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I started study of Mathematics(9 Jan 2008). The conceptual portion is very limited in TMH book. But the first problem in the examples itself is complicated. Every problem thereon is a complicated problem.
In Mathematics and Physics, the conceptual portion is going to be limited but the problems are going to be complicated. One has to sit down and do all the problems in examples and exercises to develop the sharp brain that can discover the structure in the problem given in the examination.
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. ------------------
Determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. ---------
Let (x,y) be such that Sin‾¹(ax) + Cos‾¹(y)+Cos‾¹(bxy) = π/2 Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
Equation of a circle in various forms, equations of tangent, normal and chord. Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
------------ JEE Question
Tangents are drawn from the point (17, 7) to the circle x^2 + y^2 = 169.
Statement - 1 The tangents are mutually perpendicular.
Because
Statement - 2 The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x^2 + y^2 = 338.
Equation of a straight line in various forms, angle between two lines, distance of a point from a line. Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines, centroid, orthocentre, incentre and circumcentre of a triangle. -------------------
Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
Derivatives of implicit functions, derivatives up to order two. -----------------------
definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus. Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves. ------------------- JEE question If f″(x) = − f(x) and g(x) = f′(x) and F(x) = [f(x/2)]^2 + [g(x/2)]^2 and given that F(5) = 5 then F(10) is equal to | 677.169 | 1 |
ISBN-10: 0321112504
ISBN-13: 9780321112507 sixth edition of this best-selling text balances solid mathematical coverage with a comprehensive overview of mathematical ideas as they relate to varied disciplines. This book provides an appreciation of mathematics, highlighting mathematical history, applications of mathematics to the arts and sciences across cultures, and introduces students to the uses of technology in mathematics. Exercise sets are now organized into Concept/Writing, Practice the Skills, Problem Solving, Challenge Problems/Group Activities, Research Activities. An updated Consumer Math section including updated material on sources of credit and mutual funds. Motivational, chapter-opening material demonstrates connections between math and various other disciplines. KEY MARKET For those who require a general overview of mathematics, especially in the fields of elementary education, the social sciences, business, nursing and allied health fields | 677.169 | 1 |
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Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples. If you're seeing this message, it means we're having trouble loading external resources for Khan Academy. If you're behind a web filter, please make sure that the. ACT is a mission-driven nonprofit organization. Our insights unlock potential and create solutions for K-12 education, college, and career readiness. Welcome! InterAct Math is designed to help you succeed in your math course! The tutorial exercises accompany the end-of-section exercises in your Pearson textbooks. Saylor Academy provides free and open online courses and affordable college credit opportunities to learners everywhere. Start your course today. Official site. The governing and management authority of public higher education in Georgia.
Comprehensive encyclopedia of mathematics with 13,000 detailed entries. Continually updated, extensively illustrated, and with interactive examples. CPM Educational Program is a California nonprofit 501(c)(3) corporation dedicated to improving grades 6-12 mathematics instruction. CPM's mission is to empower Tutorvista provides Online Tutoring, Homework Help, Test Prep for K-12 and College students. Connect to a Tutor Now for Math help, Algebra help, English, Science. Tacoma Community College is a community college located in Tacoma, Washington. Founded in 2015, Middle Georgia State University has grown from 2 smaller colleges to form a 4-year, residential, bachelor's degree-awarding university in Central. Application. I am ready to start my college journey, now what? ApplyTexas is a one-stop shop for applying to a public university or community college in Texas. Last Modified: Monday December 19 2016 "The Montana Office of Public Instruction provides vision, advocacy, support and leadership for schools and communities.
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Students Student Resources. Find careers that relate to your interests and learn fun facts about the economy and jobs. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. Saylor Academy provides free and open online courses and affordable college credit opportunities to learners everywhere. Start your course today. Today nearly 30,000 young men and women find their path at San Jacinto College. They nurture their aspirations with faculty mentors who know real-world success and. The Math Forum is the comprehensive resource for math education on the Internet. Some features include a K-12 math expert help service, an extensive database of math.
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The Math Forum is the comprehensive resource for math education on the Internet. Some features include a K-12 math expert help service, an extensive database of math. News Brockport Shines During Winter Gala. Nearly 400 people filled the Hyatt Regency ballroom to celebrate The College at Brockport and support its students during. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. College Navigator is a free consumer information tool designed to help students, parents, high school counselors, and others get information about over 7,000. Nate Silver's FiveThirtyEight uses statistical analysis — hard numbers — to tell compelling stories about elections, politics, sports, science, economics and. If you're seeing this message, it means we're having trouble loading external resources for Khan Academy. If you're behind a web filter, please make sure that the.
The night before final exams begin, the College community comes together for the annual Midnight Breakfast and Primal Scream. Watch.
Resources and information to support K–12 and higher education professionals in helping students prepare for college and career.
Achieve challenges you to take a new quiz highlighting the college and career readiness agenda.
Features common formulas for arithmetic, algebra, geometry, calculus, and statistics. theorem, Also, has forum board to ask questions. Available in both English and.
Today nearly 30,000 young men and women find their path at San Jacinto College. They nurture their aspirations with faculty mentors who know real-world success and. Official site. The governing and management authority of public higher education in Georgia. Features common formulas for arithmetic, algebra, geometry, calculus, and statistics. theorem, Also, has forum board to ask questions. Available in both English and. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math. Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents. ACT is a mission-driven nonprofit organization. Our insights unlock potential and create solutions for K-12 education, college, and career readiness. | 677.169 | 1 |
What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with... more...
This new work by Wilfred Kaplan, the distinguished author of influential mathematics and engineering texts, is destined to become a classic. Timely, concise, and content-driven, it provides an intermediate-level treatment of maxima, minima, and optimization. Assuming only a background in calculus and some linear algebra, Professor Kaplan presents topics... more...
This contributed volume consists of papers in the area of nonconvex optimization from researchers practicing in India. It aims to bring together new concepts, theoretical developments, and applications from these researchers. more... | 677.169 | 1 |
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Students use a set of given log functions and operations to condense the log functions into a single function. Then students evaluate their functions for a given set of criteria. This assignment requires that students have prior experience with function evaluation, notation and operations. (It is a great opportunity to revisit these important concepts and skills in a new context.)
The second part of the assessment has students expand logs using various expansion rules. The expansions also contain some evaluation components to them.
This is a ONE page product that has TWO different versions of the assessment on the top and bottom of the page. You can save on copies AND distribute multiple versions to promote academic integrity or to provide a re-take opportunity. This formative assessment would make a great bell ringer or exit quiz. | 677.169 | 1 |
Advanced Mathematical Concepts lessons develop mathematics using numerous examples, real-world applications, and an engaging narrative. Graphs, diagrams, and illustrations are used throughout to help students visualize concepts. Directions clearly indicate which problems may require the use of a graphing calculator. A full-color design, a wide range of exercise sets, relevant special features, and an emphasis on graphing and technology invite your students to experience the excitement of understanding and applying higher-level mathematics skills. Graphing calculator instruction is provided in the Graphing Calculator Appendix. Each Graphing Calculator Exploration provides a unique problem-solving situation/Glencoe 2005-01-05, 2005566042 | 677.169 | 1 |
Mathematics can be a challenging subject at times. Because the knowledge and skills for success in math are cumulative, studying math needs to be ongoing and consistent. Otherwise it is too easy to fall behind in your understanding. The way you approach studying for your math class can have a great impact on your learning experience. Here are some tips:
Before Class
Skim the section in the book. Read the headings and look at the example problems. This will help you follow along much better during the lecture.
During Class
Go to Class. Going to class is the easiest way to get an introduction to the material. Reading someone else's notes won't be as clear.
Take Good Notes. Having good notes is very useful when working on the problem set. Make sure you write down definitions and take down the example problems.
Be Active in Class. Raise your hand and ask a question if you don't understand what the professor is saying. Most likely the material will build on itself, so better to ask sooner! If you would rather not ask aloud, write your question down and ask later in office hours or by email.
After Class
Review Your Notes. Look over your notes to refresh your memory of what you've learned. Identify areas that are confusing or unclear.
Start the Problem Set. Starting the problem set after class (when the material is fresh on your mind) will help you move through the problem set faster and more efficiently then if you wait a day or two where you will have a harder time remembering a relevant class example. Do the problems you can and skim the problems for the material you haven't covered yet.
Treat homework like a test. Try to look at the problems and take a stab at solving them before looking back at your notes or book. You'll remember the problems much better if you struggle a little bit to solve them.
Attend faculty office hours, help room, SI, and other forms of tutoring. Don't wait until the last minute to get help.
Preparing for a Test
Create a 7-Day Study Plan. On Day 1 and Day 2, start with the most recent material. Review yours notes, and do lots of practice problems. On day 3 go over old homework or quizzes and work on problem areas. On Day 4 and 5, do the same for the more recent material. Since math is cumulative, you will have somewhat reviewed this information already. On Day 6, work on your trouble spots. On Day 7, do a quick review of anything you are uncertain of and then relax! Don't try to cram or get in too much last minute studying, as this will only serve to make you nervous and unsure.
Explain it to someone else. You can really tell you understand something if you are able to clearly explain it to someone else. Meet up with a classmate and review concepts together.
The night before the exam. Get plenty of sleep the night before your exam. Go easy on caffeine, sugar and carbs and be sure to eat a well-balanced meal before the exam.
Overall, be active in your studying. Math is best learned when you struggle with it and then have the "A-Ha Moment!" | 677.169 | 1 |
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By accessing this website and posing questions to a qualified guided reading activity expert or tutor, students can replicate the classroom environment and resolve any gaps in their understanding. CPM is aligned with the new standards and it does have the mathematical practices infused within instruction but it is NOT a resource that was created for the common core. Great journalism has great value, lessons and now, use: www. I hope to blog about a number of these in upcoming posts on my Math Problems of the Week by the waters of babylon answers. Then stop worrying about getting a mental block. Homework assignments typically have geometry homework help cpm org or more purposes. In this topic, you will become function-chefs. Woman who took a selfie on a flight to Spain is photobombed 'by the SAME man' who was in the background of a. I am concerned that two math teachers I respected are no longer working in the district. 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Welcome!!
Strength of materials or strengths. It is believed that strengths is one of the most difficult of junior rates. I am always glad to help both students and professionals.
Site receives many requests for help with calculations every day, and I am glad to help You. A long time I refused orders, but the tasks were to bring pleasure at one moment :).
There are two programs for the calculation of beams online on this site.
Why SOPROMAT?
СОПРОМАТ (sopromat) is a russian slang word and it means STRENGTH OF MATERIALS. | 677.169 | 1 |
In this course, educators will examine how number and operations concepts from the previous nine sessions might look when applied to situations in the classroom, focusing on the approaches students might take to mathematical tasks involving number and operations concepts. This course is specific to Grades Six through Eight.
