright Virtual University of Pakistan 47 Introduction to Cultural Anthropology SOC401 VU Lesson 21 THEORIES OF STRATIFICATION continued Prominent Theories of Stratification Theories of stratification try to explain the existing inequality of wealth in and between different cultures The Functional Theory and the Con ict Theory provide two con icting interpretations of social stratification evident around the world today The Functionalists Functionalists adopt a conservative position and maintain that social inequality exists because it is necessary for the functioning of society Functionalists emphasize the integrative nature of stratification which results in stability and social order They point out that class systems contribute to the overall well being of a society and encourage constructive endeavor Functionalists argue that differential awards are necessary if societies are to recruit the best trained and most highly skilled people for highly valued positions They maintain that highly skilled people need to be given greater rewards to act as an inventive for them to acquire the required skills For example a brain surgeon needs to spend enormous amounts of time and energy to develop his skills and help society and society must in turn reward him more than it does other people who do not have to make a similar investment in obtaining a skill Functionalists cannot account for non functional success of pop icons for example Famous personalities are often given enormous amounts of money to make public appearances due to their popularity rather than their exceptional amount of skill Functionalists ignore the barriers to participation of certain segments of society Con ict Theorists Con ict theorists assume that the natural tendency of all societies is toward change and con ict Con ict theorists believe that stratification exists because the upper classes strive to maintain their superior position at the expense of the lower classes Con ict theorists do not view stratification systems as enviable or desirable Lack of social mobility leads to exploitation crime revitalization reform and even to revolution Con ict theory is in uenced by the wirings of Karl Marx Functionalists versus Con ict Theorists Integrative aspects of stratification are beneficial for society but the exploitation of under classes does cause tensions and con ict Neither theory can alone explain the existing use and dysfunctional aspects of stratification Useful Terms Revitalization recuperation or revival Dysfunctional no longer able to function or have utility in the given circumstances Exploitation taking advantage of someone else sue to their inability to safeguard their own interests Differential Awards different remunerations or rewards Social Inequality a state of being where certain segments of society are more well off than others Suggested Readings Copyright Virtual University of Pakistan 48 Introduction to Cultural Anthropology SOC401 VU site for this lecture Which provide useful and interesting information Stratification WWWsocicanterburyacnZ resources glossary socialstshtml Copyright Virtual University of Pakistan 49 Introduction to Cultural Anthropology SOC401 VU Lesson 22 CULTURE AND CHANGE Cultural Change Although the rate of change varies from culture to culture no cultures remain unchanged Small scale cultures that are less reliant on technology are seen to change more slowly than industrialized cultures and societies However nothing is as constant as change There is no culture or society which can safeguard itself from the processes of change How Cultures Change The two principal ways that cultures change are internally through the processes of invention and innovation and externally through the process of diffusion It is generally recognized that the majority of cultural features things ideas and behavior patterns found in any society got there by diffusion rather than invention Inventions Inventions can be either deliberate or unintentional Although intentional inventors usually receive the most recognition and praise over the long run unintentional inventors have probably had the greatest impact on cultural change Consider for example the common phrase necessity is the mother of all invention which implies that often circumstances are a more compelling factor inducing innovations in society than the declared intention to make something new Because they are not bound by conventional standards many inventors and innovators tend to be marginal people living on the fringes of society Anthropologists examine the backgrounds and psychological factors that in uence innovative personalities Some of them maintain that inventors are often amongst the well off segments of society yet there are other anthropologists who present other arguments concerning innovators Diffusion The following generalizations can be made about the process of diffusion Cultural diffusion is selective in nature selectivigl not all things diffuse from one culture to another at the same rate Diffusion is a two way process 766707on1 both cultures change as a result of diffusion Cultural elements are likely to involve changes in form or function modz mz z39o a diffused cultural item will not remain exactly the same as it is to be found in its original culture Consider for example the case of Chinese food or pizza which are modified according to the taste of different countries The idea of chicken flk d topping is an example of cultural modification Cultural items involving material aspects are more likely candidates for diffusion than those involving non material aspects Diffusion is affected by a number of important variables duration and intensity of contact degree of cultural integration and similarities between donor and recipient cultures Useful Terms Variables values which are subject to change Cultural items these include both material and non material items ranging from clothing to ideas Donor a country or even an individual entity which is at the giving end of a relationship Copyright Virtual University of Pakistan 50 Introduction to Cultural Anthropology SOC401 VU Recipient a country or even an individual entity which is at the receiving taking end of a relationship Conventional standard or acceptable Intentional bez39 g motivated by an intention Intentional innovators for example clearly state that they are trying to deal With a particular problem and Will attempt to identify a solution for it Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbrepelegy A Applied Penpeez z39be by Fermrre meder Cbapz er 73 23971 Lil z brepelegy by Ember emd Peegrzbe Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Culture Change An Introduction to the Processes and Consequences of Culture Change htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 51 Introduction to Cultural Anthropology SOC401 VU Lesson 23 CULTURE AND CHANGE continued Acculturation Acculturation is a specialized form of cultural diffusion that is a result of sustained contact between two cultures one of which is subordinate to another Whereas diffusion involves a single or complex of traits acculturation involves widespread cultural reorganization over a shorter period of time There are events in history like colonization which have caused acculturation to occur in many parts of the world Some anthropologists have described situations of acculturation in which the non dominant culture has voluntarily chosen the changes Other anthropologists claim that acculturation always involves some measure of coercion and force Cultural Interrelations Because the parts of a culture are interrelated a change in one part of a culture is likely to bring about changes in other parts of the given culture This is the reason why people are often reluctant to accept change since its consequences cannot be exactly predicted nor controlled This insight of cultural anthropology should be kept in mind by applied anthropologists who are involved in planned programs of cultural change Reaction to Change In every culture there are two sets of opposing forces those interested in preserving the status quo and others desiring change The desire for prestige economic gain and more efficient ways of solving a problem are reasons why people embrace change but the threat of loss of these can lead other people to oppose change as well Barriers to Cultural Change Some societies can maintain their cultural boundaries through the exclusive use of language food and clothing Some societies can resist change in their culture because the proposed change is not compatible with their existing value systems Barriers to Cultural Change Societies resist change because it disrupts existing social and economic relationships The functional interrelatedness of cultures serves as a conservative force discouraging change Cultural boundaries include relative values customs language and eating tastes Change Agents Change agents including development workers for example facilitate change in modern times Change agents sometimes fail to understand why some people are resistant to change and should realize cultural relativity and barriers to change Useful Terms Facilitate to make easier or to promote Functional useful or practical aspects Cultural relativity the realization that cultural traits fit in logically within their own cultural environments and that since circumstances around the world differ cultures are also different Copyright Virtual University of Pakistan 52 Introduction to Cultural Anthropology SOC401 VU Status Quo The existing conditions or circumstances There are always those who are interested in maintaining the status quo since they are doing well due to it and others who oppose the status quo since it tends to exploit them or puts them in a disadvantaged position Coercion An act of force rather than that based on the need or desire of a particular individual or society Interrelations interconnections Subordinate in an inferior or subordinated position Dominant in a position of power over others Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 73 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Culture Change1 htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 53 Introduction to Cultural Anthropology SOC401 VU Lesson 24 CULTURE AND CHANGE continued The Complex Process of Change Accepting change in one part of a culture is likely to bring about changes to other parts of a culture To understand socio cultural aspects of urbanization it is important to view the rural area the urban areas and the people who move between them as parts of a complex system of change Until some decades ago anthropologists made differentiations between the mechanical solidarity of rural areas and the organic solidarity of cities Recent research notes that there is not a simple flow of migrants from rural areas to urban areas but rather a circulation of people between these areas Urbanization or the process of rural development therefore needs to take into account the fact that there is a constant criss crossing of people ideas and resources from urban to rural areas Rural migrants rely on kinsmen for land purchase dispute resolution or general household management while they go to the cities in search for cash based employment Conversely rural kinsmen may in turn obtain economic support from a urban wage earner or seek his support in finding work or a place to stay in the city for other kinsmen Planned Change Planned programs of change have been introduced into developing countries for decades under the assumption that they benefit the local people Yet a number of studies have shown that although some segments of the local population may benefit many more do not Globalization Globalization is a broad based term which is used to describe the intensification of the flow of money goods and information across the world which is seen to be taking place since the 19805 Globalization has made the study of culture change more complex due to its varied effects on various cultural processes including that of change In some cases globalization is responsible for an accelerated pace of change in world cultures In other situations the forces of globalization may stimulate traditional cultures to redefine themselves Developing countries in the attempt to better deal with the forces of globalization such as trade liberalization have begun to revamp their own economic systems in order to make them more competitive internationally This economic revamping has tremendous cultural impacts as well Globalization has resulted in diffusion of technology but also compounded existing inequalities There are human and environmental costs associated with globalization For example increased productivity has led to pollution and there are many theorists who argue that globalization has also increased the gap between the rich and the poor with those with wealth doing even better and those without it experiencing even worse poverty than before Useful Terms Globalization intensification of the flow of money goods and information across the world Urbanization the process of people moving from rural areas into the cities This phenomenon is taking place in both developed and developing countries and cultural anthropologists are very interested in studying why and how urbanization takes place and the cultural changes it brings Revamping reforming or changing Competitive the process of trying to do better than those engaging in the same activity Copyright Virtual University of Pakistan 54 Introduction to Cultural Anthropology SOC401 VU Environmental Costs the impact of a particular activity on land water or air and on various other species which inhabit the Earth alongside human beings Impacts results or effects Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 16 in Cuz um A lbropology A Applied Pmpm ive 7y Fermer andor Cbapz er 73 23971 24 z brop00gy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Culture Change2 htt anthro alomaredu chan e defaulthtm Copyright Virtual University of Pakistan 55 Introduction to Cultural Anthropology SOC401 VU Lesson 25 POLITICAL ORGANIZATION Need for Political Organization All societies have political systems that function to manage public affairs maintain social order and resolve con ict Yet the forms of these political systems are diverse sometimes embedded in other social structures Studying Political Organization Political organization involves issues like allocation of political roles levels of political integration concentrations of power and authority mechanisms of social control and resolving con icts Anthropologists recognize four types of political organization based on levels of political integration concentration specialization Political organization is found within bands tribes Chiefdoms and states Nowadays non state forms of political organization have state systems superimposed on them Types of Political Systems Societies based on bands have the least amount of political integration and role specialization Kung in Kalahari Bands Bands are most often found in foraging societies and are associated with low population densities distribution systems based on reciprocity and egalitarian social relations Tribal Organizations Tribal organizations are most commonly found among horticulturists and pastoralists Neur in Sudan With larger and more sedentary populations than are found in band societies tribal organizations do also lack centralized political leadership and are egalitarian Tribally based societies have certain pan tribal mechanisms that integrate clan members to face external threats Clan elders do not hold formal political offices but usually manage affairs of their clans settling disputes representing clan in negotiation with other clans etc Chiefdoms Chiefdoms involve a more formal and permanent political structure than is found in tribal societies Political authority in Chiefdoms rests with individuals who acts alone or with advice of a council Most chiefdom tends to have quite distinct social ranks rely on feasting and tribute as a major way of distributing goods In the late nineteenth and twentieth century many societies had Chiefdoms imposed on them by colonial powers for administrative convenience for eg British impositions in Nigeria Kenya and Australia The pre colonial Hawaiian political system of the 18th century was a typical chiefdom Useful Terms Public Affairs issues concerning the public at large instead of specific individuals only Social Order the state of being where society functions as per the expectations of people and can provide them with a sense of security Sedentary settled Copyright Virtual University of Pakistan 56 Introduction to Cultural Anthropology SOC401 VU Colonial powers at different phases of history different nations have been powerful enough to colonize other nations In the 19th century Britain was a colonial power which was able to colonize many other countries located on the African and the Asian continents Precolonial the period in history when a particular nation had not yet been colonized Allocation distribution Integration tied together or linked in a particular manner Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um A lbmpoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 23 23971 Lil z bropoogy 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Political Organization3 htt anthro alomaredu olitical defaulthtm Please use hyperlinks on the website to read the introductory materials and the information provided on bands tribes and chiefdoms Copyright Virtual University of Pakistan 57 Introduction to Cultural Anthropology SOC401 VU Lesson 26 POLITICAL ORGANIZATION continued State Systems State systems have the greatest amount of political integration specialized political roles and maintain authority on the basis of an ideology States are associated with intensive agriculture market economies urbanization and complex forms of social stratification States began to be formed 5500 years ago with the Greek city states and the Roman Empire providing impressive examples of state based political organization States have a monopoly on the use of force and can make and enforce laws collect taxes and recruit labor for military service and public works which differentiates them from other forms of political organization States are now the most prominent form of political organization found around the world today NationStates A nation is a group pf people sharing a common symbolic identity culture history and religion A state is a distinct political structure like bands tribes and chiefdoms Nation state refers to a group of people sharing a common cultural background and unified by a political structure that they consider to be legitimate Few of the world s 200 nation states have homogenous populations to fit the description of a nation state Political Organization Theories Theories explaining the rise of state systems of government have centered on the question of why people surrender some of their autonomy to the power and authority of the state There are theorists who argue that political organization is influenced by self interest and other theorists argue that self interest is not enough to give shape to political systems and that such organization often involves a certain amount of coercion Voluntaristic State Formation Some theorists suggest that those engaging in specialized labor voluntarily gave up their autonomy in exchange for perceived benefits Political integration can mediate between and protect interests of varied groups and provide them an economic superstructure required for specialization Chide 1936 Hydraulic Theory of State Formation Small scale farmers in arid or semi arid areas also voluntarily merged into larger political entities due to the economic advantage of large scale irrigation Karl Wittfogel 1957 Coercive Theory of State Formation Another explanation for state based political organization is that offered by Carneiro hold that states developed as a result of warfare and coercion rather than due to voluntary self interest Useful Terms Coercion use of force Copyright Virtual University of Pakistan 58 Introduction to Cultural Anthropology SOC401 VU Arid dry Smallscale farmers farmers possessing a little amount of land Irrigation the channeling of water from its natural route for the purposes of agriculture Monopoly dominating the production of a particular product Hydraulic water based Homogenous identical to others opposite of heterogeneous Recruit to include or to involve Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um Ambropoogy A Applied Pmpecz z39ve 7y Fermer andor Cbapz er 23 23971 Lil z bropology 7y Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Political Organization4 htt anthro alomaredu olitical defaulthtm 4 Please follow the hyperlink on the website to read about state systems Copyright Virtual University of Pakistan 59 Introduction to Cultural Anthropology SOC401 VU Lesson 2quot POLITICAL ORGANIZATION continued Need for Social Control All forms of political organization must provide means for social control Every culture has defined what are considered to be normal proper or expected ways of behaving in society These expected ways of behaving are referred to as social norms Social norms range from etiquette to laws and imply different forms of enforcement and sanctions Breaking some social norms does not result in serious consequences whereas others can result in severe punishment Consider for example the consequence of taking another person s life or of stealing something Social Norms All social norms are sanctioned to varying degrees according to the values held by different cultures Positive social norms reward people for behaving in socially expectable ways ranging from praise or social approval to awards or medals Negative social norms punish people for violating the norms ranging from disapproval to corporal punishment Maintaining Social Control Band and tribal societies Inuit and Kung maintain social control by means of informal mechanisms such as socialization public opinion lineage obligations age organizations and sanctions Societies control behavior by more formal mechanisms such as through laws and law enforcement agencies whose major function is maintaining social order and resolving conflicts Social Control Band and tribal societies Inuit and Kung maintain social control by means of informal mechanisms such as socialization public opinion lineage obligations age organizations and sanctions Societies control behavior by more formal mechanisms whose major function is maintaining social order and resolving con icts Informal Mechanisms Socialization ensures that people are taught what their social norms are Public opinion or social pressure often serves as an effective mechanism to avoid censure and rejection Age organization provides distinct age categories with defined sets of social roles Formal Mechanisms Song Duets amongst the Inuits to settle disputes Social Intermediaries like the Leopard skin Chief of the Neur in southern Sudan settles murder disputes by property settlements Moots are formal airings of disputes involving kinsmen and friends of litigants and the adjudicating bodies are ad hoc Courts and Codified Laws forbid individual use of force and provides legal frameworks established by legislative bodies interpreted by judicial bodies and implemented by administrative systems like law enforcement agencies Copyright Virtual University of Pakistan 60 Introduction to Cultural Anthropology SOC401 VU Useful Terms Administrative systems the system of government officials bureaucrats who are responsible for running public affairs Judicial systems the system of courts which interprets the laws Legislative systems systems which provide the laws for a particular society often legislatures