In each pair of one-hour workshops In the "Discovery" portion of this course, educators use real-life problems and experiments to gather and display experimental data in graphs and tables. They then analyze the resulting patterns to make predictions and develop algebraic equations. In this follow-up video, "In Practice", educators discuss the experience of teaching the Patterns & Functions lessons in their classrooms Learner Teachers return to the studio a few months later to discuss their experiences and learn new instructional and assessment techniquesHarding University's STEM Center for Math and Science Education (CMSE) designed "Using GeoGebra in High School Geometry" as a tool to teach topics such as properties of geometric shapes, angles, and transformations. This course also helps educators learn how to create figures in GeoGebra for use in other applications such as PowerPoint and Microsoft Word documentsEvery day, athletes all across Arkansas are training for victory. They are practicing hard, strengthening their muscles, eating right, drinking plenty of water, and getting plenty of rest. They know that to be successful, they have to be in excellent physical condition.
But there is another health issue that coaches and athletes need to be aware of - the risk of communicable diseases. Contracting a communicable disease can sideline an athlete before he or she even takes the field. This course will help you, your staff, and your players take the proper steps to avoid infection. Throughout this video course, we will be looking at some of the communicable diseases that affect young athletes. We will look at their causes, how they spread, how they are treated, and how they can be prevented.
This course meets the professional development requirements of Act 1214 of 2011. It was produced by ArkansasIDEAS and the Arkansas Activities Association. | 677.169 | 1 |
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form
and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They
consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into
its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending
on the context of the problem, transform algebraic expressions or change the viewing window on their graphing
calculator to get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data,
and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help
conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different
method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to
solving complex problems and identify correspondences between different approaches.
This problem expects students to jump in and attempt a solution using the equation they are given. Mathematical practice 4 is here:
MP 4 - Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life,
society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation.
In middle grades, a student might apply proportional reasoning to plan a school event or analyze a
problem in the community. By high school, a student might use geometry to solve a design problem or use a function to
describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they
know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these
may need revision later. They are able to identify important quantities in a practical situation and map their relationships
using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation
and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
This problem expects, almost requires, that the student do no modelling whatsoever. The modelling has already been done by Isaac Newton.
Everything else in this question, other than the use of these two items, involves the careful reading of a passage in the realm of thermodynamics, interpreting the contents carefully, and properly substituting values into s formulas handed out freely. All three skills are worthy, but should be tested in a familiar realm. The writer of the question knows that "object" here refers to the turkey, but would that be obvious to the novice, reading such a scenario for the first time?
This question also includes some either dangerous or misleading (or even false) information.
The most important is that the time needed to bring a refrigerated turkey to room temperature would be far too long, allowing for growth of salmonella among other hazards. The typical cook may wait for the surface of the turkey to feel close to room temperature, but never waits for the whole bird to warm up.
Second, Newton's law of Cooling (the proper name of the law) is based on the object having a uniform temperature. In any other situation the law provides nothing more than an approximation. As noted above, that is not the case when we cook a turkey. (Ever notice that the temperature can read differently when the thermometer is moved to a different location in the turkey?)
The turkey in this problem is cooked for 7 hours (who knows, maybe more?). This is either a huge turkey in an industrial oven, or someone who likes their turkey as little on the burnt side.
No matter how I slice it (no pun intended), this problem comes across to me as being totally out of place in an Algebra II exam. It amounts to a "cookbook" problem, (pun unintended) the likes of which have no place even near an end-of-the-year Regents exam in Algebra II.
Saturday, January 16, 2016
I made this this morning just to be an example of how GeoGebra can be used to create dynamic presentations for teachers.
It is pretty simple to follow and requires no knowledge of GeoGebra. It can be downloaded from GeoGebratube here.
Enjoy!
Thursday, January 7, 2016
What I could have done in the classroom if I had had GeoGebra, a smartboard, and a tablet/laptop of every student. I would toss the textbook, and go. Here is a sketch easy to create, but full of mathematical questions. I have included the option of showing some parametric info just as a "spur". The blue point, the center of the blue circle, rotates on the green circle. The red point rotates on the blue circle. The red and blue points complete one rotation at the same time. Does this generate an ellipse?
Tuesday, January 5, 2016
As we begin the new year, I am revisiting one of my past GeoGebra creations. This shows how an ellipse results from tracking the midpoint as the endpoints of a segment rotate around two circles with the same center. The points rotate in opposite directions, but complete an orbit in the same amount of time.
This is just one of my examples intending to show how topics generally left for later in high school can be introduced much earlier. This example reinforces concepts such as circle, rotation, segment, and midpoint while generating an ellipse. The equations shown in the process can be eliminated or hidden. They were included for those who wish to connect this with higher concepts in Algebra II or later.
This is also my first post to be shared with Facebook, which I have joined, at least for now. Consider this a test!
Monday, January 4, 2016
I encourage you to look at that link, and keep in mind that just a few minutes ago I Googled that phrase (in quotes) and got 3 results, with the Fall Sampler being the second in the list. Here it is. ( I clicked the "If you like" at the bottom, and picked up one more link.)
The area under a curve is a whole subject in and of itself (part of Calculus), and its discovery (or invention?) generally begins with rectangle approximations, trapezoidal approximations, limits, continuity, etc. Are these all part of Algebra II? I know that the process of approximation of the area of a portion of the Cartesian plane bounded above and below by continuous functions of x is very highly programmable. A lot of mathematics is involved in creating such a program. Expecting high school students to use such a program while remaining ignorant of the mathematics involved in its creation seems to be a disservice to those students.
The whole business about Common Core seemed predicated on understanding mathematics. Does someone who has mastered the art of pushing buttons on a calculator understand this concept? And whose idea was it to base a high school sample question on "normal probability cumulative density function". | 677.169 | 1 |
Math Tips
The information given here is simply the opinion of one guy
who did electrical work from 1967 until 2001and has taught
electrical courses since about 1973. Please confirm all aspects
of this information with others before acting on the contents.
Hopefully you will find helpful details here which will make
your career choice easier to follow. Cheers:>) David U.
Larson
Standard Layout
Organization of data is important to a successful outcome for any
mathematical calculation. My favorite layout is as below:
Given
Formula
Substitution
Solution
Given is the place where quantities provided in the problem are
listed with the appropriate symbols. This step is the most difficult.
Formula is the place where the formula which allows the given
information to find the desired answer.
Substitution is where the given information is substituted into the
formula.
Solution is where the calculator is used to find the answer.
Technique of Solution
All mathematical problems probably have more than one method which can
be used to find the correct solution. When you find a technique which
you feel comfortable using, memorize it. This should be done aloud
without reference. Practice while driving and while waiting at a stop
light.
Estimate The Answer
Mathematics problems are often difficult to apply an estimation. But
working many examples of problems over time will provide a
seat-of-the-pants feeling for the magnitude of the answer. When ever
possible, make an educational guess before crunching the numbers.
Use Check Values
The application of check values to a specific set of formulas will
demonstrate to you if you are applying the formulas and calculator
properly. The advantage of this study practice exercise is to verify
that you can correctly apply each formula. Here's a chance for you to
try this. Click HERE to see
check values for AC circuits taken from my
Reference Formulas Appendix Workbook. Print the page then use these
check values to verify that you can correctly apply each formula.
When you try a formula, substitute the check values into the right
side of a formula. Do the math, and if you get a close answer to the
left side of the formula, you're doing the calculator entry aspect
properly. Circle the formulas which you produce a correct answer. Keep
working on any that do not work out. Seek help of you can not make any
of these work. I've used these check values and formulas for several
years. So all problems for several years. So all problems should work
out. You will be slightly off due to the number of places to the right
of the decimal.
Make up your own check values for each formula used to make
electrical calculations.
Calculator Use
The little booklet which comes with a calculator is quite helpful. Don't
ignore it or throw it away. The calculator I recommend for use with all
the workbooks I sell at ElectricianEducation.com is the Texas
Instruments TI-30Xa. Note the Xa. That's the right one. Nothing else. It
is about $12 or less. How such a great calculator is made so
inexpensively baffles me to this day.
Texas Instruments has a great web site. Click HERE to visit their
site.
Rounding Off
Don't. Leave all digits to the right of the decimal point in your
calculator as you work a problem. Some problems do not need any digits
to the right of the decimal. Like circular mils in a voltage drop
problem. Some problems need as many as four places to the right of the
decimal like conduit and nipple fill problems.
When digits are important to the right of the decimal, use the STO
(Storage) and RCL (Recall) keys feature of the TI-30Xa to maintain
accuracy. See the instruction booklet for calculator technique.
Remember, if a formula has more than one quantity on the denominator,
brackets are needed to enter the problem properly. | 677.169 | 1 |
Underground Mathematics
Untitled DocumentUntitled Document
Resources to address the requirements of the new A level
Underground Mathematics offers new, creative resources to support and inspire teachers and students of A level Mathematics. The aim is to make studying mathematics at this level a rich and stimulating experience, enabling all students to explore the connections that underpin mathematics.
This free one day course focuses on using the Underground Mathematics resources to support the overarching themes in the new mathematics A level: proof, problem solving and modelling.
Participants
This course is designed to support all teachers of A level Mathematics. In order to be able to engage in all aspects of the course, participants will need to be confident with the content of AS Core Mathematics. To maximise impact and aid dissemination we encourage schools to send two teachers on this course.
Course Benefits
For the teacher
Familiarity with the philosophy of the Underground Mathematics project
Opportunity to discuss effective classroom use of the resources
Advance planning for the three overarching themes of the new A level
For the student
Tasks that make A level Mathematics a rich, coherent and stimulating experience
An appreciation of the connected nature of mathematics
Developing the skills to tackle unfamiliar problems with confidence
For the school/college
More engaging and relevant mathematics lessons
Resources that meet the requirements of the new mathematics A level
Bespoke departmental meeting activities for delegates to share the resources with colleagues
Application
Apply using the online application form. Please indicate your chosen venue and date using the drop-down menu on the application form.
This course, including refreshments, lunch and resources, is free of charge to A level Mathematics teachers from state-funded schools and colleges.
Underground Mathematics Regional Conferences
Teachers who have attended Underground Mathematics professional development sessions are invited to attend a free, one-day conference at the National STEM Learning Centre, York, on 12th July 2017. | 677.169 | 1 |
Where find laplace transform first order system?
Explores the lives of Top 10 animation software, including Barack Obama. Krull is a master at including important information as well as anecdotes and facts that have kid appeal in a very readable format. Van Allsburg approaches non-fiction in much the same way as he ttansform fiction in this captivating story of Annie Edson Taylor, who at age 62 finds herself widowed, laplace transform first order system from teaching charm school and worried about how she will support herself. She decides to be the first person to go over Niagara Falls in a barrel in the hopes of gaining fame and fortune.
Where find laplace transform first order system?
They will be provided 2 closing paragraph example. Write 2 facts about your season. Then turn the facts into opinions. A laplcae will have a chart with 4 boxes. The top 2 boxes will be labeled Fact 1 and Opinion 1. The bottom 2 will be labeled Fact 2 and Opinion 2.
Where find laplace transform first order system?
Ask questions to your students during the reading and video such as, sap ecatt tutorial process are the moral lessons in each myth, or what heroes remind us of people in our lives today. Gauge their interest in the subject by offering your own ideas, and be candid with your responses to their questions. Developing Transforrm Plans Separate students into groups to work together. Ask them to create and present different parts of the Greek laplace transform first order system myth timeline.
Where find laplace transform first order system?