or legislative assemblies are elected by the people of a particular locality ie province or a state Law enforcement agencies agencies which enforce the law like the police fro example Litigants aggrieved parties involved in a legal dispute Ad Hoc arbitrary not following any established procedure Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 12 in Cuz um A lbropology A Applied Pmpecz z39be by Fermer andor Cbapz er 23 23971 Lil z bropology by Ember cmd Peggrz39 e Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Social Control http enwikipediaorg wiki Social control Copyright Virtual University of Pakistan 61 Introduction to Cultural Anthropology SOC401 VU Lesson 28 PSYCHOLOGY AND CULTURE Psychological Development Anthropologists are interested in the psychological differences and similarities between societies and cultures of the world Cultural Anthropologists reject stereotypes based on hasty ethnocentric judgments Anthropological Queries in Psychology The major questions of relevance to cultural anthropologists attempting to understand the linkage between different cultures and what they can reveal about the human personality are 0 Do all human beings develop psychologically in the same way 0 What explains the psychological differences in personality characteristics from one society to another 0 How do people in different societies conceive of personality and psychological development 0 What types of cultural variations may be explained due to cultural factors Emotional Development Early research in anthropology was concerned mainly with supposedly universal stages of emotional development which seems to be affected by cultural differences Magma Mead found Samoan girls were much less rebellious or emotional turmoil than those in western societies In western societies adolescence is a time of turmoil that helps prepare emotionally for independence Psychological Universals The ability to make binary contrasts order phenomenon plan for the future and have an understanding of the world are universal psychological traits All people have a concept of the self they can empathize with others and feel and recognize emotions in others Cognition and Culture Recent research on psychological universals focuses on cognitive or intellectual development For example it considers how different cultures acquire thinking habits such as formal operation notions which enable a person to think of the possible outcomes of a hypothetical situation In looking for universals many researchers have discovered some apparent differences Yet most tests used in anthropological research favors thinking patterns taught in formal schools in Western cultures CrossCultural Variations Instead of focusing on uniqueness anthropologists look at psychological differences found within and between different cultures Researchers focus on child rearing practices to account for observable personality differences Some anthropologists believe that child rearing practices are adaptive and societies produce personalities according to their requirements obedience self reliance etc Copyright Virtual University of Pakistan 62 Introduction to Cultural Anthropology SOC401 VU Useful Terms Universal common in all cultures Formal schools schools organized by the public or private sector but with a standardized curriculum and professional teaching staff Variations differences Rearing bringing up Intellectual concerning the intellect and the process of thinking5 htt wwwtrini edu Nmkearl soc s html 5 Please visit the hyperlinks on the website to read selectively on topics like the nature versus nurture debate Copyright Virtual University of Pakistan 63 Introduction to Cultural Anthropology SOC401 VU Lesson 29 PSYCHOLOGY AND CULTURE continued Socialization Socialization is the term that psychologists and anthropologists use to describe the development of through the in uence of parents and others of patterns of behavior in children that conform to cultural expectations Direct and Indirect Socialization Socialization takes place both directly and indirectly Indirectly the degree to which parents like children the kinds of work children are asked to do and whether children go to school may at least partially in uence how children develop psychologically Origin of Customs Anthropologists not only seek to understand the link between personality traits and customs but also how customs were themselves developed Some anthropologists believe that societies produce the kind of customs best suited for undertaking activities necessary for the survival of society Personality Types Several anthropologists have tried to describe the in uence of culture on personality In the early 195039s for example David Riesman proposed that there are three common types of personalities around the world I The z mdz39z z39o orz39em edperrom u places a strong emphasis on doing things the same way that they have always been done Individuals with this sort of personality are less likely to try new things and to seek new experiences II Those who have z39mer dz39recz edperm7145mm are guilt oriented That is to say their behavior is strongly controlled by their conscience As a result there is little need for police to make sure that they obey the law These individuals monitor themselves If they break the law they are likely to turn themselves in for punishment III In contrast people with ber directedperrom z ier have ambiguous feelings about right and wrong When they deviate from a societal norm they usually don39t feel guilty However if they are caught in the act or exposed publicly they are likely to feel shame Abnormal Behavior Just as there are cross cultural variations in normal behavior there are also variations in abnormal behavior Abnormality is relative to a degree and a culture s ideas about mental illness and how to deal with it can also vary Applied Perspective Anthropologists are interested in understanding the possible cause of psychological differences and the possible consequences of psychological variation Anthropologists are particularly interested in how psychological characteristics may help explain statistical associations between various aspects of culture Projective Testing People tend to project their feelings ideas and concerns onto ambiguous realities Copyright Virtual University of Pakistan 64 Introduction to Cultural Anthropology SOC401 VU In Thematic Appreciation Tests subjects are shown vague drawings and asked to interpret them by projecting their own personalities An aggressive person may see a weapon in a vague drawing whereas a more industrious person may visualize a more productive tool in the same vague drawing Useful Terms Ambiguous unclear or vague Variation differences Socialization the process of learning behavior6 htt wwwtrini edu mkearl soc s html 6Please visit the hyperlinks on the website to read selectively on topics like collective behavior Copyright Virtual University of Pakistan 65 Introduction to Cultural Anthropology SOC401 VU Lesson 30 IDEOLOGY AND CULTURE Ideology An ideology is a collection of ideas An ideology can be thought of as a comprehensive vision as a way of looking at things Ideology can also be seen as a set of ideas proposed by the dominant class of a society to all members of this society For example different types of gender ideologies would describe what roles are expected of women and men in a society The ideology of economic liberalization could be seen to particularly promote the interests of the business classes Ideology in E vetyda y Life Every sociegl has an ideology that forms the basis of the public opinion or common sense a basis that usually remains invisible to most people within the society This dominant ideology appears as neutral while all others that differ from the norm are often seen as radical no matter what the actual circumstances may be In uencing Ideology Organizations that strive for power in uence the ideology of a society to provide a favorable environment for them Political organizations governments included and other groups eg lobbyists try to in uence people by broadcasting their opinions which is the reason why so often many people in a society seem to think alike A certain ethic usually forms the basis of an ideology Ideology studied as ideology rather than examples of specific ideologies has been carried out under the name systematic ideology There are many different kinds of ideology political social epistemological ethical The popularity of an ideology is in part due to the in uence of moral entrepreneurs who sometimes act in their own interests A political ideology is the body of ideals principles doctrine myth or symbols of a social movement institution class or large group that references some political and cultural plan It can be a construct of political thought often defining political parties and their policy Hegemony When most people in a society think alike about certain matters or even forget that there are alternatives to the current state of affairs we arrive at the concept of Hegemony about which the philosopher Antonio Gramsci wrote The much smaller scale concept of groupthink also owes something to his work The ideologies of the dominant class of a society are proposed to all members of that society in order to make the ruling class39 interests appear to be the interests of all and thereby achieve hegemony To reach this goal ideology makes use of a special type of discourse the lacunar discourse A number of propositions which are never untrue suggest a number of other propositions which are In this way the essence of the lacunar discourse is what is oz told but is suggested Epistemological ideologies Even when the challenging of existing beliefs is encouraged as in science the dominant paradigm or mindset can prevent certain challenges theories or experiments from being advanced Copyright Virtual University of Pakistan 66 Introduction to Cultural Anthropology SOC401 VU The philosophy of science mostly concerns itself with reducing the impact of these prior ideologies so that science can proceed with its primary task which is according to science to create knowledge There are critics who view science as an ideology itself called scientism Some scientists respond that while the scientific method is itself an ideology as it is a collection of ideas there is nothing particularly wrong or bad about it Other critics point out that while science itself is not a misleading ideology there are some fields of study within science that are misleading Two examples discussed here are in the fields of ecology and economics Useful Terms Discourse discussion or dialogue Proposition proposal or plan Paradigm standard pattern or example Doctrine set of guidelines Comprehensive complete all inclusive Ecology concerning the species found in the natural environment Moral entrepreneurs those who make up new morals according to their cultural needs Suggested Readings Please visit the following web site for this lecture which provide useful and interesting information Ideology7 http enwikipediaorg wiki Ideology 7 Please visit the hyperlinks on the website to read more about topics mentioned in the lecture Copyright Virtual University of Pakistan 67 Introduction to Cultural Anthropology SOC401 VU Lesson 31 IDEOLOGY AND CULTURE Continued Political ideologies In social studies a political ideology is a set of ideas and principles that explain how the society should work and offer the blueprint for a certain social order A political ideology largely concerns itself with how to allocate power and to what ends it should be used For example one of the most in uential and well defined political ideologies of the 20th century was communism based on the original formulations of Karl Marx and Friedrich Engels Communism is a term that can refer to one of several things a certain social system an ideology which supports that system or a political movement that wishes to implement that system As a social system communism is a type of egalitarian society with no state no private property and no social classes In communism all property is owned by the communig as a whole and all people enjoy equal social and economic status Perhaps the best known principle of a communist society is quotFrom each according to his ability to each according to his needquot As an ideology communism is synonymous for Marxism and its various derivatives most notably Marxism Leninism Among other things Marxism claims that human society has gone through various stages of development throughout its history and that capitalism is the current stage we are going through The next stage will be socialism and the one after that will be communism Therefore it should be noted that communists do not seek to establish communism right away they seek to establish socialism rst which is to be followed by communism at some point in the future Other examples of ideologies include anarchism capitalism corporate liberalism fascism monarchism nationalism fascism conservativism and social democracy Economic Ideology Karl Marx proposed a mempem mcz me model of society The bane refers to the means of production of society The mpem mcz we is formed on top of the base and comprises that society39s ideology as well as its legal system political system and religions Marx proposed that the base determines the superstructure It is the ruling class that controls the society39s means of production and thus the superstructure of society including its ideology will be determined according to what is in the ruling class39 best interests On the other hand critics of the Marxist approach feel that it attributes too much importance to economic factors in in uencing society This is far from the only theory of economics to be raised to ideology status some notable economically based ideologies include mercantilism Social Darwinism communism laisseZ faire economics and quotfree tradequot There are also current theories of safe trade and fair trade calling for a revision in terms of trade which can be seen as ideologies These ideologies call for a revision of rules based on which trade between developed and developing countries takes place Interaction between Legal and Economic Ideologies Ideologies often interact with and in uence each other in the real world Consider for example the statement 39All are equal before the law39 which is a theory behind current legal systems suggests that all Copyright Virtual University of Pakistan 68 Introduction to Cultural Anthropology SOC401 VU people may be of equal worth or have equal 39opportunities39 This is not true because the concept of private property over the means of production results in some people being able to own more mm9 more than others and their property brings power and in uence the rich can afford better lawyers among other things and this puts in question the principle of equality before the law Useful Terms Fair trade the notion that all countries should be given a fair price for the products they export through international trade Terms of trade the price which products of different countries fetch in international trade Means of production these include land labor capital investments required to produce something Inevitable unavoidable Synonymous another term carrying the same meaning Suggested Readings Please visit the following web site for this lecture which provide useful and interesting information http enwikipediaorg wiki Ideology Copyright Virtual University of Pakistan 69 Introduction to Cultural Anthropology SOC401 VU Lesson 32 ASSOCIATIONS CULTURES AND SOCIETIES What Are Associations Associations are non kin and non territorial groups found amidst all types of societies and cultures around the world Associations possess some kind of formal organizational structure and their members also have common interests and a sense of purpose which binds the varied types of societies together Cultural anthropologists are interested in examining how different cultures give shape to different types of associations and in turn what functions different types of associations perform within particular cultures Variation in Associations Associations can vary from society to society They vary according to whether or not they are voluntary and whether the qualities of members are universally ascribed variably ascribed or achieved Qualifications for Associations Achieved qualities or skills are those acquired through one s own efforts there may be hurdles in acquiring necessary skills but by and large skills have to be learnt through personal effort as they are not biologically transferable Ascribed qualities are determined at birth because of gender or ethnicity or family background A person does not need to make an effort to acquire ascribed qualities nor can effort do much in changing ascribed status since it is largely determined by forces beyond the control of individuals Universally ascribed qualities are found in all societies Gender is an example of an ascribed quality Variably ascribed qualities are unique and thus vary across cultures like ethnicity social class differences etc NonVoluntary Associations In relatively non stratified societies associations tend to be based on universally ascribed characteristics like gender and age An age set is a common form of non voluntary associations evidenced around the world even today Age Sets An age grade includes a category of people who fall into a culturally distinguished age category An age set on the other hand is a group of people of similar age and the same sex who move through some or all of life s stages together Entry into an age set is usually through an initiation ceremony and transitions to new stages are marked by succession rituals In non commercial societies age sets crosscut kinship ties and form strong supplemental bonds Age sets are prominent amongst the Nadi of Kenya for example Young warriors were given spears and shields in the past and told to bring back wealth to the community now they re given pens and paper by their elders and told to go out and do the same The Karimojong are predominantly cattle herders and number 60000 people living in northeastern Uganda who are organized via age and generation sets including 5 age sets covering 25 years Copyright Virtual University of Pakistan 7 0 Introduction to Cultural Anthropology SOC401 VU The retired generation passes on the mantle of authority to the senior generation and the junior generation recruits members until ready to assume authority and thus the society continues to function in a seamless manner Useful Terms Recruit to admit or to actively enlist Supplemental added on so as to help reinforce existing ties Characteristics identifying features Seamless continuous Organizational having features of an organization like defined roles and responsibilities Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 22 in Am bropoogy by Ember Internet Resources Please visit the following web site for this lecture which provide useful and interesting information Voluntary Associations http www fathomcom feature 122550 Copyright Virtual University of Pakistan 71 Introduction to Cultural Anthropology SOC401 VU Lesson 33 ASSOCIATIONS CULTURES AND SOCIETIES continued Regional and Ethnic Associations Regional and ethnic organizations are voluntary associations whose members possess variably ascribed characteristics Both forms of associations are usually found in societies where technological advance is accelerating bringing with it numerous forms of economic and social complexities as well Regional and Ethnic Associations Despite a variety of types regional and ethnic associations commonly emphasize helping members adapt to new conditions particularly if they are migrants Many rural migrants keep members in touch with home area traditions by the help of regional associations These associations promote improved living conditions for members who have recently migrated to urban areas in several countries where urbanization is taking place at a fast pace Examples of Regional Associations Regional associations Jermms help rural migrants adapt to urban life and Lima Peru The J Wd OJ have been seen to actively lobby the government on community issues assist members with enculturation organize fiestas and act as clearing house for flow of information Chinatowns in major cities of the world have associations performing a similar function for Chinese immigrants Ethnic Associations Ethnic Associations are based on ethic ties Such associations are particularly prominent in urban centers of West Africa Even tribal unions are commonly found in Ghana and Nigeria which superimpose the notion of ethnicity with that of tribal ties Rotating Credit Associations Such associations are based on the principle of mutual aid Each group member contributes regularly to a fund which is handed over to one member on a rotation basis Such associations are common in East South and southeastern Asia in western Africa and the West Indies Default is rare in rotating credit associations due to social pressure and the incentive is reasonable since membership ranges from 10 to 30 contributors Since no collateral is needed trustworthiness is considered essential when letting people become members of such groups Multiethnic Associations Associations with a common purpose of economic or socio political empowerment are often multi ethnic Savings and loan associations in New Guinea often link women from different tribal areas Formation of Associations Age sets arise in societies which have frequent warfare breaking out amidst them or it is found amongst groups with varying populations due to which kinship systems are not sufficient for alliance purposes Copyright Virtual University of Pakistan 72 Introduction to Cultural Anthropology SOC401 VU Urbanization and economic compulsions lack of access to credit also form associations due to the need to cooperate out of self interest Useful Terms Collateral the act of pledging an asset in order to qualify from a loan from a lending institution like a bank Empowerment to empower or reinforce the capacity of individuals Multiethnic different ethnic groups coeXisting within the same society Default being unable to pay back a loan Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 22 in Am bropoogy 7y Ember Internet Resources Please visit the following web site for this lecture which provide useful and interesting information http wwwfathomcomfeature122550 Copyright Virtual University of Pakistan 73 Introduction to Cultural Anthropology SOC401 VU Lesson 34 RACE ETHNICITY AND CULTURE Ethnicity Ethnicity refers to selected cultural and sometimes physical characteristics used to classify people into ethnic groups or categories considered to be signi cantly different from others Commonly recognized American ethnic groups include American Indians Latinos Chinese African Americans European Americans etc In some cases ethnicity involves merely a loose group identity with little or no cultural traditions in common This is the case with many Irish and German Americans In contrast some ethnic groups are coherent subcultures with a shared language and body of tradition Newly arrived immigrant groups often fit this pattern Minority versus Ethnic Group It is important not to confuse the term minority with ethnic group Ethnic groups may be either a minority or a majority in a population Whether a group is a minority or a majority also is not an absolute fact but depends on the perspective For instance in some towns along the southern border of the US people of Mexican ancestry are the overwhelming majority population and control most of the important social and political institutions but are still defined by state and national governments as a minority In small homogenous societies such as those of hunters and