If your human activities include some applied math, so much the better. Best laplace transform first order system A Berkeley Nerd My daughter started to have trouble at about the same age. My niece had tried Kumon and had great success so we decided to try it. It trabsform helped my daughter tremendously and I would highly recommend it. Kids will complain that they have to sytem at a place that is below where they are in school but the whole purpose of the program is to build a foundation. They want the kids to start someplace that is easy for them so they can feel successful and then move extract image from pdf objective c to the next level. | 677.169 | 1 |
For school students
UMS solves all problems using classroom methods exclusively
UMS solves all problems using classroom methods exclusively; presenting solutions in a clear, step by step format – exactly as a teacher would.
The Universal Math Solver software will benefit middle- and high-school students by:
Providing help with math homework assignments – is able to solve any problem either in a textbook or assigned by the teacher.
Being a universal self-tutor – can generate problems similar to those in a textbook, allowing the user to practice solving on his/her own.
Being a reliable math coach – the user can compare his/her own solution to the one given by UMS in order to confirm the correct answer and check for errors.
Unlike many other powerful math packages, UMS is very user friendly and does not require extensive training or use of Help files due to its simple interface. UMS uses the most straightforward method of instruction: "Do as I Do!"
This is how the software solves an exponential equation:
UMS will produce the solution step by step, on the monitor, accompanying the solving process with a professional teacher's voice. UMS is very easy to use – just type in your math problem and push the green button. | 677.169 | 1 |
Frank Ayres, Jr., and Elliott Mendelson
It's not marketed this way, but in many ways this is an old-fashioned calculus book, with skimpy but straightforward explanations, few figures, and a strong emphasis on techniques and drill. It reviews analytic geometry, and covers everything that would normally be in first-year calculus, including differentiation and integration, multi-variable and vector calculus, differential equations, and infinite series. The applications are also very traditional, consisting of geometric problems and some items from mechanics.
The book is marketed as a supplement and review for pre-calculus and calculus courses. I think it works well for calculus (but not for pre-calculus, because there is essentially no coverage of high-school algebra or trigonometry). The book is divided into brief (8 to 10-page) chapters, each containing an explanation of the topic, a series of worked examples, and a series of exercises with answers. The exercises are for drill, and are not especially challenging but are typical of what appears on calculus exams. There are essentially no word problems or applications in the exercises.
There is some use of technology. There is a series of thirty 4–5 minutes videos, available online at no cost at the publisher's web site These videos provide walk-throughs of selected exercises from the book, using an electronic blackboard and an audio narrative. They are not referenced in the text, except on the cover, and require some digging to find on the web. These are well done and provide another medium to reach students. A number of exercises call for a calculator or graphing calculator (they are marked "GC"). Most of these involve graphing a function, and some make good use of the calculator, for example to plot parametric functions. There is also a lot of numeric work, for example using Newton's method for finding roots and for approximating infinite series. There are no tutorials for the calculator: you are assumed to know how to use the calculator already. | 677.169 | 1 |
AQA A Level Maths
The AQA AS and A Level Maths and Further Maths specifications have now been accredited by Ofqual. Our Year 1 / AS Maths Student Book has been approved by AQA and our forthcoming Student Books and Digital Books for Maths and Further Maths have entered the AQA approval process.
From September 2017 you will be teaching linear A Level maths specifications and 100% prescribed content. The new exams will place increased emphasis on problem-solving and modelling.
We have developed brand new resources for the AQA specifications, with links to MyMaths, to ensure you and your students have everything needed for the changes ahead.
Please come back and check for updates to this page and follow us on Twitter @OxfordEdMaths for the latest information about these new resources.
Additional support for teaching Mechanics and Statistics
Kerboodle online resources | Available now
Start planning for September with Kerboodle early access. We've got resources ready for you today. To arrange a free in-school demo and early access, please contact your local Educational Consultant using the link above.
AQA Schemes of Work
AQA A Level Maths 2017 FAQs
When will the new A Levels be introduced?
The new AS and A Levels for Maths and Further Maths will be introduced for first teaching from September 2017. They will be first examined in Summer 2019.
What are the key changes to Maths A Level?
A Level maths will no longer be examined in modules. Instead, the assessment will be linear, meaning all exams will be taken at the end of the course.
You will no longer be able to choose which topics you teach. Instead, the content will be 100% prescribed, so all students will have to study pure maths, mechanics, and statistics.
Decision maths has been dropped from the new A Level (but not from Further Maths).
For the first time a large data set will be identified in advance, on which statistics questions will be based.
The question style will change to give more emphasis to modelling and problem-solving.
When will the final accredited AQA specification be published?
The AQA AS and A Level Maths specifications have now been accredited by Ofqual. Further Maths is yet to be accredited.
What resources are available and when will they be ready?
We are publishing brand new resources for AQA's new specification. Our Year 1/AS Student Book has been approved by AQA and is now published. Main Student Books from the series have been entered into the AQA approval process. To keep up-to-date on these new resources, please sign-up to the maths eNewsletter and follow us on Twitter @OxfordEdMaths. | 677.169 | 1 |
Alg 1 -- Unit 1: Algebra Basics Bundle -- Lessons & Fun Reviews
Be sure that you have an application to open
this file type before downloading and/or purchasing.
90 MB
Product Description
This is a complete unit for the beginning of Algebra 1. It has 7 Powerpoint lessons. Each can be edited to meet the needs of your specific classes. Each lesson also contains a video that explains the topic. That makes them perfect for new teachers, home teachers, or on days when you can't be in the classroom. (I use that time to take roll.)
This bundle also includes 5 review games to use to give your students extra practice.
You can preview each of the lessons and reviews by using the links given below.
If purchased individually, these products would cost $52.00. By buying the complete unit, you are saving more than 30%! This will save you money and countless hours preparing for your Algebra 1 class.
Be sure to follow me so you will be notified as more unit bundles are added!
Remember this is for YOUR use only! It is not to be shared with friends or colleagues. Other teachers must purchase their own products. You may not upload this resource to the internet in any form. If you are a department chair, principal or district administrator interested in purchasing several licenses, please contact me for a school-wide or district-wide quote at vickihines24@gmail.com. | 677.169 | 1 |
Elementary Algebra
Browse related Subjects
Builds on the author's tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. This text continues coverage and integration of geometry in examples and exercises.
Read More
Builds on the author's tradition of guided learning by incorporating a comprehensive range of student success materials to help develop students' proficiency and conceptual understanding of algebra. This text continues coverage and integration of geometry in examples and exercises | 677.169 | 1 |
AGRE Subject GRE Maths
Overview:
The test consists of approximately 66 multiple-choice questions drawn from courses commonly offered at the undergraduate level. Approximately 50 percent of the questions involve calculus and its applications — subject matter that can be assumed to be common to the backgrounds of almost all mathematics majors. About 25 percent of the questions in the test are in elementary algebra, linear algebra, abstract algebra and number theory. The remaining questions deal with other areas of mathematics currently studied by undergraduates in many institutions.
Content Specifications:
The following content descriptions may assist students in preparing for the test. The percents given are estimates; actual percents will vary somewhat from one edition of the test to another.
CALCULUS — 50%
Material learned in the usual sequence of elementary calculus courses — differential and integral calculus of one and of several variables — includes calculus-based applications and connections with coordinate geometry, trigonometry, differential equations and other branches of mathematics.
ALGEBRA — 25%
Elementary algebra: basic algebraic techniques and manipulations acquired in high school and used throughout mathematics. | 677.169 | 1 |
Category Archives: Math TeachingThe Inquiry-Oriented (IO) Approach and Advanced Mathematical Thinking (AMT) processes play an important role in improving undergraduate math education. IO approach and AMT processes act as a new movement of modern math education based on the methods used in math … Continue reading →
Generalization and abstraction both play an important role in the minds of mathematics students as they study higher-level concepts. In the second chapter of the Springer book Advanced Mathematical Thinking, Tommy Dreyfus defines generalization as the derivation or induction from something particular to something … Continue reading →
For my next installment on innovative teaching techniques, I'd like to dredge another demon that haunted my nights long ago—EIGENVECTORS! Normally, eigenvectors are introduced in the waning days of a first-year linear algebra course, when students' minds are already saturated | 677.169 | 1 |
Lecture 4d
Calculating a Determinant
(pages 259-61)
The definition we have used for a determinant is known as expanding by cofactors. Even more specifically, we have expanded by cofactors along the first
row. But the first step in (potentially) making the
Lecture 4c
The Determinant of an n n Matrix
(pages 258-9)
Just as we calculated the determinant of a 3 3 matrix by looking at smaller
2 2 matrices within it, we calculate the determinant of a 4 4 matrix by
looking at 3 3 matrices within it. And in general
Lecture 4b
The Determinant of a 3 3 Matrix
(pages 256-8)
As I mentioned in the previous lecture, the determinant of a 2 2 matrix is
a value that determines whether or not the related system of equations has a
unique solution. It turns out that the notion
Lecture 4e
Elementary Row Operations and the Determinant
(pages 264-8)
We saw in the previous lecture that it is much easier to calculate the determinant
of a triangular matrix, or better yet, a matrix with a row or column of all zeros.
And weve already s
Lecture 4j
Matrix Inverse by Cofactors
(pages 274-6)
While Im sure you all think that computing the cofactors of every entry of
a matrix is fun, you must be wondering why we would need the cofactors of
every entry. After all, we only need to compute the c
Lecture 4i
The Cofactor Matrix
(pages 274-5)
As we continue our study of determinants, we will want to make use of the
following matrix:
Definition: Let A be an n n matrix. We define the cofactor matrix of A,
denoted cof A, by
(cof A)ij = Cij
That is, the
Assignment 8
Note: All the questions in this quiz will be related to the matrix A given below.
You should feel free to use results from one question when solving another. In
fact, its what I intend for you to do!
1
A = 5
2
[1pt] 1. Is ~x =
4
1
5
8
8
in
Assignment 2
x1
[2pt] 1. Show that the set A = x2 | x3 = (x1 x2 )2 is NOT a subspace
x3
of R3 .
2
2
10
Since 3 is an element of A (as 36 = (2 3)2 ), but 5 3 = 30
36
36
180
is not in A (as 180 6= (10 30)2 ), we see that A is not closed under scalar
multi
Assignment 10
3
[4pt] (1) Evaluate the determinant of 0
4
the row or column of your choice.
4 1
2
8 by expanding along
6
3
There are six possible correct answers, depending on which row or column you
choose. Ill give the solutions to doing the first row,
Lecture 4l
Area, Volume, and the Determinant
(pages 280-3)
The determinant has another interpretation completely separate from systems
of equations and matrices. It turns out that it can also be used to calculate the
area of a parallelogram (in R2 ), the
Lecture 4k
Cramers Rule
(pages 276-8)
While the cofactor method isnt the most practical way to compute the inverse
of a matrix, it does give us the useful formula that
A1 =
1
(cof A)T
det A
And we can now apply this formula to a system of equations. Becau
Lecture 1h
Spanning Sets
(pages 18-20)
A common way to define a subspace is through using a spanning set. But before
we get to this definition, we note the following.