gatherers and pastoralists there is essentially only one ethnic group and no minorities Ethnic Categorizations For many people ethnic categorization implies a connection between biological inheritance and culture They believe that biological inheritance determines much of cultural identity If this were true for instance African American cultural traits such as quotblack Englishquot would stem from genetic inheritance This is not true The pioneering 19th century English anthropologist E B Taylor was able to demonstrate conclusively that biological race and culture is not the same thing It is clear that any one can be placed into another culture shortly after birth and can be thoroughly encultured to that culture regardless of their skin color body shape and other presumed racial features Race A race is a biological subspecies or variety of a species consisting of a more or less distinct population with anatomical traits that distinguish it clearly from other races This biologist39s definition does not fit the reality of human genetic variation today We are biologically an extremely homogenous species All humans today are 999 genetically identical and most of the variation that does occur is in the difference between males and females and our unique personal traits This homogeneity is very unusual in the animal kingdom Even our closest relatives the chimpanzees have 2 3 times more genetic variation than people Orangutans have 8 10 times more variation It is now clear that our human quotracesquot are cultural creations not biological realities The concept of human biological races is based on the false assumption that anatomical traits such as skin color and specific facial characteristics cluster together in single distinct groups of people They do not There are no clearly distinct quotblackquot quotwhitequot or other races Copyright Virtual University of Pakistan 74 Introduction to Cultural Anthropology SOC401 VU Similarity in Human Adaptations The popularly held view of human races ignores the fact that anatomical traits supposedly identifying a particular race are often found extensively in other populations as well This is due to the fact that similar natural selection factors in different parts of the world often result in the evolution of similar adaptations For instance intense sunlight in tropical latitudes has selected for darker skin color as a protection from intense ultraviolet radiation As a result the dark brown skin color characteristic of sub Saharan Africa is also found among unrelated populations in the Indian subcontinent Australia New Guinea and elsewhere in the Southwest Paci c Safeguarding Against Cultural Biases We must not let our own cultural biases get in the way of understanding the lives of other people Avoiding cultural biases is a very difficult task given the emotionally charged feelings and deep beliefs that we have concerning race and ethnicity However suspending these attitudinal barriers in order to gain a better understanding of the phenomena is well worth the effort Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz umAm bmpoogy A Applied Pmpm z39ve 7y Ferm o Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethnicity and Race An Introduction to the Nature of Social Group Differentiation and Inequality htt anthro alomaredu ethnici defaulthtm Copyright Virtual University of Pakistan 75 Introduction to Cultural Anthropology SOC401 VU Lesson 35 RACE ETHNICITY AND CULTURE continued The Complex Nature of Human Variations The actual patterns of biological variation among humans are extremely complex and constantly changingThey can also be deceptive All of us could be classified into a number of different quotracesquot depending on what genetic traits are emphasized For example if you divide people up on the basis of stature or blood types the geographic groupings are clearly different from those defined on the basis of skin color Using the B blood type for defining races Australian Aborigines could be lumped together with most Native Americans Some Africans would be in the same race as Europeans while others would be categorized with Asians Historically human quotracesquot have been defined on the basis of a small number of superficial anatomical characteristics that can be readily identified at a distance thereby making discrimination easier However focusing on such deceptive distinguishing traits as skin color body shape and hair texture causes us to magnify differences and ignore similarities between people It is also important to remember that these traits are no more accurate in making distinctions between human groups than any other genetically inherited characteristics All such attempts to scientifically divide humanity into biological races have proven fruitless Relevance of Nurture In the final analysis it is clear that people not nature create our identities Ethnicity and supposed quotracialquot groups are largely cultural and historical constructs They are primarily social rather than biological phenomena This does not mean that they do not exist To the contrary quotracesquot are very real in the world today In order to understand them however we must look into culture and social interaction rather than biological evolution Intergroup Relations How ethnic and racial groups relate to each other can be viewed as a continuum ranging from cooperation to outright exploitation and hostility 0 Pluralism Two or more groups living in harmony while retaining their own heritage and identity 0 Assimilation when one racial or ethnic minority is absorbed into the wider society Paci c Islanders assimilation into Hawaiian society provides a good example of assimilation 0 Legal Protection of Minorities While such legislation cannot ensure that minorities have equal rights they provide a measure of security against blatant forms of prejudice and discrimination 0 Population Transfer physical removal of minority to another location The ethnic Tutsi eeing Rwanda to avoid prosecution by the Hutu government is an example of population transfer 0 Longtermed Subjugation Political social and economic suppression evident in political history The example of the black majority s subjugation in South Africa under apartheid is a recent example from history 0 Genocide Mass annihilation of groups of people in Nazi Germany or in Serbia for example Copyright Virtual University of Pakistan 76 Introduction to Cultural Anthropology SOC401 VU Useful Terms Exploitation take undue advantage of another s weakness Subjugation political or social suppression Prosecution to accuse or take legal action against an individual or a group Minority a group with a lesser population in comparison to another groups within the same are Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 13 in Cuz umAm bmpoogy A Applied Pmpecz z39ve 7y Fermrro Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethnicity and Race htt anthro alomaredu ethnici defaulthtm Elbm39cz39gl cmd N Mom2km Am bropoogz39m Pmpecz z39ver htt folkuiono eirthe Ethnicit html Copyright Virtual University of Pakistan 77 Introduction to Cultural Anthropology SOC401 VU Lesson 36 CULTURE AND BELIEFS Systems of Beliefs Although all cultures have belief systems the forms these beliefs take vary widely from society to society It is often difficult to de ne belief systems cross culturally because different societies have different ways of expressing faith Anthropological Perspective on Beliefs The anthropological study of belief systems does not attempt to determine which belief systems are right or wrong Cultural anthropologists concentrate on describing various systems of belief how they function and in uence human behavior across cultures Social Function of Religion Belief systems fulfill social needs They can be powerful dynamic forces in society Beliefs provide a basis for common purpose and values that can help maintain social solidarity By reinforcing group norms they help bring about social homogeneity A uniformity of beliefs also helps bind people together to reinforce group identity Beliefs enhance the overall well being of the society by serving as a mechanism of social control and also reduce the stress and frustrations that often lead to social con ict whereby helping intensify group solidarity In most societies beliefs play an important role in social control by defining what is right and wrong behavior If individuals do the right things in life they may earn moral approval If they do the wrong things they may suffer retribution Psychological Function of Beliefs Belief systems perform certain psychological functions by providing emotional comfort by explaining the unexplainable for eg to confront and explain death A belief system also helps a person cope with stress fears and anxieties about the unknown Beliefs lift the burden of decision making from our shoulders because they tell us what is right and wrong which is of tremendous help in times of stress or crisis Even prayers provide psychological comfort and solace Moreover beliefs help ease the stress during life crises such as birth marriage serious illnesses by providing appropriate guidelines and rituals Politics and Beliefs Belief systems have played an important role in global social change through liberation theology whereby believers for social reform and justice for the poor and religious nationalism whereby religious beliefs are merged with government institutions Useful Terms Liberation freedom Rituals standardized way for performing some vital social function Retribution vengeance or payback Merged combined or put together Copyright Virtual University of Pakistan 78 Introduction to Cultural Anthropology SOC401 VU Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 14 in Cuz umAm bmpoogy A Applied Pmpecz z39ve 7y Fermrro cmd Cbapz er 25 23971 Lil z bropology 7y Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Anthropology of Religion Homepage wwwasuaeduantFacultymurphy419419WWWhtm or htt WWWindianaedu Wanthro reli ionhtm Copyright Virtual University of Pakistan 79 Introduction to Cultural Anthropology SOC401 VU Lesson 3quot LOCAL OR INDIGENOUS KNOWLEDGE What is Local Knowledge Local knowledge consists of factual knowledge skills and capabilities possessed by people belonging to a speci c area Given that local knowledge is usually geared to real life practices it can usually only be understood with reference to the situation in which it is to be applied Local knowledge is local to the extent that it is acquired and applied by people with respect to local objectives situations and problems Local knowledge may on the one hand comprise fixed and structured quotknowledgequot which can be defined or on the other hand may by virtue of its combination with the performance of actions involve a more uid process of quotknowingquot Human beings exist in a continuous flux of experiences and practices so local knowledge must include information concerning social management have forms of learning and teaching and decision making routines Local knowledge and its respective knowledge systems are rooted in local or regional cultures the respective social contexts and their economies Therefore it is important to consider these surrounding circumstances when one is considering the content of local knowledge itself Changing Definitions of Local Knowledge Originally quotindigenousquot was equivalent to quotlocalquot or quotfolkquot or when applied to knowledge quotinformal knowledgequot In the 1960s and 70s the word then took on a populist avor of grass roots politics in the sense of quotindigenousquot as opposed to state or quothighquot culture In view of the marginalization and destruction of the eco Zones inhabited by ethnic groups the term quotindigenous knowledge is being used in a context of quotnon westernquot or quotanti westernquot knowledge Local knowledge also refers to knowledge of the minorities contrasted with knowledge at the level of the nation state There are therefore various types of local knowledge Element of Exclusivity in Local Knowledge There are normally various types of public knowledge Some information is shared by all locals other information remains concealed from the majority Some items of knowledge are known only to women or only to men Within a society only a few specialists possess more in depth knowledge extending beyond laypersons knowledge in a particular field for instance specific medical or cropping expertise Using Local Knowledge Use of local knowledge for development should not be restricted to extracting information The availability of local knowledge to multinationals carries the danger of delegating power to authorities which are external to the local communities and therefore restricts establishment of competent leadership and sustainable social structures in local communities Copyright Virtual University of Pakistan 80 Introduction to Cultural Anthropology SOC401 VU There is an ongoing debate on intellectual property rights equal bene t sharing and the role of local knowledge for development Anthropologists investigate not only the behavior and the material products of people but also their thoughts and feelings In all branches of anthropology focus on the Milk view and local knowledge has increased in the last thirty years Many countries have taken political decisions to empower local institutions union councils districts etc based on the idea of giving more power to local authorities which have a closer contact with those at the grassroots level Decentralization should correspond with building local capacities Therefore local knowledge on local natural and social environments of local forest dwellers farmers is often more detailed than that of formal institutions and can be used to assure sustainable development Useful Terms Antiwestern against western values and or economic or political systems mostly instigated by experiences of exploitation Indigenous rooted in a specific locality native Decentralization delegation of authority to lower levels of administration Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Local Knowledge and Local Knowing htt wwwuni trierde uni fb4 ethno know df Copyright Virtual University of Pakistan 81 Introduction to Cultural Anthropology SOC401 VU Lesson 38 LOCAL OR INDIGENOUS KNOWLEDGE continued Scientific Knowledge vs Local Knowledge Is local knowledge ultimately equivalent to knowledge gained through science or is it structured entirely differently This is an age old topic of debate in anthropology the debate concerned with rationality and so called allemaz z39ve moder of lbougbz A corresponding practical question is if local knowledge can be utilized within the framework of scienti cally based measures Or is local knowledge a holistic counter model to science to be used to criticize measures founded on analytical science Most characterizations of local knowledge are defined in complete contrast to scientific knowledge But local and scientific knowledge are neither completely different nor entirely the same they display both commonalities and differences Similarities between Local amp Scientific Knowledge Local knowledge and knowledge derived from science are similar primarily in having an empirical and a methodological basis Both local knowledge and science use observations of the outside world which are in principle accessible and communicable While both forms of knowledge use experiments local knowledge proceeds rather from observations gained through trial and error or so called quotnatural experimentsquot ie inferences drawn from the impacts of natural changes in certain quantities Scientific knowledge on the other hand relies on controlled experiments Distinctions between Local amp Scientific Knowledge Scientific knowledge seeks information which is transferable to any spatial or social situation ie which is not context bound As a result scientists know a great deal about small sections of reality In contrast local knowledge systems seek spatially situation bound or context bound information The validity of items of local knowledge is locally restricted ie they cannot be transferred to other local contexts The potential for generalization and thus also mutual learning is in principle limited with local knowledge Owners of local knowledge are often only inadequately aware of market mechanisms Potential for Anthropological Contribution The inter cultural perspective of anthropologists enables them to re ect on and integrate both ways of knowing and for seeing where to draw the line Local knowledge out of its cultural situation loses its frame of reference and without the necessary skills to decipher it becomes meaningless The Need for Caution While local knowledge increases people s empowerment enhances the visibility of their problems is geared to subsistence and risk minimization leading to more sustainable solutions a cautious approach has to be adopted Practices which are based on local knowledge are not per re ecologically sound necessarily socially just or even democratic Neither is local knowledge equivalent to quotpeople s knowledgequot in the sense that it would always be shared by most or even all members of a group Useful Terms Copyright Virtual University of Pakistan 82 Introduction to Cultural Anthropology SOC401 VU Democratic a system based on sentiments of the majority Risk minimization measures taken to decrease given risks associated with a particular activity Subsistence survival Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Indigenous knowledge biodiversity conservation and development httpwwwciesinorgdocsOO4 173004 173html Copyright Virtual University of Pakistan 83 Introduction to Cultural Anthropology SOC401 VU Lesson 39 ANTHROPOLOGY AND DEVELOPMENT What is Development In the popular meaning of the term development is a transition towards directed change towards modernization industrialization and capitalization However major development agencies and multilateral organizations often interpret development in terms of poverty Poverty defined in relation to the absence of basic services and in income terms less than one dollar a day becomes a proxy for the absence of development and a justification for intervention Poverty and development are measured by indicators and targets some global others national which become standard devices for undertaking development But even focusing on poverty does not necessarily imply that poor people are more involved in the development planning process Often the poor cannot represent themselves they are represented It has also been noticed by anthropologists that development is often defined in negative terms not so much as the presence of something as the elimination of an unacceptable state like that of poverty Role of Anthropology in Development Anthropological studies focus on the processes of social transformation positive and negative conventionally associated with development Anthropology helps development initiatives realize the context in which their activities are to be introduced The cultural insights and the kinds of understandings that anthropology offers enables social development professional to envision what kinds of impacts particular interventions may have on particular types of social relations and institutions Comparing Development and Anthropology Development approaches and methods have much common with anthropology but there are also substantial differences What constitutes social development knowledge is determined by the need to meet policy priorities rather than the pursuit of knowledge Social development presents itself as a technical discipline using social analysis as a precondition for social transformation Like anthropological methods development is people focused and uses qualitative techniques But unlike anthropological methods requiring extended fieldwork social development methodologies are designed to fit into short timeframes Who Undertakes Development Development Organizations include multilateral agencies like the World Bank and UN agencies bilateral agencies national and international NGOs Typical partner organizations include national governments national NGOs and the lower tier community based organizations In uence of Development Notions The in uence of development extends far beyond the formal institutions charged with implementing development oriented programs Cultural attitudes informed by development aspirations are entwined in popular cultures of developed and developing countries For eg rural communities in Nepal utilize the category of developed bikas as a means of classifying people according to perceived class position and Copyright Virtual University of Pakistan 84 Introduction to Cultural Anthropology SOC401 VU social networks Wealthy individuals in developed countries provide money for communities perceived as poor via child sponsorship schemes for example Useful Terms Social Development the effort to meet basic needs and to assure access to basic human right Entwined joined or merged together Perceived considered or viewed NGOS Non government organizations Suggested Readings based on Internet Resources Students are advised to read the following paper available in PDF format from the following web site for this lecture which provides useful and interesting information Applying Anthropology in and to Development htt wwwlboroacuk de artments ss a licationsofanthro olo reen a erhtm Copyright Virtual University of Pakistan 85 Introduction to Cultural Anthropology SOC401 VU Lesson 40 ANTHROPOLOGY AND DEVELOPMENT Continued Development and Change From an anthropological point of view culture is an asset even though managing it is difficult since cultures change and do not have sharp borders Examples of development planners39 and development workers39 ignorance of local culture have had devastating repercussions on the local level What Development Anthropologists do Development anthropologists in interpret practices which are difficult for others to access who lack detailed comparative knowledge of social organization gender kinship property resources Anthropological input is often restricted to appraisal and analysis of planned outcome failures Besides international development use of applied anthropology has grown in the West as well Anthropology in the US and in South America is often associated with cultural brokerage between indigenous groups and national governments and between indigenous groups and private companies often those associated with natural resource extraction Changing Notion of Development Development necessitates a kind of social analysis of the situations which the proposed intervention will be designed to address From an anthropological view this essentially requires matching two representations of reality that of development practioners and that of local environments Research on development and culture during the past years has emphasized a culture sensitive approach in development Emphasis on people undertaking their own development instead of imposing development on them it is suggested that research into local culture is one of the most important features for ensuring participatory development Participation means that development should involve all its stakeholders Even the World Bank has recognized the compleX local environments in which development policy was supposed to operate and had failed was due to lack of participation A modified policy discourse spoke the need to include local people civil society and social networks in planning and implementation Contentions in Development If anthropology has conventionally been suspicious of unplanned changes it has been particularly distrustful of directed change and of the international development project which has had directed change as its objective The ambivalent relationship between anthropology and