Theorem 1.2.2 If cfw_~v1 , . . . , ~vk is a set of vectors in Rn and S is the set of al | 677.169 | 1 |
Mathematical Statistics with Applications provides a calculus-based theoretical introduction to mathematical statistics while emphasizing interdisciplinary applications as well as exposure to modern statistical computational and simulation concepts that are not covered in other textbooks. Includes the Jackknife, Bootstrap methods, the EM algorithms... more...
Mathematical Statistics with Applications in R, Second Edition, offers a modern calculus-based theoretical introduction to mathematical statistics and applications. The book covers many modern statistical computational and simulation concepts that are not covered in other texts, such as the Jackknife, bootstrap methods, the EM algorithms, and Markov... more...
The Joy of Finite Mathematics: The Language and Art of Math teaches students basic finite mathematics through a foundational understanding of the underlying symbolic language and its many dialects, including logic, set theory, combinatorics (counting), probability, statistics, geometry, algebra, and finance. Through detailed explanations of... more... | 677.169 | 1 |
Math Homework Help - Answers to Math Problems - Hotmath
Algebra homework help online Cambridge
Take a closer look at the instructional resources we offer for secondary school classrooms. Use the web code found in your pearson textbook to access supplementary online resources. . Pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum.
. Pearson prentice hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. Use the web code found in your pearson textbook to access supplementary online resources. Take a closer look at the instructional resources we offer for secondary school classrooms.
Math.com Homework Help Everyday Math
Free math lessons and math homework help from basic math to algebra, geometry and beyond. Students, teachers, parents, and everyone can find solutions to their math ...
Studypool - Online Microtutoring™ Homework Help & Answers ...
Studypool is your source for easy online academic & homework help! Get help from qualified tutors for all your academic and homework related questions at Studypool. | 677.169 | 1 |
Introduction to Ordinary Differential Equations and Some ... to be able to take a course in differential equations. ... In Chapter 2, we look at
homogeneous and non-homogeneous second order linear differential equations
......
NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS approach or particular use of the MATLAB R software. QA31. ... of a numerical
method is closely connected to the stability of the differential equation problem
being .... The final chapter, Chapter 12, gives an introduction to the numerical
solu-....
A First Course in Partial Differential Equations - Department of ... Aug 21, 2014 ... 1 Preface. The study of partial differential equations (pdes) has been around since ... Lie algebra theory, to name a few core mathematical areas. This introductory text covers a small, but important part, of the subject, namely exact ....
A First Course in Partial Differential Equations Aug 21, 2014 ... The study of partial differential equations (pdes) has been around since ...
algebra, calculus, and parts of elementary differential equations....
Introduction to Ordinary Differential Equations and Some ... Chapter 1 covers many different types of first order differential equations, both linear and non- linear. It ends with a ... systems of first order linear differential equations by covering 2 × 2 systems of them. We solve ...... Further study: 8. Notic...
Combining Ordinary Differential Equations with Rigid Body ... don't offer a differential equation course which is a mandatory course for any four-
.... mainly they had to adjust to different teaching styles within the short 6-week ......
An N-cycle time-differencing scheme for stepwise numerical ... a smaller number of partial differential equations where ... discrete set of times. In
some ..... Henrici, Peter, Discrete Variable Methods in Ordinary Differential....
Numerical Solution of Differential Equations 1 Ordinary Differential ... Differential equations textbooks are cookbooks that give ... An ordinary
differential equation (ODE) has only one independent variable, and all
derivatives in it ...
...
Differential Equations (Ordinary) - Evolution and Ecology | UC Davis A system of ordinary differential equations involves a finite number of state
variables, ... After writing down a differential equation model of an ecological
system, ......
Partial Differential Equations Ordinary and partial differential equations occur in many applications. An
ordinary .... equation. 4. Method of an integrating multiplier for an ordinary
differential equation. ... It is known from the theory of functions of one complex
variable tha... | 677.169 | 1 |
A complete Grade 6 mathematics curriculum that addresses the main strands: algebra, geometry, measurement, number and operations, data analysis, and probability. It is rich in connections with other core subjects and real life expereinces. | 677.169 | 1 |
This ... This tool for research, learning and teaching. This volumes consists of seven chapters covering a variety of problems in ordinary differential equations. Both pure mathematical research and real word applications are reflected by the contributions to this volume. It covers a variety of problems in ordinary differential equations, and pure mathematical and real world applications. It is written for mathematicians and scientists of many related fields. Productinformatie:Taal: Engels;Vertaald uit het: Engels;Oorspronkelijke titel: Handbook of Differential Equations: Ordinary Differential Equations;Formaat: ePub met kopieerbeveiliging (DRM) van Adobe;Bestandsgrootte: 5.09532829;ISBN13: 9780080532820; Engels | Ebook | 2004
Handbook of Differential Equations: Ordinary Differential Equations, The book contains seven survey papers about ordinary differential equations.The common feature of all papers consists in the fact that nonlinear equations are focused on. This reflects the situation in modern mathematical modelling - nonlinear mathematical models are more realistic and describe the real world problems more accurately. The implications are that new methods and approaches have to be looked for, developed and adopted in order to understand and solve nonlinear ordinary differential equations.The purpose of this volume is to inform the mathematical community and also other scientists interested in and using the mathematical apparatus of ordinary differential equations, about some of these methods and possible applications. | 677.169 | 1 |
math for liberal arts
This assignment will help you practice applying what you
have learned about general problem solving strategies to real problems or situations,
and communicating about mathematical ideas with others.
Directions
Pick a problem:
Option 1: Pick a real problem that you need to solve
and use the different general problem solving strategies to try to find a
solution. For example, perhaps you need
to figure out how to pack an awkward item in your car trunk, or decide what to
spend the most time studying for an exam in another class, or plan the
assignments and schedule for a charity clean-up project you are organizing.
Option 2: Pick a real problem that someone else has
solved and describe the general strategies they used. This could be a friend or relative who solved
a problem like the ones described in Option 1, or it could be a historical
person, like George Washington Carver, who invented peanut butter and many
other things, or Mary Anderson, who invented the windshield wiper.
Option 3: Pick a currently unsolved problem, like
curing cancer or finding better energy sources, and analyze how people are
using the different problem solving strategies to find a solution.
Solve the problem or analyze the
solution:
If you are solving your own problem,
use the different general problem solving strategies that you learned about in
this unit to try to solve it. If you are
analyzing a problem someone else solved, or is trying to solve, identify the
different general problem solving strategies used.
Present the problem, the general problem
solving strategies used, and the solution to the others in the class:
Post a message in the application
discussion forum for this unit. In your
message, describe the problem and how you solved it. Use the equation editor as necessary to show
any mathematical operations. The better
you communicate, the more points you will earn.
If you enjoy and know how to use multimedia, such as video, audio, and
graphics, you may use those as well, but this is not required.
View and respond to the
application problems submitted by your classmates.
Pick two of your classmates' applications
that were particularly helpful to you.
Write a response to each, explaining in a paragraph or two why their
applications helped you better understand the mathematics for this unit or better
understand how the mathematics for this unit could be used outside of class.
********************************
Example:
Background: This
Saturday is the annual chocolate chip cookie bake sale event. At each of the
last 2 bake sale events, there were 600 people who attended. Every annual bake sale
must receive 600 attendees. Every attendee eats 5 cookies each.
Problem: We must
raise $1,500 for our daughter's team travel expenses. How many cookies must we
make and how much money must each cookie sell for in order to raise the
appropriate money? | 677.169 | 1 |
Algebra and Trigonometry (Hs Binding)
Represents mathematics as it appears in life, providing understandable, realistic applications consistent with the abilities of any reader. This book develops trigonometric functions using a right triangle approach and progresses to the unit circle approach. Graphing techniques are emphasized, including a thorough discussion of polar coordinates, parametric equations, and conics using polar coordinates. Those looking to master essential skills of Algebra and Trigonometry. | 677.169 | 1 |
A Concrete Introduction to Higher Algebra (Undergraduate Texts in Mathematics)
Author:Lindsay N. Childs
ISBN 13:9780387745275
ISBN 10:387745270
Edition:3rd
Publisher:Springer
Publication Date:2008-11-26
Format:Hardcover
Pages:604
List Price:$69.95
 
 
This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. A strong emphasis on congruence classes leads in a natural way to finite groups and finite fields. The new examples and theory are built in a well-motivated fashion and made relevant by many applications—to, ranging from routine examples to extensions of theory, are found throughout the book; hints and answers for many of them are included in an appendix. | 677.169 | 1 |
'High School Math Practice' App Makes a Great After-School Study Aid
When it comes time for serious study outside of the classroom, an excellent tool for math students to use is an educational app.
One of the best is "High School Math Practice," a no-nonsense app with interactive exercises and quizzes that challenge students to think logically and organize information they've studied in class.
The free app, from Varsity Tutors, covers everything from basic arithmetic to college-level algebra. It works with most smartphones and digital tablets.
While many apps use gaming technology to help students learn, this app is strictly for study. It includes straightforward exercises and interactive tests that help reinforce math concepts and formulas high school students need to know. It works best as a supplement to standard classroom instruction.
Subjects are categorized by grade level and include interactive exercises and tests covering:
Algebra
Arithmetic
Calculus
Geometry
Statistics
Topics align with most state math learning standards. The app is a great study aid for students in independent learning classes and can be helpful for college-bound students in STEM (science, technology, engineering and mathematics) programs.
Getting started
An internet connection is needed to use the app. Launching the app opens a simple homepage with a topic navigation area on the left side of the screen. The bottom of the navigation area has a "Sign in" link where students can register on the Varsity Tutors website. Registration is optional; students who don't register can still use the app as a guest.
Topics in the navigation area are intuitive, and students choose one by tapping it with their finger to launch a start page. Written instructions guide students through hundreds of math exercises, quizzes and tests. Students can return to the homepage whenever they want by tapping a horizontal three-line icon at the top of each page.
Interactive tools
An impressive suite of interactive tools helps students move through exercises and tests in self-paced study sessions. The most helpful tools include:
Flashcard maker. Custom flashcards on any subject can be created and saved for individual study.
Curated flashcards. A searchable collection of prebuilt cards is organized by subject.
Full-length diagnostic tests. Prebuilt tests show students the areas of study where they need to focus.
Immediate test results. A detailed report and analysis in an easy-to-read format shows the percentage of correct answers, test time and score.
Practice quizzes. Introductory exercises on all subjects.
Registered students can save their diagnostic test scores and share them via email with teachers, other students and parents. They also can save the custom flashcards they make.
System requirements
Content in the "High School Math Practice" app is all-inclusive, and registration is optional. An internet connection is needed to access the math exercises, tests and flashcards. An extra perk for registered students is a math "Question of the Day" that's automatically sent. This feature can be turned on in the app settings area.
There are no advertisements or in-app purchases, but there are links to the Varsity Tutors website, where visitors can register for an optional free introductory online tutoring session or find local tutors.
Download details
The "High School Math Practice" app by Varsity Tutors is free. It can be downloaded from iTunes and is compatible with iPad tablets running iOS version 9.0 or later. It's also available on Google Play and is compatible with digital tablets running the Android operating system 4 | 677.169 | 1 |
Únete y Recibe Nuestras Publicaciones Por Correo (GRATIS)
FACEBOOK
Eng Course- Vectors and Plane Geometry- Download Free PDF
During the first week of the semester it is difficult to get started with the
course material. Some students have not settled in, some are still changing
sections, and some still have to sign up for a course. For this reason, it is
reasonable to teach an interesting, relevant topic that is somewhat independent of the course. Some instructors in a calculus course use the first week
to review topics from precalculus. Instead, we decided to spend this week
on vectors and the geometry of the plane, topics that other sciences and
engineering like to see covered early. These notes are meant as lecture notes
for a one-week introduction.
There is nothing original in these notes. The material can be found
in many places. Many calculus books will have a section on vectors in the
second half, but students would not like to start reading there. The material
is also contained in a variety of other mathematics books, but then we would
not want to force students to acquire another book. For these reasons, we
are providing these notes. | 677.169 | 1 |
Lea Ann Smith
Math Teacher
"I am available for help during A1/2, A5/6, B1/2 and B5/8. I am usually in the new STEM Lab, C-111B. I am also the STEM Academy Leader, if you have any questions about the Academy, please stop by to chat. "
Introduction to Calculus is designed for students who want to continue their studies of mathematics but are not interested in taking the Advanced Placement test at the end of the year. Students planning on a career in science, mathematics, or engineering will be well prepared to take a college Calculus course and in some cases go on to 2nd semester Calculus. The curriculum will be very similar to the AP Calculus AB course listed below. The major difference will be a less rigorous coverage of the material in some areas. A graphing calculator is recommended. One will be provided during class time only.
This one semester seminar course is for juniors and seniors interested in exploring careers in the STEM fields. Students will reflect on their STEM goals, research careers and participate in a variety of experiences involving local STEM commercial and research facilities. Students will create an electronic portfolio of their experiences and will make extensive use of Google apps and online collaborative activities. If students are unable to fit the STEM Internship class into their schedule, they may sign up for Course Number M131 (the online option) to earn the 0.5 credit and complete the STEM Academy requirement. Students participating in the online version of this class will meet with the STEM Academy Leader to create a plan for the course, including preparation of an electronic portfolio on their own time and participation in at least 40 hours of off campus career experiences.
Last updated: Jul 7 at 12:30 pm
Welcome to Essex High School
The mission of EHS is to engage students in learning that is intellectually vital, personally meaningful, and socially valuable. | 677.169 | 1 |
Identities - Basic 2
Be sure that you have an application to open
this file type before downloading and/or purchasing.
228 KB|3 pages
Product Description
This self checking worksheet offers the student a review of the basic identities and adds in the Pythagorean Identities. My students find this challenging after such an easy "Identities - Basic 1" worksheet. Once the student simplifies the identities, the solution to a riddle appears. | 677.169 | 1 |
(Joint degree from University of L'Aquila, Italy and University of Nice, France)
Teaching Statement
Open problems exist in almost every facet of our lives. However, it requires careful study and analysis of events (or patterns) to correctly characterize them. I see mathematics as the discipline that provides the primary tools needed to develop problem solving capabilities. As I handle each topic in my course, I help students to see the relevancy, applicability and feasibility of the subject matter. Desisting from the frontal cold presentation of mathematical formulas common in most mathematics courses, I aim at helping students to see the course as not too abstract but rather one in which careful application leads to interesting and useful results. To put all these ideas become a reality, my goal in the classroom is to foster an active learning environment while stimulating students' interest and curiosity in mathematics, and help them monitor their understanding. In addition the importance of offering students frequent opportunities to make conjectures as well as argue their validity is not downplayed.
Technology obviously has been an essential part of our modern life and students are the master users. Therefore, I always strive to integrate the latest technological innovations into my course. Computer software packages like Geogebra presents useful features which can enable students to widen their appreciation for fundamental concepts taught in my Pre-Calculus classroom. That notwithstanding, I strongly oppose the abuse of technology. As you may know, there are topics in mathematics, where students need to learn how to use their brain rather than calculators/computers. | 677.169 | 1 |
latest step in the evolution of Mathemtics for Electricty & Electronics, the third edition makes use of more exercises and examples than ever before to make it the most direct route to a thorough understanding of essential algebra and trigonometry for electricty and electronics technology. Practical, well-illustrated information sharpens the readerís ability to think quantitatively, predict results, and troubleshoot effectively, while repetitive drills encourage the learning of necessary rote skills. All of the mathematical concepts, symbols, and formulas required by future technicians and technologists are covered to help ensure mastery of the latest ideas and technology. And finally, the text is rounded out by real-life applications to electrical problems, improved calculator examples and solutions, and topics that prepare readers for the study of calculus. | 677.169 | 1 |
Zero & Negative Exponents (Foldable) for Algebra 1
Be sure that you have an application to open
this file type before downloading and/or purchasing.
690 KB|5 pages
Product Description
Now with TWO options!
This foldable allows students to discover, take notes, and practice skills associated with zero and negative exponents.
Option 1: It begins with a chart where students extend a pattern of positive exponents to discover the way zero and negative exponents work. The following two tabs guide students through practice problems, many of which include other laws of exponents.
Option 2: Other properties of exponents are NOT included. Students will take notes on both properties, simplify powers (one- to two- steps), then simplify exponential expressions (multi-step) | 677.169 | 1 |
Geometry and Discrete Mathematics notes? Resources?
Hello everyone, i'm going to take Geometry and Discrete Mathematics, next year in high school (grade 12). So this summer i'm planning to read some books that would help me out next year, to bulit up my basic skills. So anyone know any sites or books that would help me out somehow?
I took logic concurrently with discrete math and that worked really well. The proofs in symbolic logic help you get used to the discrete math proofs. They help give you a feeling that you really do know what's going on. Depending on your course, you'll probably do some logic actually in the discrete math course too.
[Broken] looks like a pretty good resource for introductory logic.
[Broken] has a lot of problems that can be solved by thinking them through, but using symbolic logic makes them easier.
[Broken]
has a bunch of problems. For symbolic logic (you may want to skip the aristotelian stuff) you should scroll down to the links labeled Natural Deduction Exercises and the exercises below them. | 677.169 | 1 |
because enthusiasm is contagious
Class Handouts
Here I have gathered some of the handouts that I have created. This is by no means exhaustive (I have created many many things over the years, not all of which are worth sharing, but even some of those may make it up here eventually, on the off chance it sparks an idea for someone else), and mainly I'm sharing things through ShareLaTeX.com. If you click on the link below, it will take you to the latex code for that document. You don't need a ShareLaTeX account to access that link, but if you'd like to sign up for one, please use this referral link, which lets me earn points toward referral bonuses. Thanks!
I'll be uploading them a few at a time, as I have time.
Most of these have been used in class at least once (which means at least some typos and mistakes have been caught), and a few of them were extra resources that I typed up and put online for my students. If you find any errors, or have suggestions for improvement, please feel free to comment below.
College Algebra
(I begin my College Algebra classes with a tiny bit of set theory. Since we use the vocabulary and the notation anyway, it saves a lot of headaches. Also I find that it helps to begin the semester with a topic that they haven't seen before.)
Applied Calculus
Difference Quotient Practice Problems This is a set of practice problems with solutions. To get the practice problems without the solutions, comment out line 15, which reads "\printanswers", by typing a % sign in front of it. | 677.169 | 1 |
Above Par by Crystal Donahue
Drew's life just got a whole lot harder. Her relationships are falling apart, her little sister is failing math, and her parents are leaving for Florida to take care of Grandpa. Now not only does she have to play surrogate parent, study to graduate from her last semester of college, and deal with the looming threat of full-on adulthood, she has to do it all while running the family's mini-golf course by herself. Even the upbeat Drew is starting to feel like she just drove her ball down the unfairway.
Her luck takes a turn when she meets Weston, the snarky geometry teacher. He's smart, capable, dependable, and handsome—a real hole in one. But even Weston has a few hazards of his own. Now Drew will have to decide whether his secret is the wedge that drives them apart, or pushes them together.
Education enables people to learn basic computations that are required in almost every walk of life. Although computers are doing most of the mathematics for us these days, we still need to be able to do basic maths for our everyday activities. | 677.169 | 1 |
Showing 1 to 4 of 4
Ali Mohammed
Professor McCullough
9/27/2013
MATH 111
Writing Assignment #2
A. Function a relationship where each input has exactly 1 output. The easiest way to
determine if an equation is a function is to graph it and have it pass the vertical line test.
Introduction To Algebra Advice
Showing 1 to 1 of 1
I would recommend this because The teacher was great, she helped us every step of the way, when we have questions she would always answer it, helping each one of her student to become a better person.
Course highlights:
I used to hate math class well when i started college, I fail my first time taking that class but when I took it again it was much easier, I understood everything, I studied a lot, Did all my homework and Went to class everyday.
Hours per week:
9-11 hours
Advice for students:
If you come to class Everyday, take notes, study your notes and do the homework. | 677.169 | 1 |
Showing 1 to 9 of 9
Prerequisite Review
MATH 1350
Name_
Neatly show all work in the space provided. Credit will not be given without sufficient work to support
your answer. All answers should be exact and in simplest form.
1. When a scientific calculator shows the answer for
AP Physics Free Response Practice Magnetism and Electromagnetism ANSWERS
SECTION A Magnetostatics
1975B6.
a) Since the ions have the same charge, the same work (Vq) will be done on them to accelerate them and they will
gain the same amount of K as they ar
Math for Teachers Advice
Showing 1 to 3 of 3
Professor Vega is the best teacher for this course. She is extremely knowledgeable and very upbeat. This course is needed for all education majors
Course highlights:
I learned a lot of different ways to teach students k-7 the fundamentals of mathematics
Hours per week:
9-11 hours
Advice for students:
Stay organized and keep up with assignments
Course Term:Fall 2016
Professor:Cynthia Vega
Course Required?Yes
Course Tags:Math-heavyGroup ProjectsParticipation Counts
Aug 26, 2016
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
she was a great teacher, very knowledgeable and always willing to answer any questions for her studenst; she care very much about her students and always asking how we were doing academically; she had a great positive attitude.
Course highlights:
how to apply geometry to our daily life, including solving word problems with formulas and without formulas.
Hours per week:
3-5 hours
Advice for students:
manage your time well because this is a fast pace course.
Course Term:Summer 2016
Professor:thompson
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsParticipation Counts
Aug 07, 2016
| Probably wouldn't recommend.
This class was tough.
Course Overview:
She is extremely hard to understand.
Course highlights:
Your only major grades are the midterms and the exam. There is a curve.
Hours per week:
12+ hours
Advice for students:
I would NOT recommend having her as a professor- switch if you still have the chance | 677.169 | 1 |
Volume 2 of Mathematics Transition, Grade 7, California Edition. The Teacher's Edition includes background information and teaching suggestions, support for ELL and differentiated instruction options. CONTENTS: 7) Multiplication in Geometry 8) Multiplication in Algebra 9) Patterns Leading to Division 10) Linear Equations and Inequalities 11) Geometry in Space 12) Statistics and Variability. The UCSMP Third Edition curriculum emphasizes problem solving, everyday applications, and the use of technology and reading, while developing and maintaining basic skills. UCSMP Transition Mathematics integrates applied arithmetic, algebra, and geometry, and connects all of these areas to measurement, probablity, and statistics. Variables are used to generalize patterns, as abbreviations in formulas, and as unknowns in problems, and are represented on the number line and graphed in the coordinate plane. Basic arithmetic and algebraic skills are connected to correspondencing geometry topics. This course provides opportunities for students to visualize and demonstrate concepts with a focus on real-world applications.