development has its origins in the colonial systems of governance British anthropology strove to be useful to practical men of colonial administration in the 1930 s to access public funds In France anthropological methods were used to improve colonial government This history accounts for the suspicion with which anthropology is still viewed in many countries which have a fairly recent history of colonial domination Copyright Virtual University of Pakistan 86 Introduction to Cultural Anthropology SOC401 VU A New Role for Anthropologists The involvement of anthropology in development did not end with the dawning of the post colonial era The inclusion of the discipline in the institutional structures of international development from the late 1970 s on has created a number of anthropological positions within development agencies Induction of anthropologists in development agencies in the 1980 s and 1990 s coincided with a new people oriented discourse in international development and a renewed focus on social exclusion and marginality Useful Terms Contentions controversies or opposing points of view Conventionally standardized way of doing something Natural resource extraction extraction of resources from the natural environment from the land or the sea for productive purposes Post Colonial the time period commencing after the colonization period is over although the in uence of colonizing countries may still remain after they have physically vacated a colony Ambivalent ambiguous or lacking a clear cut def1nition Suggested Readings using Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information The cultural process of development Some impressions of anthropologists working in development htt wwwvalthelsinkif1 kmi ulkais WPt 1998 W898HTM Copyright Virtual University of Pakistan 87 Introduction to Cultural Anthropology SOC401 VU Lesson 41 ANTHROPOLOGY AND DEVELOPMENT Continued Expectations from an Anthropologist Commonly it is expected that an anthropologist can assist development programmes by bringing in the anthropological perspective Anthropologists are expected to address social rather that technical aspects of development programs It is anticipated that an anthropologist should take care of the soft elements of the project This is a diffuse expectation which can imply many tasks The anthropologist can be expected to report on for example the division of labour in an area or why cultivators prefer a special crop In the latter case the anthropologist collaborates with an agronomist on the given project An anthropologist is expected to give answers to certain questions which should lead to action for eg to drill a well it is necessary to form a water group which will contribute labor and or take the responsibility of maintaining the well after it has become operational Anthropologists entered the field of development when development organizations acknowledged that things often did not work out according to expectations because of cultural factors Anthropologists can help in this regard given their understanding of cultural similarities and differences Anthropology s Contribution to Development Anthropologists have highlighted an appreciation of local knowledge and practices Anthropologists argue that indigenous knowledge practices and social institutions must be considered if local resource management and development plans are to work Interaction between so called experts in the modern sector and people representing local specific knowledge can result in the creation of new knowledge and be a starting point for development activities In an anthropological sense culture is integrated in society and social development and is thus heterogeneous dynamic and holistic Anthropologists have shown that people are not an undifferentiated mass A first step of development workers is to get the whole picture of norms and values and maybe their ideals in a specific area The second step is to look for the variations in the heterogeneity of what first looks like a homogeneous mass of people Hierarchies are found everywhere It is of utmost importance to recognize hierarchies in the process of planned change The manner in which certain groups are left outside the decision making process also deserves attention Requirements amp Rewards of Anthropological Input Research into culture and development requires time It involves considering the interaction and interchange of different kind of knowledge and learning between development agents the so called experts and people representing local knowledge all this also requires much effort and resources Much work done by the anthropologist is anticipatory in nature Anthropological experience helps anticipate potential both negative and positive changes A well done cultural analysis of development initiatives also helps to anticipate conflicts which can be addressed before they become serious problems Useful Terms Hierarchies segmented responsibilities accompanied by differences in rewards and prestige Copyright Virtual University of Pakistan 88 Introduction to Cultural Anthropology SOC401 VU Undifferentiated lacking differentiation similar Integrated tied or connected to each other Operational functional or workable Internet Resources In addition to reading from the textbook please visit the following web site for this lecture Which provide useful and interesting information Addressing livelihoods in Afghanistan http WWWareuorgpk publications livelihoods Addressing20Livelihoods pdf Copyright Virtual University of Pakistan 89 Introduction to Cultural Anthropology SOC401 VU Lesson 42 CULTURAL ANTHROPOLOGY AND ART What is Art Art can be de ned as the process and products of applying certain skills to any activity that transforms matter sound or motion into a form that is deemed aesthetically meaningful to people in a society Yet there is no universal definition of art Art re ects the human urge to express oneself and to take pleasure from aesthetics The creative process of art is enjoyable produces an emotional response and conveys a message Verbal art includes myths and folktales Myths tend to involve supernatural beings whereas folktales are more secular in nature Like other art forms verbal arts are connected to other aspects of a culture Art and Anthropology Art plays a useful social function and is prominent in ceremonies and customs of most cultures The forms of artistic expression of relevance to cultural anthropologists include graphic and plastic arts such as painting carving and weaving music dance and verbal art such as myth and folklore Examples of Art Painting sculpture and ceramics are common forms of western art Religiously inspired art forms are also impressive including architecture Smaller societies also have distinct art forms the Nubian body decorations Eskimo body tattooing and Navajo sand paintings are examples of art Relevance of Art Art contributes to the well being of individuals and society For individuals art provides emotional gratification to the artist and the beholder From the social perspective art strengthens and reinforces social bonds and cultural themes acts as a mechanism of social control and is a symbol of high status particularly in complex societies Differences in Art Forms Major differences in art forms are found between different cultures of the world In small scale societies of foragers pastoralists or shifting cultivators with nomadic or semi nomadic residence patterns the art in these societies either involves performing arts song dance or story telling or is highly portable jewelry tattooing Judging the Quality of Art In modern societies what constitutes good art is largely determined by the professional art establishment experts critics academics In societies lacking professional art establishments artistic standards are less elaborate and more diffuse and democratic relying on public reaction Complex societies with specialization and sophisticated institutions invest in elaborate buildings larger than life canvases kept in museums Copyright Virtual University of Pakistan 90 Introduction to Cultural Anthropology SOC401 VU Art and Politics It is possible to see symbols of political power expressed via art In Polynesia leadership based on centralized chiefdoms results in chiefs using permanent tattoos to reflect their hereditary high status In Melanesia on the other hand power is more uid and the big men indicate their authority using temporary body paints Useful Terms Tattoo form of body art which illustrates onto the skin using permanent ink Canvas the cloth on which paintings are done Diffuse spread out Art establishment art experts critics and academics Museums and other art institutions are also part of this establishment Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 15 in Cuz umAm bropoogy A Applied Pmpecz z39be by Fermrro cmd Cbapz er 26 23971 Lil z bropology by Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Art and Anthropology wwwanthroarcheartorg or wwwartandanthropologycom or wwwaugieedu deptz art Copyright Virtual University of Pakistan 91 Introduction to Cultural Anthropology SOC401 VU Lesson 43 CULTURAL ANTHROPOLOGY AND ART continued Functionalist Perspectives Concerning Art Manilowski tended to emphasize how various cultural elements function for the psychological well being of the individual Radcliff Brown stressed how a cultural functional element of art functions to contribute to the well being or continuity of society Psychological Benefits of Art For the artist artistic impressions enable expression of emotional energy in a concrete and visible manner The creative tension released via artistic expression brings personal gratification Works of art evoke emotional responses from their viewers which can be positive or negative but do help relieve stress Art and Social Integration Art functions to sustain longevity of the society in which it is found Art is connected to other parts of the social system and used to evoke positive feelings for its rulers Even in ancient Aztec and Egyptian civilizations the ziggurats and pyramids served to provide a visual reinforcement of the awesome power of the rulers Art forms like music also help reinforce social bonds and cultural themes Martial music on the other hand helps rally people against a common enemy Story telling also passes on social values from one generation onto the next whereby helping social integration Art and Social Control A popular perception concerning artists is that they are non conformist visionary and aloof Art often reinforces existing socio cultural systems It also instills important cultural values and in uences people to behave in socially appropriate ways Art can buttress inequalities of existing stratification systems In highly stratified societies state governments use art for maintaining the status quo and to solicit obedience and respect Art as a Status Symbol Acquiring art objects provides a convincing way to display one s wealth and power Possessing art objects implies high prestige due to its uniqueness Art in ancient Egypt was the personal property of the pharos Art galleries often exhibit personal collections obtained from high ranking members of society Art as a Form of Protest Art functions as a vehicle for protest resistance and even revolution Various artists have attempted to raise the consciousness of their countrymen through their poems painting and plays and helped instigate socio political changes Useful Terms Consciousness the feelings sentiments and thoughts of a person or of a given people Copyright Virtual University of Pakistan 92 Introduction to Cultural Anthropology SOC401 VU Acquiring obtaining Status quo the existing system Ziggurats ancient places of worship in the South American continent renowned for their architectural design Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 15 in Cuz umAm bropoogy A Applied Pmpecz z39ve 7y Fermrro cmd Cbapz er 26 23971 Lil z bropology 7y Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Art and Anthropology wwwanthroarcheartorg or wwwartandanthropologycom or wwwaugieedu dept art Copyright Virtual University of Pakistan 93 Introduction to Cultural Anthropology SOC401 VU Lesson 44 ETHICS IN ANTHROPOLOGY Ethical Condemnation Since the 1960s cultural anthropology has been the target of critical attacks both from within and without the discipline The condemnation of anthropology and anthropologists by postmodernism literary theory and post colonialism among others has been directed at its status as a science and its participation in the oppression of minorities and justification of colonialism Critics assert that anthropology has been used solely to objectify oppressed peoples and that it cannot be considered a science Anthropologists are blamed for asserting domination over his or her subject due to negative and inaccurate representations formed by the critics Anthropology is charged with ignoring history in studying non Western societies and so anthropologists have been blamed for treating cultures as isolated from neighbors and the world at large Anthropologists can also reinforce biases and stereotypes by using flawed methodology in their works Orientalism By studying the orient the scholar separates him or herself from the culture they study and recreate it as another world Said believes that Asians are confined by the Oriental label that has been constructed by the European scholar It is natural for the human race to divide itself into quotusquot and quotthemquot It is this division that leads to hostility The separation that arises due to scholarly study only strengthens this hostility Response of Anthropologists In order to continue the study of culture anthropology developed the term relativism which stated that all cultures were equal but not necessarily alike Cultural anthropology could not however accept relativism because issues of morality became controversial The study of anthropology became obsessed with data analysis in order to avoid moral judgment Classic anthropologists feared domination of the discipline by psychology and sociology therefore anthropology had to be redefined in order to shift the focus of the discipline back to the study of culture Past research existed only on exotic cultures and the theories developed from that research were used to try to define modern or first world culture Several problems arose from this movement Few people were interested in studies in cities or familiar places the exotic areas broke the rule that all cultures are equal and therefore these areas drew the attention of anthropologists Another problem was that all previous studies were done on societies with no recorded history and therefore no changes in patterns or traditions were observed Defending Anthropological Integrity Leading and influential anthropologists generally believed in uniformity in the actions and nature of humankind not in the idea of self and the Other They wanted to study all forms of culture at home and abroad to discover similarities There are several examples of anthropologists who recognized the importance of borrowing diffusion and regional and global interactions in shaping society Anthropology should base their criticisms on a careful scrutiny of facts Copyright Virtual University of Pakistan 94 Introduction to Cultural Anthropology SOC401 VU Using Criticism Constructively Questions and ideas put forth by anthropology s critics must be used to help avoid misperceptions and poorly founded opinions from passing on as common knowledge to the new generation of anthropologists Reexamination of the prevalent attitudes in anthropology can move away the notion of anthropologists as authoritarian figures to humanistic scientific scholars interested in comparing and contrasting cultures Assuring Anthropological Integrity Objectivity and functional analysis combined with today s knowledge of psychology that is the key to comprehensiveness and objectivity in anthropology Useful Terms Objectivity unbiased observation of facts Authoritarian monopolized exertion of power Prevalent existing or in current use Scrutiny study or careful observation Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 28 in Am bropoogy by Ember Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Ethics in Anthropology htt www ublicanthro 010 or ournals En a in Ideas Rt ano Peterslhtm Copyright Virtual University of Pakistan 95 Introduction to Cultural Anthropology SOC401 VU Lesson 45 RELEVANCE OF CULTURAL ANTHROPOLOGY Change and the Future of Anthropology Change is occurring at such an accelerated pace that it is difficult to keep up with all the changes in the world today The recent revolution in transportation and telecommunications and the resulting increase in communications and travel are diffusing cultures at a much greater rate today than ever before Some argue that cultural anthropology will loose importance in the future since it is only a matter of time when all cultures will be homogenized Yet few cultural anthropologists are studying pristine cultures as the discipline is adapting to the realities of this changing world Concern for survival of indigenous cultures and the study of complex societies is now the new focus area for many cultural anthropologists There is also greater emphasis on using anthropological perspectives to deal with developmental problems There is little evidence to suggest that the world is becoming a cultural melting pot so despite cultural changes there is enough diversity in the world to keep cultural anthropologists occupied for a long time to come Ensuring Cultural Survival Cultural patterns and in some cases people themselves have been eradicated as a direct result of progress and economic development The indigenous population of Tasmania in 19th century by white settlers for sheep herding is a tragic example of cultural extinction The 1884 Berlin Conference was a civilized way of dividing spoils of Africa but not safeguarding rights of indigenous people and numerous conflicts on the African continent are based on this insensitive division and lumping together of different ethnic groups The Brazilian Amazon shelters the largest population of the world s still indigenous people But by building roads through the Amazonian frontier the Brazilian government has introduced diseases such as in uenza and measles amongst the indigenous communities Contemporary Anthropologists Anthropological research has great relevance for the public at large Consider for example the role archaeology played in society during the nineteenth century Books on the subject were widely read Darwin s work for example significantly changed beliefs on human history and development of the modern world Throughout this era of advancements academic archaeology was on the rise This movement finally phased out the participation of amateurs in the field creating a more elitist and inaccessible discipline While professionalization has certainly had numerous benefits including developments in quotmethod theory and culture historical knowledgequot its negative aspects are causing a significant deterioration of popular interest in archaeology A movement towards popularization through accessible writing must take place in order to involve the public and rekindle active interest in archaeology and indeed in other branches of anthropology Accessibility glorifies the field of anthropology rather than denigrates it Nowadays rather than writing holistic ethnographies cultural anthropologists bring to the study of cities and complex societies a more nuanced sensitivity towards understanding and dealing with the issue of ethnic diversity Copyright Virtual University of Pakistan 96 Introduction to Cultural Anthropology SOC401 VU Anthropologists practicing quotaction anthropologyquot collaborate with other disciplines concerning the development of culture and how it relates to current pertinent issues Useful Terms Holistic ethnographies overarching description concerning all aspects of life of a given community Ethnic diversity different ethnic groups or the differences within or between them Pertinent relevant or important Nuanced having various aspects Suggested Readings Students are advised to read the following chapters to develop a better understanding of the various principals highlighted in this hand out Chapter 17 in Cultural Anthropology A Applied Pmpecz z39ve by Ferrarro Internet Resources In addition to reading from the textbook please visit the following web site for this lecture which provide useful and interesting information Intellectuals and the Responsibilities of Public Life An Interview with Chomsky htt www ublicanthro 010 or ournals En a in Ideas choms htm Copyright Virtual University of Pakistan | 677.169 | 1 |
Mathematical Models with Applications
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An innovative course that offers students an exciting new perspective on mathematics, Mathematical Models with Applications explores the same types of problems that math professionals encounter daily. The modeling process--forming a theory, testing it, and revisiting it based on the results of the test--is critical for learning how to think mathematically. Demonstrating this ability can open up a wide range of educational and professional opportunities for students. Mathematical Models with Applications has been designed for students who have completed Algebra I or Geometry and see this as the final course in their high school mathematics sequence, or who would like additional math preparation before Algebra II. Mathematical Models with Applications ListServ As a service to instructors using Mathematical Models with Applications, a listserv has been designed as a forum to share ideas, ask questions and learn new ways to enhance the learning experience for their students.Mathematical Models with Applications has been designed for students who have completed Algebra I or Geometry and see this as the final course in their high school mathematics sequence, or who would like additional math preparation before ...
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Mathematical Models with Applications
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COMAP
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Macmillan - 2001-06-15
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Proof
Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics. Mathematical Association of America. Paperback. Condizione libro: new. BRAND NEW, Proofs without Words: Exercises in Visual Thinking: v. 1, Roger B. Nelsen, B9780883857007 Paperback. Condizione libro: New. Not Signed; book. Codice libro della libreria ria9780883857007_rkm
Descrizione libro The Mathematical Association of3857007 | 677.169 | 1 |
Fun Self-Discovery Tools
1-5 Representing Functions
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Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Reason quantitatively and use units to solve problems.Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. | 677.169 | 1 |
How Math is Different from Other Subjects
1. Math requires different study processes. In other courses, you learn and understand the material, but you seldom have to actually APPLY IT. You have to do the problems.
2. Math is a linear learning process. What is used one day is used the next, and so forth. (In history you can learn chapter 2 and not 3 and do OK on 4. In math, you must understand the material in chapter 1 before you go on to chapter 2.)
3. Math is much like a foreign language. It must be practiced EVERY DAY, and often the VOCABULARY is unfamiliar.
4. Math in the university is different from math in high school. Instead of going to class everyday, in college you go only two or three times a week. What took a year to learn in high school is now covered in only fifteen weeks.