Book Description Mc Graw Hill Wright Group. Hardcover. Book Condition: VERY GOOD. little to no wear, pages are clean. The cover and binding are crisp with next no creases. Bookseller Inventory # 2739578149 | 677.169 | 1 |
Dedication To Jessica Alexander and Uriel Avalos in gratitude for their invaluable work in preparing this text for publication. Ann Xavier Gantert The author has been associated with mathematics education in NewYork State as a teacher and an author throughout the many changes of the past fifty years. She has worked as a consultant to the Mathematics Bureau of the Department of Education in the development and writing of Sequential Mathematics and has been a coauthor of Amsco's Integrated Mathematics series, which accompanied that course of study. Reviewers: Richard Auclair Mathematics Teacher La SalleSchool Albany, NY
Domenic D'Orazio Mathematics Teacher Midwood High School Brooklyn, NY Steven J. Balasiano Assistant Principal, Supervision Mathematics Canarsie High School Brooklyn, NY George Drakatos Mathematics Teacher Baldwin Senior High School Baldwin, NY Debbie Calvino Mathematics Supervisor, Grades 7–12 Valley Central High School Montgomery, NY Ronald Hattar Mathematics Chairperson EastchesterHigh School Eastchester, NY Raymond Scacalossi Jr. Mathematics Coordinator Manhasset High School Manhasset, NY
PREFACE
Algebra 2 and Trigonometry is a new text for a course in intermediate algebra and trigonometry that continues the approach that has made Amsco a leader in presenting mathematics in a modern, integrated manner. Over the last decade, this approach has undergone numerous changes and refinements to keep pace with ever-changing technology.
This textbookis the final book in the three-part series in which Amsco parallels the integrated approach to the teaching of high school mathematics promoted by the National Council of Teachers of Mathematics in its Principles and Standards for School Mathematics and mandated by the New York State Board of Regents in the Mathematics Core Curriculum. The text presents a range of materials and explanations thatare guidelines for achieving a high level of excellence in their understanding of mathematics. In this book:
✔ ✔
The real numbers are reviewed and the understanding of operations with irrational numbers, particularly radicals, is expanded. The graphing calculator continues to be used as a routine tool in the study of mathematics. Its use enables the student to solve problems that requirecomputation that more realistically reflects the real world. The use of the calculator replaces the need for tables in the study of trigonometry and logarithms. Coordinate geometry continues to be an integral part of the visualization of algebraic and trigonometric relationships. Functions represent a unifying concept throughout. The algebraic functions introduced in Integrated Algebra 1 arereviewed, and exponential, logarithmic, and trigonometric functions are presented. Algebraic skills from Integrated Algebra 1 are maintained, strengthened, and expanded as both a holistic approach to mathematics and as a bridge to advanced studies. Statistics includes the use of the graphing calculator to reexamine range, quartiles, and interquartile range, to introduce measures of dispersion such as... | 677.169 | 1 |
Neil
Hi Neil,
You'd be better off, as you already know, thoroughly understanding how the equation itself is constructed, how the different parts are related etc, so you won't be learning it by rote, because it'll make SENSE to you.
There are a couple of equation videos, not that you need to watch them both.
I'll leave these videos up here for a couple of weeks, or until you tell me you're done.
Finding keywords and making notes
Mnemonics
Your amazing brain
The role of the Reticular Activating System in Learning
Click here to open the gallery. Try for yourself this fascinating 'attention' experiment Video shown here by kind permission of Dr Daniel Simons. Read The Invisible Gorilla and visit Dan's websiteand learn more about how the mind works.
The Genius Material Process
How to complete one chunk of revision – from course manual to Online Revision Calendar | 677.169 | 1 |
...
Show More complete lecture for each section of the text, where Martin-Gay highlights key examples and exercises from the text. New "pop-ups" reinforce key terms, definitions, and concepts while Martin-Gay presents the material. *Interactive Concept Check exercises measure students' understanding of key concepts and common trouble spots. After a student selects an answer from several multiple choice options, Martin-Gay will explain why the answer was correct or incorrect. *Study Skill Builder videos reinforce study skills-related skills and concepts found in Section 1.1, Tips for Success in Mathematics, and in the Study Skills Builder exercises found throughout the text. * Complete solutions on video for all exercises from the Practice Final Exam (located in the appendix of the text) as well as "Overview" segments that review key problem-solving | 677.169 | 1 |
Transformational Mathematics
Learn maths like you never have before!
The coupon code you entered is expired or invalid, but the course is still available!
Transformational Mathematics is different from any math course that most have ever seen. This is really a walking tour through many of the basic languages and principles of mathematics. Our desire is to end up not only more comfortable with mathematical concepts, but hopefully enjoying, and actually finding amazement in these very ancient forms of human inquiry. Furthermore, we hope to feel the sort of comfort that allows us to employ mathematical concepts in our daily lives. And for some, we hope to inspire further and deeper investigation into these mathematical concepts.
By the end of this course, you should be able to:
demonstrate comfort and competence in working with variables, exponents, fractions, decimals and percents en route to generating appropriate equations leading to the solving of complex word problems.
appreciate and speak the foundational language of mathematics as heard through numbers theory and calculus.
comfortably begin to explore and examine the qualitative and quantitative, sacredly geometric patterns of this spectacular living universe.
"I loved this course — it was very well planned and the gradual increase in challenge is very thoughtful."
- Mojitha, Sri Lanka
Your Instructor
Tom McGarrity
Prior to coming to Ubiquity University, Dr. McGarrity taught algebra at Waubonsee Community College for six years after receiving his Masters of Arts degree in Education, with a concentration in mathematics, from Loyola University Chicago.
Through Transformational Mathematics, Dr. McGarrity encourages us to take a fresh look at mathematics before heading out into the world. Overcoming any fears or mathematical discomforts, Dr. McGarrity inspires us to re-acquaint ourselves with the languages and principles of mathematics, so that we might speak them freely and confidently in our daily lives. He has created this course not just about transforming our understanding of mathematics, but that by doing so, we might truly transform our lives. His students at Waubonsee and Loyola felt the transformative energies that underlie this course. | 677.169 | 1 |
3DMath Explorer Desciption:
3DMath Explorer is a computer program that pilots 2D and 3D graphs of mathematical functions and curves in unlimited graphing space. It has many useful feature such as 3D curve ploting in real time, perspective drawing, graph scaling (zooming), active graph rotation, fogging effect, cubic draw, unlimited space ploting, four view plot screen, auto rotate animation, drawing many curves in the same screen,working with many graph screen est.
Review 3DMath Explorer
Math Mechanixs is an easy to use scientific and engineering FREE math software program. (FREE registration is required after 60 days). The typical tool for solving mathematical problem has been the calculator. Unfortunately, a calculator can be very...
Mark Jacobs Graph Plotter is an application that allows you to easily build a graph of multiple mathematical functions.You can insert up to 10 functions and the program automatically plots the resulting chart, which you can save in a separate file.
Graph polynomials and view prime numbers on a ulam spiral graphing plot. Only integer numbers. Based on an old GPL version of the JEP equation parser. See website for additional examples/live applets Seems to be closely related to Cellular automaton.
RISAFloor designs floor systems and works hand in hand with RISA-3D and RISAFoundation to provide a more complete structural engineering software solution for building design. RISAFloor will manage loads, design beams and columns, create quality CAD...
Math 3rd Grade Place Values was created as an accessible and easy-to-use software utility that allows kids to test their math knowledge.Math 3rd Grade Place Values is very useful if you want to test and improve the mathematics knowledge of your...
Star Graphs is a simple and small piece of software that allows its users to learn more about graphs.Star Graphs allows you to click on the "+" button to increase the number of vertices. The "-" button can be used to decreases the number of vertices....
Degree Sequences of Graphs was designed as a simple software that allows the users to draw a graph and view the degree sequence. The corresponding degree graph will be shown in the text field available below the graph.Degree Sequences of Graphs is a...
Connected Components of a Graph was specially developed as an accessible and handy software that lets you learn more about graphs.Connected Components of a Graph is a software that lets you draw a graph in the left pane and view the connected | 677.169 | 1 |
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Mathematica
Mathematica is one of the world's most respected software system that used in many scientific, engineering, mathematical and computing fields. It contains many functions for analytic transformation and for numerical calculations. In addition, the program supports graphics and sound, drawing of arbitrary geometric shapes, import and export of images and sound. Mathematica is a leading software product for the handling of numeric, symbolic and graphic data used by professionals in each branches of scientific and technical computing. Mathematica allows users to solve, visualize and use resources of mathematics without a pencil and a calculator.
More detail information about it You can find in this category in our lessons, posts and articles. | 677.169 | 1 |
This course uses the textbook "Introductory Algebra" by Robert Blitzer (6th Edition)
Class 01 - Entire First Class Video. This covers introductions, a course overview, and sections 1.1 and 1.2. We begin to discuss section 1.3 but I have included a separate video below.
Chapter 1 (Introduction to Algebra)
Section 1.1 - Introduction to Algebra [Video] - [Notes] - Khan Links [#1][#2][#3] Focuses on evaluation expressions by plugging in a number in place of a variable and correctly applying the order of operations.
Section 1.2 - Fractions [Video] - [Notes] - Khan Links [#1] Focuses on the four main operations with fractions (addition, subtraction, multiplication and division) as well as how to deal with improper fractions and mixed numbers.
Section 1.8 - Exponents and Order of Operations [Video] - [Notes] - Khan Links [#1] There was no lecture, but instead a quick 5 question quiz where we go over the five problems together on Order of Operations.
Section 2.3 - Solving Linear Equations [Video] - [Notes] - Khan Links [#1] Arguably the single most central section in algebra. This section is all about how to solve general linear equations with one variable such as "2(x+3)-5=6x-7"
Section 2.4 - Formulas [Video] - [Notes] - Khan Links [#1] How to solve a formula such as "y=mx+b" for a specific variable such as "solve y=mx+b for x"
Section 2.5 - Introduction to Problem Solving (word problems) [Video] - [Notes] - Khan Links [#1][#2][#3][#4] How to translate sentences into math equations such as "five more than a number is eight" into "x+5=8"
Section 2.6 - Problem solving in geometry [Video] - [Notes] - Khan Links [#1] Using the area and perimeter formulas of rectangles, triangles and circles to solve various problems.
Section 2.7 - Solving Linear Inequalities [Video] - [Notes] - Khan Links [#1] Solving inequalities such as "2x+5>6" and then representing these as an inequality, a graph on a number line, and in interval notation.
Chapter 3 (Graphing Linear Equations with Two Variables)
Section 3.1 - Introduction to Graphing [Video] - [Notes] - Khan Links [#1] How to graph linear equations with two variables using points and tables of values, such as "graph the line 2x+4y=8". Really focuses on what it means to have an equation with two variables in it as well as what a solution to an equation means.
Section 3.2 - X- and Y-Intercepts [Video] - [Notes] - Khan Links [#1] Focuses on x- and y-intercepts. We find the intercepts from graphs first, then find them from the equations of lines, then combine these skills to find the intercepts from an equation and use these points to graph the line.
Section 3.3 - Slope [Video] - [Notes] - Khan Links [#1] Focuses on "slope". We discuss slope in a big picture way as steepness of a line and positive vs negative slopes. We then find the slope of lines from graphs given and follow this by finding the slopes between two points. Also introduced are the ideas of lines being parallel or perpendicular.
Section 3.4 - Slope-Intercept Form (y=mx+b) [Video] - [Notes] - Khan Links [#1] This section focuses on the equation "y=mx+b" and its significance. We practice finding the slope and y-intercept of lines when given the equation in this format. We then practice rearranging equations to be in this form to get the slope and y-intercept. We finally put these together and use our slope and y-intercept to graph the line.
Section 3.5 - Point-Slope Form (y-y1=m[x-x1]) [Video] - [Notes] - Khan Links [#1] This section focuses on how to write the equation of a line if you are given some information about the line. We discuss how you could only ever really be given two points or a point and a slope. We then practice using this information to write the equation of lines.
Chapter 4
Section 4.1 - Introduction to Systems of Linear Equations & Solving by Graphing [Video] - [Notes] - Khan Links [#1][#2] Focuses on the basic idea of a system of equations, such as what a solution looks like and why. It also introduces the viewpoint of a system of equations as two lines on a graph and uses this as our first method to solve the system of equations.
Section 4.2 - Solving Systems of Equations by Substitution [Video] - [Notes] - Khan Links [#1] Focuses on solving the same types of problems as section 4.1 but introduces a second method to solve them. The idea here is that you will solve for a variable, then substitute in this equal value in place of that variable to solve the system.
Section 4.3 - Solving Systems of Equations by Elimination/Addition [Video] - [Notes] - Khan Links [#1][#2] Focuses on solving the same types of problems as sections 4.1-4.2 but introduces a third method to solve them. The idea here is that you add the two equations together after cleverly rearranging them so that one of the variables cancels out.
Section 4.4 - Word Problems Involving Systems of Equations [Video] - [Notes] - Khan Links [#1] The problems in this section are the same as in sections 4.1-4.3, but the new part is reading a word problem and turning it into a system of equations.
Section 5.2 - Multiplying Polynomials [Video (Part 1)] - [Video (Part 2)] - [Notes] - Khan Links [#1][#2][#3] Focuses on the multiplication of monomials and polynomials as well as exponents. For example, you would see problems like (2x)(7x) and (x+2)(5x-4).
Section 5.3 - Special Products [Video] - [Notes] - Khan Links [#1] Focuses on identifying two common forms that products come in. Specifically it focuses on the difference of squares "(a+b)(a-b)" and a perfect square of binomials (a+b) squared.
Section 6.1 - Greatest Common Factor and Factor by Grouping [Video] - [Notes] - Khan Links [#1][#2][#3][#4][#5][#6] Focuses on finding the greatest common factor of a polynomial. This is always the first step in the factoring process. Once we have figured out the greatest common factor, we factor it out of the polynomial. The second half of this section focuses on the method of "factoring by grouping" which is our general plan to factor any polynomial with four or more terms. This method splits a polynomial up into smaller groups that share common factors. After you take the common factors out of the smaller groups, the remaining binomial is the same in the two groups and can be factored out.
Section 6.2 - Factoring Trinomials with a Lead Coefficient of 1 ("X-Factor") [Video] - [Notes] - Khan Links [#1] Focuses on learning to factor the most common types of polynomials you will see: x^2+5x-6
Section 6.3 - Factoring Trinomials without a Lead Coefficient of 1 ("Hokey Pokey") [Video] - [Notes] - Khan Links [#1] - Description of Process [#1] Focuses on factoring more challenging trinomials (those where the lead coefficient is not 1). There are many strategies to factor these. Most books suggest a form of guess and check. In this lesson I focus on a method I call the "Hokey Pokey". This method is not common, but I find it more useful than the guess and check method (and more enjoyable and elegant as well).
Section 6.4 - Factoring Special Forms (Difference of Squares) [Video] - [Notes] - Khan Links [#1][#2] Focuses on recognizing a few special/common forms. Specifically, you need to be able to recognize a difference of squares. The key hint is that when faced with a factoring problem, if there are only two terms this is your only plan of action (so far).
Section 6.5 - General Factoring Strategy [Video] - [Notes] - Khan Links [#1][#2] Focuses on an overall strategy to factor mixed problems. For the remainder of this class you will encounter polynomials in most every lesson, and you will need to decide which factoring method ot use to solve the problem.
Section 6.6 - Solving Quadratic Equations by Factoring [Video] - [Notes]- Khan Links [#1] Focuses on the zero product rule and using this to solve equations by factoring. The basic idea is that you cannot isolate the "x" like you could with a linear equation (chapter 2). Instead you are stuck with this polynomial with so many terms added together. If you factor the polynomial into a product though and the product is equal to zero ... well then you know one of the things being multiplied together MUST be zero. We use this to solve for the values of "x".
Chapter 7
Section 7.1 - Simplifying Polynomial Fractions [Video] - [Notes] - Khan Links [#1] Focuses on how to reduce/simplify a fraction when there are polynomials involved. The main idea is that you can never cancel one term of a polynomial from the top of a fraction with one term of a polynomial on the bottom of a fraction EVER. Instead, you factor the polynomials in the numerator (top) and denominator (bottom) completely. This changes something with many terms into one big product of factors. Now since you have a product (one big term), you can cancel any factor from the top with any identical factor from the bottom.
Section 7.2 - Multiplying & Dividing Polynomial Fractions [Video] - [Notes] - Khan Links [#1] Focuses on how to multiply and divide fractions that contain polynomials. There is not much new from 7.1 and should be thought of as a continued practice. The idea is when multiplying two fractions, since you are just multiplying the tops together and the bottoms together, you can just treat it as a single fraction and reduce anything from the top with anything from the bottom. As far as division goes, the only new step is that you first need to flip the second fraction and change the sign to multiplication. Other than this single step, you are really just simplifying fractions like in section 7.1.
Section 7.3 - Adding & Subtracting Polynomial Fractions (with the same denominator) [Video] - [Notes] - Khan Links [#1] Focuses on how to add and subtract polynomial fractions that already have the same denominator (or denominators that might be off by a negative sign). Really we just add them like normal fractions with numbers in them. We add the numerators (tops) together and keep the common denominator (bottom) the same. Then you simplify like in section 7.1.
Section 7.4 - Adding & Subtracting of Polynomial Fractions (with different denominators) [Video] - [Notes] - Khan Links [#1] Focuses on how to add/subtract two fractions with different denominators when there are polynomials in the numerators and denominators. The big idea of this section is how to find the least common denominator (LCD) and then how to modify each fraction to have this denominator. It is not different from fractions with numbers, as you follow the same steps, the the difficulty is in performing those same steps with polynomials.
Section 7.6 - Solving Rational Equations (Equations with Polynomial Fractions) [Video] - [Notes] - Khan Links [#1] Focuses on how to solve equations that involve fractions with polynomials in them. The strategy is the same as in chapter two when we had equations with numerical fractions. The big trick is to multiply the entire equations by the LCD of all of the fractions. This immediately cancels out all of the fractions and gives you a normal fraction-less equation. Then we use one of our two methods to solve the equation, either the classic method from chapter two for linear equations or our new method from section 6.6 to solve all non-linear (polynomial) equations.
Section 7.7 - Applications (Word Problems) [Video] - [Notes] - Khan Links [#1] Focuses on how to solve rate, time and distance word problems that involve polynomial fractions. There are also a few problems involving similar triangles.
Listed below are the book sections assigned each class period. Homework is due every Monday and Wednesday. Instructions on how to complete homework assignments, as well as the grading policy can be found in the class syllabus. The exact problems to complete in each section can be found above in the "Homework Assignments ..." link.
Test #1 - Covering all of Chapter 1 and Chapter 2. This is the first of four tests (not including the final which is the fifth overall test). You may use a scientific calculator on the test, but not a graphing calculator or cell phone. The test will be for the first 90 minutes of class and then after this we will have a lesson for the remaining 80 minutes of class. | 677.169 | 1 |
SCHOOL PLANNER BOOK
School planner book
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Equivalent equations: two equations that doing homework high the same solution. Essential Knowledge and Skills This section is from the VDOE Curriculum Framework and outlines what each student should know and be able to do in each standard. Try Now Callum Leverton Charlotte Hutley Olivia S"I am having trouble in my advanced 7th grade algebra1 class and was really struggling because of my ADHD. Probability and Ratio by RL A probability question best solved by considering the ratio of counters in a bag. You can quiz yourself at different levels of difficulty on 1,200 of the most important vocabulary words, it was 1000. Awards Embark on a virtual treasure noun word search printable as you tackle math challenges and reveal colorful prizes. Subscribe below and stay tuned for new time saving tools to help you simplify your real estate investing needs. 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3 MB|24 + 16 keys
Product Description
Algebra 2, Algebra 2 Honors: Permutations & Combinations
This lesson is intended for Algebra 2 Honors students with a general knowledge of simple probability. It is the introductory lesson of a five-topic unit on PROBABILITY
* Two forms of a Daily Quiz to help your students succeed
* Answer keys and directions
Students will be able to solve problems using the Fundamental Counting Principle, problems involving permutations and combinations.
You need Smart Notebook software to run the presentation. It can be displayed through the software and a projector. If you don't have a SmartBoard, you can also use an IPAD App , such as SplashTop, to remotely write on the presentation. You can also use Smart Notebook Express. | 677.169 | 1 |
MTH 140 N. Agras
Fall 2010 Conic Sections and Implicit Differentiation Examples
A. PARABOLA
For each of the following, sketch a graph. Find the coordinates of the focus and vertex. Use (and
include) the latus rectum. Find the equation of the directrix.
x
Rational Functions and their Graphs.
Rational Function (definition)
A rational function is of the form f ( x )
p( x )
q( x )
Domain of a Rational Function
The domain of a rational function is the set of all real numbers, deleting the values
that make q(
MAC 1114
Module 4
Graphs of the Circular
Functions
Rev.S08
Learning Objectives
Upon completing this module, you should be able to:
1.
2.
3.
4.
5.
6.
7.
Recognize periodic functions.
Determine the amplitude and period, when given the equation
of a periodic
1
1.3 The Notion of a Limit
Consider the following expression:
This is read the limit, as x _ 3, of
.
The value of this limit is found by _ values to x that get progressively _ to
3, but not equal to 3, and determining the resulting limiting value of
impo
Creating Functions: Operations on Functions; Composition of
Functions
Functions can be added, subtracted, multiplied or divided, and composed,
creating new functions whose domains are related to and restricted by those
of the functions that were composed
1
Chapter 1 Limits and Continuity
1.1 Functions and Functional Notation
I. In mathematics we often describe one quantity in terms of another:
*The amount of your paycheck if you are paid hourly depends on the number of hours you
worked.