8 May 2009, Friday 14:25 | 677.169 | 1 |
This preface is addressed to readers who are interested in computing but who seldom either consult manuals or read prefaces. So, I will be brief. Computing requires an integrated approach, in which scientific and mathematical analysis, numerical algorithms, and programming are developed and used together. The purpose of this book is to provide an introduction to analysis, numerics, and their applications. I believe that a firm grounding in the basic concepts and methods in these areas is necessary if you wish to use numerical recipes effectively. The topics that I develop extensively are drawn mostly from applied mathematics, the physical sciences, and engineering. They are divided almost equally among review of the mathematics, numerical-analysis methods (such as differentiation, integration, and solution of differential equations from the sciences and engineering), and dataanalysis applications (such as splines, least-squares fitting, and Fourier expansions). I call this a workbook, since I think that the best way to learn numerically oriented computing is to work many examples. Therefore, you will notice and, I hope, solve many of the exercises that are strewn throughout the text like rocks in the stream of consciousness. I try to introduce you to a technique, show you some of the steps, then let you work out further steps and developments yourself. I also suggest new and scenic routes rather than overtraveled highways. There are occasional diversions from the mainstream, often to point out how a topic that we are developing fits in with others in computing and its applications. The programming language in which I present programs is C. This language is now used extensively in systems development, data-acquisition systems, numerical methods, and in many engineering applications. To accommodate readers who prefer Fortran or Pascal, I have used only the numerically oriented parts of C and I have v
vi
structured the programs so that they can usually be translated line-by-line to these other languages if you insist. The appendix summarizes the correspondences between the three languages. All programs and the functions they use are listed in the index to programs. The fully-developed programs are usually included in the projects near the end of each chapter. This book may be used either in a regular course or for self-study. In a onesemester course for college students at the junior level or above (including graduate students), more than half the topics in the book could be covered in detail. There is adequate cross-referencing between chapters, in addition to references to sources for preparation and for further exploration of topics. It is therefore possible to be selective of topics within each chapter. Now that we have an idea of where we're heading, let's get going and compute! William J. Thompson Chapel Hill, July 1992
CONTENTS
1. Introduction to Applicable Mathematics and Computing 1.1 What is applicable mathematics? 1
1
Analysis, numerics, and applications 2 Cooking school, then recipes 3 Diversions and new routes 4 Roads not taken 4
1.2 Computing, programming, coding 5 The C language for the programs 6 Learning to program in C 7 Translating to Fortran or Pascal from C 8 The computing projects and the programs 9 Caveat emptor about the programs 10 The index to computer programs 11
1.3
One picture is worth 1000 words 11 Why and when you should use graphics 11 Impressive graphics, or practical graphics 12
1.4 Suggestions for using this book 12 Links between the chapters 13 The exercises and projects 13
References for the introduction 14 General references 14 References on learning and using C 15
Project 2: Program to convert between coordinates 45 Stepping into the correct quadrant 45 Coding, testing, and using the program 46
References on complex numbers 49 3. Power Series and Their Applications 3.1 Motivation for using series: Taylor's theorem 51 The geometric series 52 Programming geometric series 53 Alternating series 56 Taylor's theorem and its proof 58 Interpreting Taylor series 59
Project 3: Testing the convergence of series 85 Coding and checking each series expansion 85 Including the hyperbolic functions 91 File output and graphics options 92 The composite program for the functions 92 Using the program to test series convergence 97
References on power series 98
51
CONTENTS
4. Numerical Derivatives and Integrals 4.1 The working function and its properties 100
ix
99
Properties of the working function 100 A C function for Homer's algorithm 103 Programming the working function 106
9.5 Some practical Fourier series 337 The square-pulse function 338 Program for Fourier series 340 The wedge function 343 The window function 345 The sawtooth function 347
9.6 Diversion: The Wilbraham-Gibbs overshoot 349 Fourier series for the generalized sawtooth 350 The Wilbraham-Gibbs phenomenon 353 Overshoot for the square pulse and sawtooth 356 Numerical methods for summing trigonometric series 359
9.7
Project 9A: Program for the fast Fourier transform 360 Building and testing the FFT function 360 Speed testing the FFT algorithm 364
9.8
Project 9B: Fourier analysis of an electroencephalogram 365 Overview of EEGs and the clinical record 365 Program for the EEG analysis 368 Frequency spectrum analysis of the EEG 372 Filtering the EEG data: The Lanczos filter 373
10.4 Project 10: Computing and applying the Voigt profile 411 The numerics of Dawson's integral 412 Program for series expansion of profile 413 Program for direct integration of profile 418 Application to stellar spectra 419
References on Fourier integral transforms 419 EPILOGUE
421
APPENDIX: TRANSLATING BETWEEN C, FORTRAN, AND PASCAL LANGUAGES
423
INDEX TO COMPUTER PROGRAMS
429
INDEX
433
COMPUTING FOR SCIENTISTS AND ENGINEERS
Previous Home Next
Chapter 1
INTRODUCTION TO APPLICABLE MATHEMATICS AND COMPUTING
The major goal for you in using this book should be to integrate your understanding, at both conceptual and technical levels, of the interplay among mathematical analysis, numerical methods, and computer programming and its applications. The purpose of this chapter is to introduce you to the major viewpoints that I have about how you can best accomplish this integration. Beginning in Section 1.1, is my summary of what I mean by applicable mathematics and my opinions on its relations to programming. The nested hierarchy (a rare bird indeed) of computing, programming, and coding is described in Section 1.2. There I also describe why I am using C as the programming language, then how to translate from to Fortran or Pascal from C, if you insist. The text has twelve projects on computing, so I also summarize their purpose in this section. Section 1.3 has remarks about the usefulness of graphics, of which there are many in this book. Various ways in which the book may be used are suggested in Section 1.4, where I also point out the guideposts that are provided for you to navigate through it. Finally, there is a list of general references and many references on learning the C language, especially for those already familiar with Fortran or Pascal.
1.1
WHAT IS APPLICABLE MATHEMATICS?
Applicable mathematics covers a wide range of topics from diverse fields of mathematics and computing. In this section I summarize the main themes of this book. First I emphasize my distinctions between programming and applications of programs, then I summarize the purpose of the diversion sections, some new paths to 1
2
INTRODUCTION
familiar destinations are then pointed out, and I conclude with remarks about common topics that I have omitted. Analysis, numerics, and applications In computing, the fields of analysis, numerics, and applications interact in complicated ways. I envisage the connections between the areas of mathematical analysis, numerical methods, and computer programming, and their scientific applications as shown schematically in Figure 1.1.
FIGURE 1.1 Mathematical analysis is the foundation upon which numerical methods and computer programming for scientific and engineering applications are built.
You should note that the lines connecting these areas are not arrows flying upward. The demands of scientific and engineering applications have a large impact on numerical methods and computing, and all of these have an impact on topics and progress in mathematics. Therefore, there is also a downward flow of ideas and methods. For example, numerical weather forecasting accelerates the development of supercomputers, while topics such as chaos, string theory in physics, and neural networks have a large influence on diverse areas of mathematics. Several of the applications topics that I cover in some detail are often found in books on mathematical modeling, such as the interesting books by Dym and Ivey, by Meyer, and by Mesterton-Gibbons. However, such books usually do not emphasize much computation beyond the pencil-and-paper level. This is not enough for scientists, since there are usually many experimental or observational data to be handled and many model parameters to be estimated. Thus, interfacing the mathematical models to the realities of computer use and the experience of program writing is mandatory for the training of scientists and engi neers. I hope that by working the materials provided in this book you will become adept at connecting formalism to practice.
1.1
WHAT IS APPLICABLE MATHEMATICS?
3
By comparison with the computational physics books by Koonin (see also Koonin and Meredith), and by Gould and Tobochnik, I place less emphasis on the physics and more emphasis on the mathematics and general algorithms than these authors provide. There are several textbooks on computational methods in engineering and science, such as that by Nakamura. These books place more emphasis on specific problem-solving techniques and less emphasis on computer use than I provide here. Data analysis methods, as well as mathematical or numerical techniques that may be useful in data analysis, are given significant attention in this book. Examples are spline fitting (Chapter 5), least-squares analyses (Chapter 6), the fast Fourier transform (Chapter 9), and convolutions (Chapter 10). My observation is that, as part of their training, many scientists and engineers learn to apply data-analysis methods without understanding their assumptions, formulation, and limitations. I hope to provide you the opportunity to avoid these defects of training. That is one reason why I develop the algorithms fairly completely and show their relation to other analysis methods. The statistics background to several of the data-analysis methods is also provided. Cooking school, then recipes Those who wish to become good cooks of nutritious and enjoyable food usually go to cooking school to learn and practice the culinary arts. After such training they are able to use and adapt a wide variety of recipes from various culinary traditions for their needs, employment, and pleasure. I believe that computing — including analysis, numerics, and their applications — should be approached in the same way. One should first develop an understanding of analytical techniques and algorithms. After such training, one can usually make profitable and enlightened use of numerical recipes from a variety of sources and in a variety of computer languages. The approach used in this book is therefore to illustrate the processes through which algorithms and programming are derived from mathematical analysis of scientific problems. The topics that I have chosen to develop in detail are those that I believe contain elements common to many such problems. Thus, after working through this book, when you tackle an unfamiliar computing task you will often recognize parts that relate to topics in this book, and you can then probably master the task effectively. Therefore I have not attempted an exhaustive (and exhausting) spread of topics. To continue the culinary analogs, once you have learned how to make a good vanilla ice cream you can probably concoct 30 other varieties and flavors. I prefer to examine various facets of each topic, from mathematical analysis, through appropriate numerical methods, to computer programs and their applications. In this book, therefore, you will usually find the presentation of a topic in this order: analysis, numerics, and applications. The level I have aimed at for mathematical and computational sophistication, as well as for scientific applications, is a middle ground. The applications themselves do not require expert knowledge in the fields from which they derive, although I
4
INTRODUCTION
give appropriate background references. Although I present several topics that are also in my earlier book, Computing in Applied Science (Thompson, 1984), the preliminary and review materials in this book are always more condensed, while the developments are carried further and with more rigor. The background level necessary for the mathematics used in this book is available from mathematics texts such as that by Wylie and Barrett, and also from the calculus text by Taylor and Mann. Diversions and new routes In order to broaden your scientific horizons and interests, I have included a few diversion sections. These are intended to point out the conceptual connections between the topic under development and the wider world of science, technology, and the useful arts. The diversions include a discussion of the interpretation of complex numbers (in Section 2.5), the relationships between recursion in mathematics and computing (Section 3.4), the development of computers and the growing use of spline methods in data analysis (Section 5.6), and the Wilbraham-Gibbs overshoot in the Fourier series of discontinuous functions (Section 9.6). Other connections are indicated in subsections of various chapters. Although these diversions may not necessarily be of much help in making you an expert in your field of endeavor, they will help you to appreciate how your field fits into the larger scientific landscape. This book also uses some seldom-traveled routes to reach known destinations, as well as a few tracks that are quite new. These routes and their results include linearized square-root approximations (Section 3.3), the relations between maximum likelihood, least squares, and Fourier expansion methods (Sections 6.1, 6.2, 9.2), algorithms for least-squares normalization factors (Section 6.4), logarithmic transformations and parameter biases (Section 6.5), generalized logistic growth (Section 7.3), a novel approach to catenaries (Section 8.2), the discrete and integral Fourier transforms of the complex-exponential function (Sections 9.2 and 10.2), and the Wilbraham-Gibbs overshoot (Section 9.6). Roads not taken Because this book is directed to a readership of mathematicians, scientists, engineers, and other professionals who may be starting work in fields from which examples in this book are drawn, there are many byways that I have not ventured upon. Rather, I emphasize principles and methods that are of both general validity and general applicability. One road not taken is the one leading to topics from linear algebra (except incidentally) and matrix manipulation. Mathematics texts are replete with examples of methods for 3 x 3 matrices, many of which will usually fail for matrices of typically interesting size of 100 x 100. I believe that matrix computation is best taught and handled numerically by using the powerful and somewhat advanced methods specially developed for computers, rather than methods that are simply extensions of methods suitable for hand calculations on small matrices.
1.2
COMPUTING, PROGRAMMING, CODING
5
Symbolic calculation, also misleadingly termed "computer algebra," is not discussed here either. I believe that it is dangerous to just "give it to the computer" when mathematical analysis is necessary. Machines should certainly be used to solve long and tedious problems reliably and to display results graphically. However, the human who sets up the problem for the computer should always understand clearly the problem that is being solved. This is not likely to be so if most of one's training has been through the wizardry of the computer. I have the same objection to the overuse of applications programs for numerical work until the principles have been mastered. (Of course, that's one reason to set oneself the task of writing a book such as this.) In spite of my warnings, when you have the appropriate understanding of topics, you should master systems for doing mathematics by computer, such as those described in Wolfram's Mathematica. Finally, I have deliberately omitted descriptions of computational methods that are optimized for specialized computer designs, such as vectorized or parallel architectures. You should leam and use these methods when you need them, but most of them are developed from the simpler principles that I hope you will learn from this book. You can become informed on both parallel computation and matrix methods by reading the book by Modi. 1.2 COMPUTING, PROGRAMMING, CODING When computing numerically there are three levels that I envision in problem analysis: program design, coding, and testing. They are best described by Figure 1.2.
FIGURE 1.2 Computing, programming, and coding form a nested hierarchy. An example of this nesting activity is that of converting between coordinates (Section 2.6).
6
INTRODUCTION
In Figure 1.2 the activity of computing includes programming, which includes coding. The right side of the figure shows the example of converting between Cartesian and polar coordinates — the programming project described in Section 2.6. The aspects of computing that lie outside programming and coding are numerical analysis and (to some degree) algorithm design. In the example, the formulas for calculating coordinates are part of numerical analysis, while deciding how quadrants are selected is probably best considered as part of algorithm design. At the programming level one first has to decide what one wants to calculate, that is, what output the program should produce. Then one decides what input is needed to achieve this. One can then decide on the overall structure of the program; for example, the conversions for Cartesian and polar coordinates are probably best handled by separate branches in the program. At this level the choices of computing system and programming language usually become important. Finally, as shown in Figure 1.2, one reaches the coding level. Here the program is built up from the language elements. In the C language, for example, functions are written or obtained from function libraries, variable types are declared, and variables are shared between functions. Detailed instructions are coded, the interfaces to files or graphics are written, and the program is tested for corectness of formulas and program control, such as the method used to terminate program execution. If you think of the activity of computing as a nested three-level system, as schematized in Figure 1.2, then you will probably produce better results faster than if you let your thoughts and actions become jumbled together like wet crabs scuttling in a fishing basket. In the following parts of this section, I make remarks and give suggestions about programming as described in this book. First, I justify my choice of C as the language for preparing the programs, then I give you some pointers for learning to program in C, and for translating the programs in this book to Fortran or Pascal from C (if you insist on doing this). Next, I summarize programming aspects of the projects that occur toward the end of each of Chapters 2 through 10, and I remark on the correctness and portability of these programs. Finally in this section, I draw your attention to a convenience, namely, that all the programs and functions provided in this book are listed by section and topic in the index to computer programs before the main index. The C language for the programs I decided to write the sample programs in C language for the following reasons. First, C is a language that encourages clear program structure, which leads to programs that are readable. My experience is that C is easier to write than Pascal because the logic of the program is usually clearer. For example, the use of a topdown structure in the programs is closer to the way scientists and enginners tackle real problems. In this aspect C and Fortran are similar. The C language is more demanding than Fortran, in that what you want to do and the meanings of the variables must all be specified more accurately. Surprisingly, scientists (who pride themselves on precise language) often object to this demand from a computer. I estimate
1.2 COMPUTING, PROGRAMMING, CODING
7
that C is the easiest of the three languages in which to write programs that are numerically oriented. Ease is indicated by the time it takes to produce a correctly executing program in each language. The second reason for using C in this book is that it is intermediate in complexity between Fortran and Pascal, as illustrated by the comparison chart in the appendix. That is, there are very few elements in the Fortran language (which makes it simple to write but hard to understand), most of the elements of Fortran are in C, and some of the elements of Pascal are also in C. For data handling, C is much more powerful and convenient than Fortran because facilities for handling characters and strings were designed into the original C language. The third reason for my choice of C is that it is now the language of choice for writing applications programs for workstations and personal computers. Therefore, programs that you write in C for these machines will probably interface easily with such applications programs. This is a major reason why C is used extensively in engineering applications. A fourth reason for using C is that its developers have tried to make it portable across different computers. Lack of portability has long been a problem with Fortran. Interconnectivity between computers, plus the upward mobility of programs developed on personal computers and workstations to larger machines and supercomputers, demand program portability. Since very few large computer systems have extensive support for Pascal, C is the current language of choice for portability. One drawback of C lies with input and output. Some of the difficulty arises from the extensive use of pointers in C, and some inconvenience arises from the limited flexibility of the input and output functions in the language. For these reasons, I have written the input and output parts of the sample programs as simply as practicable, without any attempt to produce elegant formats. Since you probably want to modify the programs to send the output to a file for processing by a graphics program, as discussed in Section 1.3, for this reason also such elegance is not worthwhile in the sample programs. Complex numbers are not part of the C language, although they are used extensively in numerical applications, as discussed in Chapter 2. Our calculations that use complex variables convert the complex numbers to pairs of real numbers, then work with these. Extensive practice with programming using complex numbers in this way is given in Sections 2.1 and 2.6. In Numerical Recipes in C, Press et al. also discuss (Chapter 1 and Appendix E) handling complex numbers in C. Learning to program in C This book does not claim to be a guide to learning the C programming language. It will, however, provide extensive on-the-job training for the programming of numerical applications in C. If you wish to learn how to program in C, especially for the numerically oriented applications emphasized herein, there are several suitable textbooks. Starting with books which do not assume that you have much familiarity with progamming then moving upward, there are Eliason's C, a Practical Learning Guide and Schildt's Teach Yourself C, then the text by Darne11 and Margolis, C, a
8
INTRODUCTION
Software Engineering Approach. Many of C's more subtle and confusing aspects are described by Koenig in C Traps and Pitfalls. If you are familiar with other programming languages and wish to use the C programs in this book, there are several texts that should be of considerable help to you. For general use there is the book by Gehani, which emphasizes the differences between C and other procedural programming languages. Gehani also discusses the advanced aspects of C as implemented on UNIX systems. A second cross-cultural book that will help you with the language barrier is Kerrigan's From Fortran to C, which has extensive discussions and examples of how to learn C and to reprogram from Fortran. The book by M?ldner and Steele, C as a Second Language, and that by Shammas, Introducing C to Pascal Programmers, are especially suitable for those who are familiar with Pascal but wish to learn to program effectively in C. Finally, for detailed references on the C language there are C: A Reference Manual by Harbison and Steele, and The Standard C Library by Plauger. You should also consult the programming manuals for C provided with the implementation of C for your computing environment, and the manuals that explain the connection between C and your computer's operating system. The references on learning to program in the C language are listed together in the reference section at the end of this chapter. The appendix provides examples of translating between C, Fortran, and Pascal that are drawn from the C programs in the first chapters. Translating to Fortran or Pascal from C By choosing to present the example programs and the project programs in C language, I know that I will have made a few friends but I may have alienated others. Especially for the latter, I have tried to decrease their animosity by avoiding use of some of the useful constructions in C that are sometimes not available or are awkward to implement in other numerically oriented procedural languages. This should make the programs easier to translate on-the-fly into Fortran or Pascal. Among my main concessions are the following. In arrays the [0] element is usually not used by the program, so that the used elements of the array range from [1] upward. The only confusion this may cause when programming is that the array size must be declared one larger than the maximum element that will ever be used. For example, if you want to be able to use elements 1...100, then the maximum array size (which is always defined as MAX) should be 101. This labeling starting with [1] usually also makes the correspondence between the mathematics and the coding simpler, because most of the summation and iteration indices in formulas (k or j) begin with unity: any zeroth-index value in a summation or iteration has usually to be treated specially. I use the [0] element in an array only if this tightens the connections between the mathematical analysis, the algorithm, and the code. In summations and indexing, the C construction of ++ to denote incrementing by one, and similarly - - for decrementing, is not used except in for loops. Although general avoidance of ++ and - - is less efficient, it is less confusing when translating to Fortran or Pascal, which do not allow such useful constructions.
1.2 COMPUTING, PROGRAMMING, CODING
9
The for loop in C is such a practical and convenient programming device that I use it without concession to Fortran programmers, who are often confined to the much clumsier DO loop. However, I use the for loop in a consistent style to which you can readily adapt. I have avoided go to statements, so there are no statement labels in the programs. Consequently, you will have to go elsewhere if your favorite computing recipes include spaghetti. (The come from statement, which might rescue many a programmer from distress, is also scrupulously avoided.) These omissions do not make C programs difficult to write or to use. There are a few operators in C, especially the logical operators, that look quite different and may be confusing to Fortran and Pascal programmers. I explain these operators where they appear in programs, especially in the early chapters. They are listed with their Fortran and Pascal counterparts at the end of the appendix on translating between C, Fortran, and Pascal. In C the exit function terminates execution when it is called. (Technically, it terminates the calling process. All open output streams are flushed, all open files are closed, and all temporary files are removed.) There is a conventional distinction, which we follow, between exit (0) and exit (1) . The first is for successful termination and graceful exit, while the second is to signal an abnormal situation. In some computing environments the process that refers to the terminating program may be able to make use of this distinction. Within the text, the font and style used to refer to names of programs, functions, and variables is 10-point Monaco (since all programming involves an element of gambling). All the programs and functions are listed in the index to computer programs, which is discussed below. The computing projects and the programs Several of the exercises and projects, as described in Section 1.1, require that you modify programs that are in the text. By this means you will practice what is so common in scientific and engineering computing, namely the assembling of tested and documented stand-alone function modules to make a more powerful program tailored for your use. One advantage of this method is that, provided you are careful how you make the modifications, you will usually be able to check the integrity of the program module by comparison with the stand-alone version. The sample programs, both in the text and in the projects, are written for clarity and efficiency of writing effort. In particular, when there are choices between algorithms, as in the numerical solution of differential equations, the different algorithms are usually coded in-line so that it is easy for you to compare them. Therefore, if you wish to transform one of the chosen sections of in-line code into a function you will need to be careful, especially in the type declarations of variables used. I have not attempted to make the sample programs efficient in terms of execution speed or use of memory. If you want to use a particular computing technique for production work, after you have understood an algorithm by exploring with the pro-
10
INTRODUCTION
grams provided, you should use a program package specially developed for the purpose and for the computer resources that you have available. At an intermediate level of efficiency of your effort and computer time are the programs available (in C, Fortran, and Pascal) as part of the Numerical Recipes books of Press et al. My view of the connections among materials in this book, the C language, the Numerical Recipes books, systems such as Mathematica, and their uses in scientific applications is summarized in Figure 1.3.
FIGURE 1.3 Connections among topics in this book, C language, the Numerical Recipes books, the Mathematica system, and scientific applications.
In Figure 1.3 the lines are connectors, not arrows. They indicate the strongest two-way connections between the topics and books (names written in italics). Some significant links have been omitted, mostly for topological reasons. For example, many of the scientific applications examples in this book do not require C programs or use of the Mathematica system. Also, much of the latter is programmed in C, and it can convert its symbolic results into C (or Fortran) source code, as described in Wolfram's book on Mathematica. Caveat emptor about the programs The sample programs included in this book have been written as simply as practical in order that they could readily be understood by the human reader and by the compiler. In order to keep the programs easy to read, I have not included extensive checking of the allowed range of input variables, such as choices that control program options. My rule of thumb has been to put in a range check if I made an input
1.3 ONE PICTURE IS WORTH 1000 WORDS
11
error while testing a program, or if lack of a check is likely to produce confusing results. There are checks for array bounds if they are simple to code and do not interrupt program flow. Errors of use or input that are diagnosed by our C programs always begin with a double exclamation, ! !, followed by an explanation of the error. Program execution will often continue after some reasonable fix-up is attempted. A typical fix-up is just to request another input for the troublesome variable. Because the programs are written to be translated easily to Fortran or Pascal, as described in a previous subsection and shown in the appendix, I have tried to avoid nonstandard parts of C. The compiler that I use claims to follow ANSI standards. I also checked for compatibility with the C language as described in the second edition of Kernighan and Ritchie's book. In spite of all these precautions, I have two words of advice: caveat emptor — let the buyer beware. The programs are supplied as is and are not guaranteed. For each program you use, I suggest that you make at least the checks I have indicated. If you can devise other tests of program correctness, I encourage you to do so. The index to computer programs Because this book has many computer programs with associated functions, I have included an annotated index to all the programs and functions. They are listed, by order of appearance, in the index that immediately precedes the regular index. The programs and functions also appear alphabetically by name in the regular index.
1.3
ONE PICTURE IS WORTH 1000 WORDS
In this book graphical output is usually suggested as a way to improve the presentation of results, especially in the projects. Since graphics are so hardware dependent, my suggestions for graphics in the projects are necessarily vague. You should familiarize yourself as much as practicable with techniques of graphical presentation. If you have access to a powerful system that combines graphics and mathematics, such as Mathematica as described by Wolfram or by Wagon, you may wish to develop some of the projects by using such a system. Why and when you should use graphics Tufte, in his two books on displaying and envisioning quantitative information, has given very interesting discussions and examples of effective (and ineffective) ways of displaying quantitative information from a variety of fields. In numerical applications of mathematics, graphics are especially important because of the enormous number of numbers that come spewing out of the computer in a stream of numerical environmental pollution. If there are many values to be compared, or if you want to show trends and comparisons (as we usually do), it is worth the effort to write a graphical interface for your program. If there are just a few check values to output,
12
INTRODUCTION
it is not worth the extra coding and possible lack of clarity that graphics may produce. If you have access to Mathematica or some other system that combines mathematics, numerics, and graphics, your learning will be enhanced if you combine the three elements when working the exercises and projects. Wagon's book provides many examples of graphics techniques that would be useful in conjunction with this workbook. Impressive graphics, or practical graphics In many books that relate to computing you will see elegant and impressive graphics that have been produced by long runs on powerful computers using special-purpose programs. Although these illustrations may improve your comprehension, and perhaps inspire you to become a computer-graphics artist, their production is usually not practicable for most computer users. Therefore, I urge you to find a simple graphics system that interfaces easily to your programming environment, that is readily available to you, and that is inexpensive to use. For example, there are about a hundred line drawings in this book. They were all produced by using only two applications programs (one for graphics and one for drafting). The graphics program used input files that the C programs produced, so the numbers were seldom touched by human hand, and the graphics output was produced on the same laser printer that printed the text. Many of the programs in this book produce simple output files. I most often used these files for input to graphics, and sometimes for preparing tables. If you make a similar interface and use it often to produce graphics (perhaps through the intermediary of a spreadsheet), I think it will improve your comprehension of the numerical results, without burdening you with much coding effort. If you have convenient access to a state-of-the-art graphics system, it may be useful for a few of the projects in this book. Just as I believe that an approach to numerical computing that is completely recipe-based is unwise, I believe that using computer-graphics systems without an understanding of their background is similarly unwise. A comprehensive treatment of many aspects of computer graphics is provided in the treatise by Foley et al. Methods for preparing high-resolution graphics, and how to implement them in Pascal, are described in the book by Angell and Griffith.
1.4 SUGGESTIONS FOR USING THIS BOOK This book may be used for both self-study and for class use. I have some suggestions that should help you to make most effective use of it. First I indicate the conections between the remaining nine chapters, then there are remarks about the exercises and the projects.
1.4 SUGGESTIONS FOR USING THIS BOOK
13
Links between the chapters Because we cover a large amount of territory and a variety of scientific and engineering landscapes in this book, it is useful to have an indication of the connections between its nine other chapters. Table 1.1 summarizes the strength of the links between the chapters. TABLE 1.1 Cross-reference chart for use of this book. Key: chapter above is necessary preparation chapter above is desirable preparation chapter above is optional preparation
Complex variables Power series Numerical derivatives and integrals Fitting curves through data Least-squares analysis of data Introduction to differential equations Second-order differential equations Discrete Fourier transforms and series Fourier integral transforms For example, if you plan to work through Chapter 7 (introduction to differential equations), use of Chapters 2, 5, and 6 is optional, Chapter 3 (power series) is desirable, whereas Chapter 4 (numerical derivatives and integrals) is necessary preparation. Within each chapter you should read not only the text, but also the exercises, which are embedded in the text. Exercises containing equations are especially important to be read, since these equations often form part of the development. Therefore, read over every exercise, even if you don't work it through in detail. The exercises and projects Since this book has an overwhelming number of exercises, many of them nontrivial, a guide to use of the exercises is appropriate. It will be clear to you that I always insert an exercise whenever I don't want to show you all the steps of a development. This is not laziness on my part, because I assure you that I have worked through every step. Rather, an exercise provides a checkpoint where you should pause, take stock of what you have been reading, then test your understanding by trying the exercise. If you have difficulty with this exercise, reread all of the subsection containing the exercise, even past the troublesome exercise. Then work the exercise
14
INTRODUCTION
once more. I believe that by doing this you will have a realistic estimate of your progress in comprehension. If you are using this book for self-study, this procedure should help you considerably. Some exercises are more than checkpoints, they are crossroads where concepts and techniques developed previously are focused on relevant and interesting problems. This type of exercise, which often appears in a project toward the ends of chapters, is always indicated. Such exercises provide very good tests of your overall comprehension of the material in the current and previous chapters. The projects, of which there is at least one per chapter after this introductory chapter, are designed to bring together many of the aspects of analysis and numerics emphasized within the chapter. They provide you with opportunities to explore the numerics and the science by using the number-crunching and graphical powers of computers. Programming aspects of the projects are discussed in Section 1.2.
A REVIEW OF COMPLEX VARIABLES The purpose of this chapter is to review your understanding of complex numbers and complex variables, and to summarize results that are used extensively in subsequent chapters. Complex variables are treated elegantly and completely in many mathematics texts, but a treatment in which the aim is to develop intuition in the scientific applications of complex numbers may have much more modest goals, and is best done from a geometrical perspective, rather than from an analytic or algebraic viewpoint. We start with the algebra and arithmetic of complex numbers (in Section 2.1) including a simple program, then turn in Section 2.2 to the complex-plane representation because of its similarities to plane-polar coordinates and to planar vectors. The simplest (and most common) functions of complex variables-complex exponentials and hyperbolic functions- are reviewed in Section 2.3. Phase angles, vibrations, and complex-number representation of waves, which are all of great interest to scientists and engineers, are summarized in Section 2.4 before we take a diversion in Section 2.5 to discuss the interpretation of complex numbers. Project 2, which includes program Cartesian & Polar Coordinate Interconversion for converting between plane-polar and Cartesian coordinates, concludes the text of this chapter. This program serves to emphasize the ambiguities in calculating polar coordinates from Cartesian coordinates, and it will be useful in the later programming applications. References on complex numbers complete the chapter. The discussion of complex variables is limited to the above topics, and does not develop extensively or with any rigor the topics of analytic functions in the complex plane, their differentiation, or their integration. Although several derivations later in this book, especially those involving integrals, would be simplified if the methods of contour integration were used, the methods of derivation used here are usually direct and do not require the extra formalism of contours and residues. Readers who have experience with functions of a complex variable will often be able to substitute their own methods of proof, which may be more direct than those provided here. 17
Next
18
COMPLEX VARIABLES
Many of the examples and exercises in this chapter anticipate steps in our later developments that use complex variables, especially the material on Fourier expansions (Chapters 9 and 10). Since we always have a reference to the later material, you may wish to look ahead to see how the complex variables are used. 2.1
ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS
In this section we summarize the algebraic properties of complex numbers, their properties for numerical computation, the relation between complex numbers and plane geometry, and the special operations on complex numbers — complex conjugation, modulus, and argument. When you have reviewed this section, you will have the conceptual and technical skills for the more extensive developments of complex numbers that are presented in the remainder of this chapter. In particular, Project 2 — the program for converting between coordinates (Section 2.6) -requires most of the ideas from this section. If you are experienced with complex variables, you may try the project before working this section. If you have difficulties with the mathematics in the project (as distinguished from the programming involved), return and rework this section. The algebra of complex numbers We indicate a complex number, z, symbolically by (2.1) in which x and y are understood to be both real numbers. The sign + in this formula does not mean arithmetic addition, although it has many rules that are similar to those for addition. You will have already encountered yet another meaning of + as a sign used in the addition of vectors, which is also distinct from arithmetic addition. In (2.1), the symbol i has the property that (2.2) with a unique value being assumed for i itself. In engineering contexts it is more usual to find the symbol i replaced by the symbol j, thus avoiding possible confusion when complex numbers are used to describe currents (i) in electrical circuits, as in our Section 8.1. We will use i, recalling its origins in the initial letter of the historical term "imaginary." Complex numbers may also be thought of as pairs of numbers, in which the order in the pair is significant. Thus we might write (2.3) analogously to vectors in a plane. Just as the coordinates (x, y) and (y,x) are usually distinct, so are the analogous complex numbers. The notation in (2.3) avoids
2.1 ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS
19
ambiguities in using the + sign and in the necessity of inventing a symbol satisfying (2.2). Further, many of the rules for manipulating complex numbers have a strong similarity to those for vectors in a plane if the notation (2.3) is used. Although we will write our results for complex-number algebra and arithmetic in the notation (2.1), you are invited to try the number-pair notation in Exercise 2.1 (c). This notation is also used in some programming languages that allow complexarithmetic operations. The rules for manipulating complex numbers must be consistent with those for purely real numbers ( y = 0) and for purely imaginary numbers (x = 0). In the following, let z = x + iy generically, and let z1 = x1+ iy1, z2 = x2 + i y2 represent two particular complex numbers. Then the following properties hold: (2.4) Negation of a complex number is defined by (2.5) which is often written casually as (2.6) A complex number is zero only if both its real and imaginary parts are zero, which is consistent with zero being the only solution of the equation z = -z. Addition or subtraction of two complex numbers is accomplished by (2.7) Multiplication of complex numbers is performed by (2.8) Reciprocal of a complex number is defined by (2.9) which has the property that z (1/z) = 1, as for the arithmetic of real numbers. Division of one complex number into another is based on the reciprocal of the divisor, and is therefore undefined if the divisor is zero: (2.10)
20
COMPLEX VARIABLES
In order to check your comprehension of these rules for complex arithmetic, try the following exercise. Exercise 2.1 (a) Verify that the rules (2.7) through (2.10) are consistent with those for real arithmetic by checking them for y1 = y2 = 0. (b) Check the consistency of (2.7) through (2.10) for purely imaginary numbers by setting x1= x2 = 0 and noting the condition on i, (2.2). (c) Use the notation for complex numbers as ordered-number pairs, as indicated by (2.3), to write down the preceding complex-arithmetic rules, (2.4) through (2.10). n Now that we have summarized the formal basis of complex-variable algebra, it is time to consider complex arithmetic, especially for computer applications.
Programming with complex numbers Few computer languages are designed to include complex-variable types in their standard definition. They are available in Fortran, but not in C or Pascal. In Wolfram's Mathematica system for doing mathematics by computer, which has both symbolic and numeric capabilities, complex numbers can be handled readily. An introduction to their use is provided in Section 1.1 of Wolfram's book. To appreciate why computer hardware is not built and computer software is not designed to assume that numbers they handle are complex, consider the following exercise. Exercise 2.2 Show that the total number of real-arithmetic operations needed for complexnumber addition and subtraction is 2, the number for multiplication is 6, and the number for division is 11 or 14, depending on whether or not the divisor in (2.9) is stored. n We now show a simple program in C language for performing complex arithmetic by the rules given in the preceding subsection. The purpose of this program is twofold: if you are unfamiliar with the C language the program will provide a simple introduction, while it will also develop your understanding of complex arithmetic. The program Complex-Arithmetic Functions takes as input x1, y1, x2, y2 for the components of two complex numbers z1 and z2. After checking that both numbers are nonzero, it calls the functions for addition, subtraction, multiplication, and division, namely CAdd, CSub, CMult, and CDiv, then prints the results before returning for more input. Here is the program.
The program reveals an immediate difficulty with modifying a language to include complex variables, in that two values must be returned by a complex-variable
2.1 ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS
23
function. In C this cannot be done simply by a conventional function (which returns just one value, at most). One can get around the problem by using the indirection (dereferencing) operator, written as an asterisk (*) preceding a variable name, as used for each of the two real-variable values returned by the program functions. Here are some suggestions for exploring complex numbers by using the program Complex-Arithmetic Functions. Exercise 2.3 (a) Use several pairs of real numbers for both inputs ( y1 = 0, y2 = 0 ) in order to verify that the complex numbers contain real numbers as special cases. (b) Input purely imaginary numbers ( xl = 0, x2 = 0 ) to the program and verify the correctness of the arithmetic. (c) Show by a careful analytical proof that if the product of two complex numbers is zero, then at least one of the complex numbers is identically zero (both real and imaginary parts zero). Prove that if one of a pair of complex numbers is zero, their product is zero. Verify this by using the program. n With this background of algebraic and arithmetic properties of complex numbers, we are prepared to review some more formal definitions and properties. Complex conjugation, modulus, argument In complex-variable algebra and arithmetic one often needs complex quantities that are related by reversal of the sign of just their imaginary parts. We therefore have the operation called complex conjugation. In mathematics texts the notation for this operation is often denoted by a bar, - , while other scientists often use an asterisk, as * . In the latter notation the complex-conjugate value of z is z*=x-iy
(2.11)
z=x+iy
(2.12)
if and only if
From the definition of complex conjugation we can readily derive several interesting results. Exercise 2.4 (a) Prove that (2.13) where the notation Re stands for "real part of."
24
COMPLEX VARIABLES
(b) Similarly, prove that
(2.14) where Im denotes "imaginary part of." (c) Derive the following properties of complex conjugation: (2.15)
(2.17) which show that complex conjugation is distributive over addition, subtraction, multiplication, and division. n The identity zz*=x2 y2
(2.18)
shows that z z * is zero only if z is identically zero, which is an example of the condition from Exercise 2.3 (c) for vanishing of the product of two complex numbers. The frequent occurrence of z z * and its connection with vectors in two dimensions lead to the notation of the modulus of a complex number z, denoted by (2.19) Thus mod z indicates the magnitude of z if we picture it as a vector in the x - y plane. Another name for modulus is absolute value. For example, the modulus of a real number is just its value without regard to sign, that is, its absolute value. The modulus of a pure imaginary number is just the value of y without regard to sign. The modulus of a complex number is zero if and only if both its real and its imaginary part are zero. The argument of a complex number is introduced similarly to polar coordinates for two-dimensional vectors. One defines the arg function for a complex variable by the requirement that (2.20)
and the requirement that
25
2.1 ALGEBRA AND COMPUTING WITH COMPLEX NUMBERS
(2.21)
which are necessary and sufficient conditions for definition of arg z to within a multiple of 27 Exercise 2.5 Explain why the commonly given formula (2.22) is not sufficient to specify arg z uniquely, even to within a multiple of 27
n
In Section 2.6, in the program to convert between coordinates, we return to this problem of ambiguous angles. The argument of a complex number is sometimes called the phase angle, or (confusingly) the amplitude. One aspect of the argument relates to the analogy between complex variables and planar vectors. If the pair (x,y) formed the components of a vector, then arg z would be the angle that the vector makes with the positive x axis. For example, arg (Re z) = ± arg i = /2, and arg (-i) = - / 2 . A program for complex conjugate and modulus For applications with complex variables it is worthwhile to have available programs for complex functions. We provide here a program that invokes functions returning complex conjugate and modulus values. The more-involved coding for the argument function is provided in the programming project, Section 2.6. Here is the program Conjugate & Modulus Functions.
Some remarks on programming the functions CConjugate and CModulus in the C language are in order: 1. Note that complex conjugation performs an operation on a complex number, albeit a simple one. So it does not return a value in the sense of a C function value. Therefore, the "function" CConjugate is declared to be "void." The value of the
2.2 THE COMPLEX PLANE AND PLANE GEOMETRY
27
complex conjugate is returned in the argument list of CConjugate as xc and yc, which are dereferenced variables (preceded by a *, which should not be confused with complex conjugation, or even with / * and * / used as comment terminators). 2. On the other hand, CModulus is declared as "double" because it is a function which returns a value, namely mod (inside CModulus), which is assigned to zmod within the main program. Note that CModulus might also be used within an arithmetic statement on the right-hand side of the = sign for zmod. 3. The program continues to process complex-number pairs while the input number pair is nonzero. If the zero-valued complex number (x = 0 and y = 0 ) is entered, the program exits gracefully by exit (0) rather than with the signal of an ungraceful termination, exit (1). With this background to programming complex conjugation and modulus, plus the program for arguments in Section 2.6, you are ready to compute with complex variables. Exercise 2.6 Run several complex-number pairs through the program Conjugate & Modulus Functions. For example, check that the complex conjugate of a complex-conjugate number produces the original number. Also verify that the modulus values of (x, y) and (y, x) are the same. ?
2.2
THE COMPLEX PLANE AND PLANE GEOMETRY
In the preceding section on algebra and computing with complex numbers we had several hints that there is a strong connection between complex-number pairs (x, y) and the coordinates of points in a plane. This connection, formally called a "mapping," is reinforced when we consider successive multiplications of z = x + i y by i itself. Exercise 2.7 (a) Show that if z is represented by (x, y), then i z is (-y, x), then i2z is (-x,-y), i3z is (y,-x), and i4z regains (x , y). (6) Verify that in plane-polar geometry these coordinates are displaced from each other by successive rotations through / 2, as shown in Figure 2.1. n The geometric representation of complex numbers shown in Figure 2.1 is variously known as the complex plane, the Argand diagram, or the Gauss plane. The relations between rotations and multiplication with complex numbers are summarized in the following subsections.
28
COMPLEX VARIABLES
FIGURE 2.1 Rotations of complex numbers by /2 in the complex plane. Note that rotations do not change the length of a complex number, as indicated by the dashed circle.
Cartesian and plane-polar coordinates Before reviewing the connections between the complex plane and plane geometry, let us recall some elementary relations between Cartesian and plane-polar coordinates. If the polar coordinates are r, the (positive) distance from the origin along a line to a point in the plane and the angle (positive in the anticlockwise direction) that this line makes with the x axis, then the Cartesian coordinates are given as
in a right-handed coordinate system. The polar coordinates are indicated in Figure 2.1. Inverting these equations to determine r and which is not as trivial as it may look, is discussed in the first part of Section 2.6. In the complex plane we may therefore write z as (2.24) so that the modulus, which is also the radius, is given by
2.2 THE COMPLEX PLANE AND PLANE GEOMETRY
29
(2.25) and the polar angle with respect to the x axis is given by (2.26) The principal value of the polar angle is the smallest angle lying between and Such a specification of the principal value allows unique location of a point in the complex plane. Other choices that limit may also be encountered, for example, the range 0 to 27. Complex conjugation is readily accomplished by reflecting from to since (2.27) In the language of physics, z, and its complex conjugate are related through a parity symmetry in a two-dimensional space. With this angular representation of complex variables, we can derive several interesting results.
De Moivre's theorem and its uses A theorem on multiplication of complex numbers in terms of their polar-coordinate representations in the complex plane was enunciated by Abraham De Moivre (16671754). We derive his theorem as follows. Suppose that we have two complex numbers. the first as (2.28) and the second as (2.29) Their product can be obtained by using the trigonometric identities for expanding cosines and sines of sums of angles, to obtain (2.30) From this result we see that multiplication of complex numbers involves conventional multiplication of their moduli, the r1r2 part of (2.30), and addition of their angles.
30
COMPLEX VARIABLES
Therefore, multiplication in the complex plane, as well as addition, can readily be shown, as in Figure 2.2.
FIGURE 2.2 Combination of complex numbers in the complex plane. The complex numbers and their sum are indicated by the dashed lines, while their product is shown by the solid line.
Equation (2.30) can be generalized to the product of n complex numbers, as the following exercise suggests. Exercise 2.8 (a) Prove by the method of mathematical induction that for the product of n complex numbers, z1,z2, ..., zn, one has in polar-coordinate form (2.31) (b) From this result, setting all the complex numbers equal to each other, prove that for the (positive-integer) nth power of a complex number (2.32) which is called De Moivre's theorem. n This remarkable theorem can also be proved directly by using induction on n. Reciprocation of a complex number is readily performed in polar-coordinate form, and therefore so is division, as you may wish to show.
2.3 FUNCTIONS OF COMPLEX VARIABLES
31
Exercise 2.9 (a) Show that for a nonzero complex number, z, its reciprocal in polar-coordinate form is given by (2.33) (b) From the result in (a) show that the quotient of two complex numbers can be written in polar-coordinate form as
(2.34) where it is assumed that r2 is not zero, that is, z2 is not zero. n Thus the polar-coordinate expressions for multiplication and division are much simpler than the Cartesian-coordinate forms, (2.8) and (2.10). Although we emphasized in Exercise 2.2 that complex-number multiplication and division in Cartesian form are much slower than such operations with real numbers, these operations may be somewhat speedier in polar form, especially if several numbers are to be multiplied. An overhead is imposed by the need to calculate cosines and sines. Note that such advantages and disadvantages also occur when using logarithms to multiply real numbers.
2.3 FUNCTIONS OF COMPLEX VARIABLES In the preceding two sections we reviewed complex numbers from algebraic, computational, and geometric viewpoints. The goal of this section is to summarize how complex variables appear in the most common functions, particularly the exponential, the cosine and sine, and the hyperbolic functions. We also introduce the idea of trajectories of functions in the complex plane. Complex exponentials: Euler's theorem In discussing De Moivre's theorem at the end of the preceding section we noticed that multiplication of complex numbers may be done by adding their angles, a procedure analogous to multiplying exponentials by adding their exponents, just the procedure used in multiplication using logarithms. Therefore, there is probably a connection between complex numbers in polar-coordinate form and exponentials. This is the subject of Euler's theorem. A nice way to derive Euler's theorem is to write (2.35)
32
COMPLEX VARIABLES
then to note the derivative relation
(2.36) But the general solution of an equation of the form
(2.37) is given by (2.38) Exercise 2.10 Show, by identifying (2.36) and (2.37) with the result in (2.38), that = i. Then choose a special angle, say = 0, to show that = 1. Thus, you have proved Euler's theorem, (2.39) which is a remarkable theorem showing a profound connection between the geometry and algebra of complex variables. n It is now clear from Euler's theorem why multiplication of complex numbers involves addition of angles, because the angles are added when they appear in the exponents. Now that we have the formal derivation of Euler's theorem out of the way, it is time to apply it to interesting functions. Applications of Euler's theorem There are several interesting and practical results that follow from Euler's theorem and the algebra of complex numbers that we reviewed in Section 2.1. The trigonometric and complex-exponential functions can be related by noting that, for real angles, (2.40) which follows by taking complex conjugates on both sides of (2.39). On combining (2.39) and (2.40) for the cosine we have
(2.41) while solving for the sine function gives
2.3 FUNCTIONS OF COMPLEX VARIABLES
33
(2.42) Both formulas are of considerable usefulness for simplifying expressions involving complex exponentials. Exercise 2.11 Use the above complex-exponential forms of the cosine and sine functions to prove the familiar trigonometric identity (2.43) the familiar theorem of Pythagoras. n Although our derivation of Euler's theorem does not justify it, since is assumed to be real in the differentiation (2.36) the theorem holds even for being a complex variable. Thus the Pythagoras theorem also holds for complex . Some remarkable results, which are also often useful in later chapters, are found if multiples of /2 are inserted in Euler's identity, (2.39). Derive them yourself. Exercise 2.12 Use Euler's theorem to show that (2.44) which gives the values for successive rotations in the complex plane by r/2. Compare these results with Figure 2.1. n The exponential form is generally much more symmetric and therefore is easier to handle analytically than are the cosine and sine functions, with their awkward function changes and sign changes upon differentiation compared with the simplicity of differentiating the complex exponential. This simplicity is very powerful when used in discussing the solution of differential equations in Chapter 8, and also in deriving the fast Fourier transform (FFT) algorithm in Chapter 9.3. An interesting application of the complex-exponential function that anticipates its use in the FFT algorithm is made in the following exercise. Exercise 2.13 Consider the distributive and recurrence properties of the complex-exponential function defined by (2.45) (a) Prove the following properties of powers of E(N):
34
COMPLEX VARIABLES
(2.46) (2.47) for any a, b, and for any p 0. (b) Using these results, show that if N = 2V, where v is a positive integer, then, no matter how many integer powers of E(N) are required, only one evaluation of this complex-exponential function is required. n As a further topic in our review of functions of complex variables, let us consider the hyperbolic and circular functions. Hyperbolic functions and their circular analogs Exponential functions with complex arguments are required when studying the solutions of differential equations in Chapters 7 and 8. A frequently occurring combination is made from exponentially damped and exponentially increasing functions. This leads to the definition of hyperbolic functions, as follows. The hyperbolic cosine, called "cosh," is defined by (2.48) while the hyperbolic sine, pronounced "sinsh," is defined by (2.49) If u is real, then the hyperbolic functions are real-valued. The name "hyperbolic" comes from noting the identity (2.50) in which, if u describes the x and y coordinates parametrically by (2.5 1) then an x - y plot is a rectangular hyperbola with lines at /4 to the x and y axes as asymptotes. Exercise 2.14 (a) Noting the theorem of Pythagoras, (2.52)
2.3 FUNCTIONS OF COMPLEX VARIABLES
35
for any (complex) u, as proved in Exercise 2.11, explain why the cosine and sine functions are called "circular" functions. (b) Derive the following relations between hyperbolic and circular functions (2.53) a n d (2.54) valid for any complex-valued u. n These two equations may be used to provide a general rule relating signs in identities for hyperbolic functions to identities for circular functions: An algebraic identity for hyperbolic functions is the same as that for circular functions, except that in the former the product (or implied product) of two sinh functions has the opposite sign to that for two sin functions. For example, given the identity for the circular functions (2.55) we immediately have the identity for the hyperbolic functions (2.56) Exercise 2.15 Provide a brief general proof of the hyperbolic-circular rule stated above. n Note that derivatives, and therefore integrals, of hyperbolic and circular functions do not satisfy the above general rule. The derivatives of the hyperbolic functions are given by (2.57) and by (2.58) in both of which the real argument, u, is in radians. There is no sign change on differentiating the hyperbolic cosine, unlike the analogous result for the circular cosine.
36
COMPLEX VARIABLES
The differential equations satisfied by the circular and hyperbolic functions also differ by signs, since the cosine and sine are solutions of (2.59) which has oscillatory solutions, whereas the cosh and sinh are solutions of
(2.60) which has solutions exponentially increasing or exponentially decreasing. Exercise 2.16 (a) Prove the two derivative relations (2.57) and (2.58) by starting with the defining equations for the cosh and sinh. (b) Use the relations between hyperbolic and circular functions, (2.53) and (2.54), to compute the derivatives of the hyperbolic functions in terms of those for the circular functions. (c) Verify the appropriateness of the circular and hyperbolic functions as solutions of the differential equations (2.59) and (2.60), respectively. n To complete the analogy with the circular functions, one also defines the hyperbolic tangent, called "tansh," by (2.6 1) which is analogous to the circular function, the tangent, defined by
(2.62) Among these six hyperbolic and circular functions, for real arguments there are three that are bounded by ±1 (sin, cos, tanh) and three that are unbounded (sinh, cosh, tan). Therefore we show them in a pair of figures, Figures 2.3 and 2.4, with appropriate scales. By displaying the bounded hyperbolic tangent on the same scale as the sine function in Figure 2.3, we notice an interesting fact- these two functions are equal to within 10% for | x | < 2, so may often be used nearly interchangeably. The explanation for their agreement is given in Section 3.2, where their Maclaurin series are presented. Figure 2.4 shows a similar near-coincidence of the cosh and sinh functions for x > 1.5, where they agree to better than 10% and the agreement improves as x increases because they both tend to the exponential function.
FIGURE 2.4 The unbounded circular and hyperbolic functions, tangent, hyperbolic cosine, and hyperbolic sine. For x greater than about 1.5, the latter two functions are indistinguishable on the scale of this figure. The tangent function is undefined in the limit that the cosine function in the denominator of its definition (2.62) tends to zero. For example, in Figure 2.4 values of the argument of the tangent function within about 0.1 of x = ± /2 have been omitted.
38
COMPLEX VARIABLES
Trajectories in the complex plane Another interesting concept and visualization method for complex quantities is that of the trajectory in the complex plane. It is best introduced by analogy with particle trajectories in two space dimensions, as we now summarize. When studying motion in a real plane one often displays the path of the motion, called the trajectory, by plotting the coordinates x (t) and y (t), with time t being the parameter labeling points on the trajectory. For example, suppose that x (t) = A cos ( t) and y (t) = B sin ( ), with A and B positive, then the trajectory is an ellipse with axes A and B, and it is symmetric about the origin of the X- y coordinates. As t increases from zero, x initially decreases and y initially increases. One may indicate this by labeling the trajectory to indicate the direction of increasing t. The intricate Lissajous figures in mechanics, obtained by superposition of harmonic motions, provide a more-involved example of trajectories. Analogously to kinematic trajectories, in the complex plane real and imaginary parts of a complex-valued function of a parameter may be displayed. For example, in Section 8.1 we discuss the motion of damped harmonic oscillators in terms of a real dimensionless damping parameter Expressed in polar-coordinate form, the amplitude of oscillation is (2.63) where the complex "frequency" (if x represents time) is given by (2.64) The trajectory of v depends on the range of square root in (2.64). Exercise 2.17 (a) Show that if
and on the sign associated with the
1, which gives rise to damped oscillatory motion, then (2.65)
and that the trajectory of v+ is a counterclockwise semicircle in the upper half plane, while the trajectory of v- is a clockwise semicircle in the lower half plane. In both trajectories is given as increasing from -1 to +l. (b) Suppose that >1, which produces exponential decay called overdamped motion. Show that v± is then purely real and negative, so the trajectory lies along the real axis. Show that v- increases from -1 toward the origin as increases, while v+ decreases toward as increases. n The complex-plane trajectory, with as displayed in Figure 2.5.
as parameter, expressed by (2.64) is therefore
2.3 FUNCTIONS OF COMPLEX VARIABLES
39
FIGURE 2.5 Frequency trajectory in the complex plane according to (2.64) as a function of the damping parameter
As a final note on this example, there is no acceptable solution for v- if < - 1 and if x > 0 is considered in (2.63), since y (x) is then divergent. Another interesting example of a trajectory in the complex plane arises in the problem of forced oscillations (Section 10.2) in the approximation that the energy dependence is given by the Lorentzian (2.66) where c is a proportionality constant and the complex Lorentzian amplitudes L± are given by (2.67) where and are dimensionless frequency and damping parameters. The analysis of this example is similar to the first one.
40
COMPLEX VARIABLES
FIGURE 2.6 Frequency trajectory in the complex plane for the Lorentzian amplitude described by (2.67).
Exercise 2.18 (a) Show that the Lorentzian amplitudes in (2.67) satisfy (2.68) so that the trajectories of L± lie on circles of radius l/2 in the complex plane. (b) Investigate the details of the trajectory by showing that L+ describes the anticlockwise semicircle in the lower half plane, while L- describes the clockwise semicircle in the upper half complex plane. For both of these trajectories the directions are for going from to n The trajectories of the Lorentzian amplitudes are shown in Figure 2.6. They are discussed more completely in Chapter 3 of the text by Pippard in the context of Cole-Cole plots of complex-valued dielectric constants as a function of frequency. In both of our examples the trajectories in the complex plane lie on circles. This is neither a mere coincidence nor is it uncommon, as the following exercise should convince you.
2.4 PHASE ANGLES, VIBRATIONS, AND WAVES
41
Exercise 2.19 Consider the following complex function z, called a linear fractional transformation of the real variable p according to
(2.69) , and are complex constants, with in which 0. Now consider the z' that is obtained from z by the shift and scaling transformation function
(2.70) By analogy with the result in Exercise 2.18, argue that z' lies on a circular trajectory and therefore that the original z in (2.69) lies on a circular trajectory. n Thus, the functions (2.67) and (2.69) are both special cases of the more general circular trajectory given by (2.70). From these examples we see that the notion of a trajectory in the complex plane is useful for visualizing the properties of complexvalued functions.
2.4 PHASE ANGLES, VIBRATIONS, AND WAVES The angle in the complex plane often has interesting interpretations in scientific applications, particularly in the context of vibrations and waves. In this section we summarize some of the main results. An encyclopedic treatment is provided in Pippard's book on the physics of vibration. The topics introduced here are developed and applied throughout this book. In particular, Section 8.1 discusses free-motion and resonant vibrations in mechanical and electrical systems, then the quantum oscillator is considered briefly in Section 8.5. In Chapters 9 and 10 we develop Fourier expansions, emphasizing the complex-exponential treatment for the discrete, series, and integral expansions. Phase angles and phasors Suppose that we have a (real) angle = , where is a constant angular frequency, = 2 f (with f the frequency) and t denotes time. Then (2.71)
42
COMPLEX VARIABLES
describes in the complex plane uniform circular motion of the point z1, while the projections onto the real and imaginary axes (x and y) describe simple harmonic motions. Exercise 2.20 Prove that the motion of z1 is periodic by showing that (2.72) where the period T = 2
n
If a second uniform circular motion in the complex plane is described by (2.73) then this motion has the same period as that described by z1, but at a given time z2 has its phase advanced by over the phase of zl. Whether one refers to a positive value of as a lag or a lead depends on the scientific field in which one is working. If > 0, in mechanics the phase of z2 is said to lug that of zl, whereas in electrical-circuit applications z2 is said to lead zl. A complex-plane diagram showing the magnitude of z and a relative phase (with t usually suppressed) is called a vibration diagram or phasor diagram. Its use gives a visualization of complex-variable relationships which often improves comprehension and interpretation. Vibrations and waves We can broaden the discussion of phases to include both spatial as well as temporal variation in the amplitude of a complex vibration. For example, a wave that has constant amplitude of unity at all points along the x direction and at all times t can be described by (2.74) in which the wavenumber, k, is given in terms of wavelength, ?, by
(2.75) Although the wavenumber is physically a less intuitive quantity than is the wavelength, computationally and in most applications k is a much simpler quantity to deal with. Note that k has the dimensions of an inverse length, just as the angular frequency, has the dimensions of inverse time. Thus, the argument of the exponential function in (2.74) is dimension-free, as should be the argument of any function in mathematics.
2.5 DIVERSION: INTERPRETING COMPLEX NUMBERS
43
Recall also that the wave described by (2.74) is monochromatic (unique values of and k) and that points of constant phase have an associated phase velocity, vp, given by (2.76) Exercise 2.21 Discuss from the viewpoint of wave motion why vp in (2.76) is called the phase velocity. n The superposition of such waves of constant phase to build up a dispersive wave in which components with different frequencies transport energy at different speeds is an extension of the Fourier expansions in Chapters 9 and 10. A comprehensive and lucid discussion is given by Baldock and Bridgeman in their book on wave motion. In Chapters 9 and 10 on Fourier expansions we make detailed study of phenomena described in terms of x or in terms of the complementary variable k, or in terms oft and its complementary variable Clear expositions of the relations between complex exponentials and vibrations are given in detail with many applications in Pippard's omnibus book. Vibrations and waves are described very completely at an introductory level in the book by Ingard.
2.5 DIVERSION: INTERPRETING COMPLEX NUMBERS The development of the interpretation of complex numbers provides an example of the consequences of education and of the dominance of scientific thought by mathematical representations. Since many scientists claim that a phenomenon is not understood until it can be described mathematically, it is interesting to discuss the relation between "the queen and servant of science" and the natural sciences. Are complex numbers real? Before the quantum physics revolution of the 192Os, scientists usually apologized for using complex numbers, since they provided only mathematically convenient shortcuts and shorthand for problem solving. Indeed, Leonard Euler of Euler's theorem in Section 2.3 coined the Latin "imaginarius" for the quantity i = . The first major use of complex numbers was made by C. P. Steinmetz (1865-1923), a research electrical engineer who used them extensively (as we do in Section 8.1) to simplify the analysis of alternating-current circuits. In quantum mechanics, for example in the Schr?dinger equation that we use in Section 8.5, the wave function is fundamentally a complex variable that is not a shorthand for two real quantities such as magnitude and phase. Many quantities derived from wave functions, such as scattering amplitudes, are also intrinsically complex-valued. This leads to the scientific use of "analytic continuation," a concept and technique familiar in mathematics but of more recent use in the natural sciences.
44
COMPLEX VARIABLES
Analytic continuation We have displayed in Figures 2.3 and 2.4 the circular functions and the hyperbolic functions, respectively. In terms of complex variables, however, these hyperbolic functions are essentially just the circular functions evaluated for purely imaginary arguments, or vice versa. It is therefore interesting, and sometimes useful, to think of there being just a single set of functions, say the circular functions, which may be evaluated along the real axis (then they are the conventional trigonometric functions) or they may be evaluated along the imaginary axis (then they are the hyperbolic functions, within factors of i), or they may be evaluated for the argument which takes on a value anywhere in the complex plane. When we make this last bold step off either the real or the imaginary axis and into the complex plane we are making an analytic continuation of the functions. The concept of analytic continuation and some understanding of the techniques applied to it are best appreciated by working the following exercise. Exercise 2.22 Consider the behavior of the complex-valued function of complex-variable argument, Z, defined as follows:
(2.77) (a) Show that for a real argument A is just the hyperbolic cosine function discussed in Section 2.3, while for purely imaginary z it is the circular cosine function. (b) Sketch the graph of cos x along the real axis of the complex-z plane and the graph of cosh y along the imaginary axis of the same plane. They look quite different, don't they? (c) Devise a graphical representation of A (z) that is suitable for arbitrary complex z, and make some representative sketches of the function thus graphed. One possible form of representation is to sketch contours of constant Re A and of constant ImA. n In scientific research analytic continuation is often a useful technique. As an example, experiments on wave scattering (such as in acoustics, optics, electromagnetism, and subatomic physics) are, at best, obtained in the range of scattering angles from zero to . How would the data look if they could be analytically continued into the complex-angle plane? Similarly, data obtained at real energies or frequencies may be interesting to extrapolate to complex energies or complex frequencies. Indeed, we explore this possibility in discussing the Lorentzian resonances in Section 10.2.
2.6 PROJECT 2: PROGRAM TO CONVERT BETWEEN COORDINATES
2.6
45
PROJECT 2: PROGRAM TO CONVERT BETWEEN COORDINATES
The program Cartesian & Polar Coordinate Interconversion developed in this project serves both to develop your understanding of the relations between these two coordinate systems and to give you practice with writing programs in C. The conversion from plane-polar to Cartesian coordinates is straightforward and unambiguous. Given r and one has immediately (as discussed in Section 2.2) (2.78) which can be programmed directly.
Stepping into the correct quadrant The transformation from Cartesian coordinates to polar coordinates is less direct than the inverse transformation just considered. The required formulas are
(2.79) which is straightforward to compute, and, in terms of the atan or tan-l function, (2.80) which is ambiguous. This formula does not uniquely determine the quadrant in which lies because only the sign of the quotient in (2.80) is available after the division has been made. The relative signs of the circular functions in the four quadrants indicated in Figure 2.7 may be used to determine the angle uniquely from the signs of x and y. In some programming languages, including C, two functions are available for the inverse tangent. In one, such as the atan (t) in C language (with t the argument of the function), the angle is usually returned in the range /2 to /2, and the user of this function has to determine the appropriate quadrant by other means. In the second function for the inverse tangent, such as atan2 (y, x) in the C language, the angle is located in the correct quadrant by the function itself. If we were to use at atan2 in the program, the conversion from Cartesian to polar representation would be very direct. For practice in C and to reinforce your understanding of plane-polar coordinates we use the simpler function atan.
46
COMPLEX VARIABLES
FIGURE 2.7 Signs of the circular functions in each quadrant of the Cartesian plane.
If we begin with an angle calculated into the first quadrant by using the absolute value of y/x, then multiples of /2 have to be added or subtracted to get the angle into the correct quadrant. Computationally, the best way to get at the accuracy of your | 677.169 | 1 |
Learn the fundamentals of number theory from former MATHCOUNTS, AHSME, and AIME perfect scorer Mathew Crawford. Topics covered in the book include primes & composites, multiples & divisors, prime factorization and its uses, simple Diophantine equations, base numbers, modular arithmetic, divisibility rules, linear congruences, how to develop number sense, and much more. The text is structured to inspire the reader to explore and develop new ideas. In addition to the instructional material, the book contains hundreds of problems. The solutions manual contains full solutions to nearly every problem, not just the answers. This book is ideal for students who have mastered basic algebra, such as solving linear equations. Middle school students preparing for MATHCOUNTS, high school students preparing for the AMC, and other students seeking to master the fundamentals of number theory will find this book an instrumental part of their mathematics libraries. About the author: Mathew Crawford is the founder and CEO of MIST Academy, a school for gifted students, in Birmingham, Alabama. Crawford was a perfect scorer at the national MATHCOUNTS competition in 1990, and a member of the national championship team (Alabama) in 1991. He was a 3-time invitee to the Math Olympiad Summer Program, a perfect scorer on the AIME, and a 2-time USA Math Olympiad honorable mention. ISBN: 978-0-9773045-4-7 Text: 336 pages. Solutions: 144 pages. Paperback. 10 7/8 x 8 3/8 x 5/8 inches. Author : Mathew Crawford ISBN : 1934124133 Language : English Edition : 2nd Publication Date : 2008 Format/Binding : Paperback Book dimensions : 10.9x8.3x0.9 Book weight : 0.02 | 677.169 | 1 |
GCSE Mathematics – Aiming for an A or Better
Grade Criteria and exemplar examination questions to get a Grade A or A* in the following topics:Surds,Recurring Decimals,Limits of Accuracy, Indices,Proportionality,Rearranging Formulae,Algebraic Fractions,Using Graphs
Quadratic Equations,Simultaneous Equations
Algebraic Proofs,Circle Theorems,Trigonometry– for triangles which are not right-angled,Vectors,Similar Triangles
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GCSE Mathematics: Surds Grade Criteria Can Do To get a grade B you must be able to: Work out the square roots of some decimal numbers To get a grade A you must be able to: Simplify surds by rewriting with the smallest integer inside the root To get a grade A* you must be able to: Simplify surds by rationalising denominators Manipulate & simplify expressions involving surds Question 1 [i] [ii]…read more
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GCSE Mathematics: Limits of Accuracy Grade Criteria Can Do To get a grade B you must be able to: Find limits of accuracy for numbers given to whole number accuracy To get a grade A you must be able to: Find limits of accuracy for numbers given to d.p. or s.f.…read more
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GCSE Mathematics: Indices Grade Criteria Can Do To get a grade C you must be able to: Multiply and divide numbers in index form To get a grade A you must be able to: Know how to use indices rules for negative & fractional powers Question 4…read more
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GCSE Mathematics: Proportionality Grade Criteria Can Do To get a grade A you must be able to: Find formulae that describe direct & inverse proportionality (or variation) and use them to solve problems To get a grade A* you must be able to: Solve variation problems involving 3 variables Question 5 [i] [ii]…read more
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GCSE Mathematics: Rearranging Formulae Grade Criteria Can Do To get a grade B you must be able to: Rearrange more complicated formulae To get a grade A you must be able to: Rearrange formulae where the subject appears twice To get a grade A* you must be able to: Rearrange formulae where the subject appears with a power Question 6…read more | 677.169 | 1 |
Rings and Homology
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This concise text is geared toward students of mathematics who have completed a basic college course in algebra. Combining material on ring structure and homological algebra, the treatment offers advanced undergraduate and graduate students practice in the techniques of both areas. After a brief review of basic concepts, the text proceeds to an examination of ring structure, with particular attention to the structure of semisimple rings with minimum condition. Subsequent chapters develop certain elementary homological theories, introducing the functor Ext and exploring the various projective dimensions, global dimension, and duality theory. Each chapter concludes with a set of exercises.Copyright Copyright Ac 1964 by Holt, Rinehart and Winston, Inc. All rights
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