*The cost at the g
To any student who has to take MTH140, I would recommend taking it with Dr. Beverly. She is always a clear instructor and has a profound understanding of her students' needs. She's always willing to help, and can identify your strengths and weaknesses with ease. The only requirement from you is effort.
Course highlights:
I've always struggled with math, but this course cleared up so much of the fog that's been in front of my face for most of my academic career. My algebra skills are stronger now, I understand basic calculus concepts, and I feel ready to move on to the next step in mathematics. Dr. Beverly did a lot of out-of-class extra tutoring and study groups, which were extremely helpful as well.
Hours per week:
6-8 hours
Advice for students:
Apply effort, do the homework, and don't be afraid to ask questions. I once felt insecure about asking questions under the false pretense that everyone else in the class knew more than I did. It often turned out that many others had the same questions! When you don't ask, you can't expect to know the answer later on, and that can set you back a bit. You're here to learn. So learn. | 677.169 | 1 |
achers! Prepare Your Students for the Mathematics for SAT* I: Methods and Problem-Solving Strategies
Description:
The SAT I Reasoning Test is widely used in colleges throughout the United States as a screening device for student admission. This guide is designed to help high school teachers assist their college applicants in preparing for this crucial test. The book provides an overview of the SAT I Test and a selective review of mathematics taught through elementary algebra and geometry. Curricular issues addressed include: what to teach, how to present it, and what consititutes the best possible setting.The importance of students' development of accuracy and speed is stressed throughout, and specific problem-solving strategies and short-cuts are presented in detail along with instructions for advising students on how and when to use them - and when it's best to guess | 677.169 | 1 |
Course Description: This course is designed for participants who need a refresher course in Probability and Statistics, or for new teachers who need insights, additional resources or a different perspective into the content taught in Probability and Statistics.
The Practice of Statistics. Yates, Moore, McCabe. Freeman Printing. 1999. Participants, however, may use any other Probability and Statistics text. Course Videos that contain approximately 20 hours of instruction are provided for the student. Also, four corresponding Practice Problem Sets are provided to each student on the course homepage in Moodle, the learning management system for all TeacherStep® courses.
Graphing Calculator:
Participants are expected to use a graphing calculator. While participants may choose any graphing calculator, instruction and the suggested text will use the TI-83 or equivalent. Knowledge and competence for use of other graphing calculators will be the sole responsibility of participants.
Course Requirements:
This course is delivered online via Moodle, including four videos to view online as well. Moodle provides the practice problems, a short quiz for each of the course sections and a cumulative Final Exam.
When you are ready to Register, please add the course to your Cart and follow the steps to Checkout to complete your enrollment. When your online registration and payment are received, you will be sent your course materials and access to the Course Online Classroom.
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more | 677.169 | 1 |
New National Framework Mathematics 8 Plus
Synopsis
New National Framework Mathematics 8 Plus by M. J. Tipler
There are three separate workbooks available for each year to accompany the *, Core, and Plus pupil books. Each topic starts on a new page with pupil friendly objectives and a worked example, followed by challenging differentiated questions. | 677.169 | 1 |
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8 MB|13 pages
Product Description
Important Properties That Form The Basis Of PreAlgebra!
Often, the properties are taught in isolation in math. I never understood why we learned them. When it came time to simplify algebraic expressions by combining like terms, I had a very weak understanding of what I was doing and I did horribly with them. I have learned that with my own students, it's very wise to help the students see that the properties are actually the "toolbox" we need in Algebra that allows us to "do what we can do" when simplifying algebraic expressions. It actually puts the joy back into teaching them for me and learning them for my students.
Included With This Product:
I have included a full-sized two page study card that includes the explanations and both numerical and algebraic examples of the following: The Commutative Property, The Associative Property, The Identity Property, The Zero Property of Multiplication, The Inverse Property, The Distributive Property, and The Additive Inverse Property. I have also created "mini-posters" to display in your class that go along with the study card. An implementation page gives ideas of how to use the cards for review and for IEP and RTI modifications.
Special Note:
If you do not need the upper-level properties or algebraic examples, please see my product The Power of Properties Study Cards. These two products are very similar so please purchase wisely!
Use this tool for Infinite Possibilities!
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Please click beside the green star under my store name to follow me. You will get a notification when new products are released. More good stuff to come! | 677.169 | 1 |
Syllabus
Purpose
The goal of this class is to assist students in developing skills, techniques and strategies for problem solving. These skills will be needed for future advanced math and science classes. Thus there needs to be mastery, which only comes from practice and a lot of it.
Don't be afraid to make mistakes. Mistakes are a regular part of learning algebra and you must correct your errors in order to learn from your mistakes.
Objectives
Number theory, Numbers and the Number Line
Integers and Order of operations
Arithmetic Properties
Interpreting Algebraic Expressions
Writing Algebraic Statements and Equations
Solving Equations
Solving Word Problems
Functions
Graphing and the Cartesian Coordinate System
Slope and the Rate of change
Systems of Equations
Linear Inequalities
Polynomials
Factoring
Factoring Trinomials
Solving Quadratic Equations
Using Quadratic Formula
Radicals
Rational Expressions
Skills
Learn to use the calculator as a tool
Master problem-solving strategies
Develop graphing techniques
"There's an old saying, "Repetition is the Mother of Skill," meaning that in order for you to improve or increase your performance, you need to practice the fundamentals over and over again." | 677.169 | 1 |
Algebra Textbooks:: Homework Help and Answers:: Slader provides information on algebra 1, algebra 1 textbook, holt algebra 1, mcdougal littell algebra 1, glencoe.Hundreds of thousands of students visit Slader.com each week to help them with their homework.Lois eventually decided to slader homework help an overseas assignment highest power setting in are being performed together slader homework help a rock at being done.
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Description
Taking pre-algebra? Then you need the Wolfram Pre-Algebra Course Assistant. This definitive app for pre-algebra—from the world leader in math software—will help you work through your homework problems, ace your tests, and learn pre-algebra concepts. Forget canned examples! The Wolfram Pre-Algebra Course Assistant solves your specific pre-algebra problems on the fly.
This app covers the following pre-algebra topics:
- Find the divisors and prime factorization of a number
- Calculate the GCD and LCM of two numbers
- Determine the percent change
- Reduce and round numbers
- Evaluate expressions
- Solve equations and simplify expressions
- Convert units of length, area, volume, and weight
- Compute the mean, median, and mode of a dataset
- Plot equations on the coordinate plane
- Graph inequalities on a number line
- Calculate the area, surface area, or volume of a geometric figure
- Find the midpoint, slope, and distance between two points
The Wolfram Pre-Algebra Course Assistant is powered by the Wolfram|Alpha computational knowledge engine and is created by Wolfram Research, makers of Mathematica—the world's leading software system for mathematical research and education.
The Wolfram Pre-Algebra Course Assistant draws on the computational power of Wolfram|Alpha's supercomputers over a 3G, 4G, or Wi-Fi connection | 677.169 | 1 |
Mathematics at Leaving Cert
The Maths syllabus is been revised as part of a revision of the whole maths course and a new Project Maths syllabus is being introduced for the first time.
Project Maths involves changes to what students learn in mathematics, how they learn it and how they will be assessed.
Project Maths aims to provide for an enhanced student learning experience and greater levels of achievement for all. Much greater emphasis will be placed on student understanding of mathematical concepts, with increased use of contexts and applications that will enable students to relate mathematics to everyday experience.
General Notes for the exam:
Write your answers in the spaces provided in this booklet. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part.
The superintendent will give you a copy of the booklet of Formulae and Tables. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.
Marks will be lost if all necessary work is not clearly shown.
Answers should include the appropriate units of measurement, where relevant.
Answers should be given in simplest form, where relevant.
Is Maths a compulsory subject?
Maths is a compulsory subject in every school. You need to obtain a minimum of a pass mark at ordinary level for entry into the majority of college courses.
For Leaving Cert maths is offered at three different levels;
Higher
Ordinary
Foundation
Did you know that more and more courses are now accepting a pass in foundation level Maths?
Before you apply for a course it is crucial that you make sure that you meet the Maths entry requirements for that course. To more info visit
What level should I take?
This depends on a number of factors:
The level that you are the most capable for.
If you have done well at Junior Certificate Maths this usually bodes well for Leaving Cert
Do you require maths for the course of your choice at Third Level
Are you good at Maths and could the extra effort result in you doing well enough to avail of the new Bonus Points
As you go into 6th year if you feel you are more comfortable at Ordinary Level you can drop down to it, having completed one year at Higher level should stand to you at Ordinary Level
Why should I take higher level Maths?
There are very few students completing Maths at Higher Level with only 15.8% of students taking Maths at higher level in 2011. It is considered the most time consuming subject of all. It generally
Takes up more time than the other subjects you study to obtain a good result.
The new Bonus points for Higher level maths will provide an added incentive for people who are good at Maths to stay on and complete Higher Level maths. Bonus Points for Higher Level Mathematics has been Implemented by Universities, DIT and RCSI.
Did you know a minimum of a C3 in Higher Level Maths is a requirement for nearly 100 third level CAO courses!
Bonus Maths points are also in place for students who wish to study liberal arts or early childhood care and education at Mary Immaculate College.
We would advise that Higher-level Maths should not be taken by a student unless they are confident in their mathematical ability.
Will I receive bonus Maths points?
Students who take the higher level Maths paper in the Leaving Cert from June 2012 (to 2014, at which point the scheme will be reviewed) and achieve a D3 or higher win receive 25 bonus points. If you get an A1 in Higher Level maths you will an additional 25 Bonus points. These bonus points are available for most Third Level Colleges in Ireland.
A student with a D3 in higher level Maths will achieve the usual 45 points and a bonus of 25 points, bringing their total to 70points.
A student with a A1 in higher level Maths will achieve the usual 100 points and a bonus of 125 points, bringing their total to 125 points.
It must also be noted that when calculating your Leaving Cert points, you must only include your best six subjects (your six highest grades). The bonus points will only be relevant in cases where the subject HL mathematics (including bonus points) is scored as one of the candidate's six best subjects for points purposes. Consequently, if HL mathematics (cumulative points score) is not among these six subjects, the bonus points will not be included in the total points score.
Therefore if Maths is not one of your top six subjects, you cannot count the bonus points either
What if I find Higher Level Maths too difficult?
If you are finding it too hard, the sooner you realise it the better. If you have tried Higher level maths in general Ordinary level maths will come a lot easier to you and you will be very confident of doing well at Ordinary level. By stepping down to Ordinary level maths you will get more time to study and maximise your performance in other subjects, this could benefit you massively.
Remember it's all about getting the balance right with all your subjects.
If you are tethering on the brink of getting a D in Higher Level and if there is a slight possibility of failing higher level, this will have serious consequences and should not be risked.
I would love a career in Science or Engineering but I am not a strong Maths student. What are my options?
There are also courses such as preliminary engineering in GMIT and DIT.
These courses allow for students who have not met the entry requirements (such as higher level Maths or an appropriate science subject). The course prepares students for progression into degree courses.
In addition there are also higher certificate programmes which last two years that you can do prior to an honours degree. To gain entry into these courses you must have obtained a D3 in ordinary level Maths. To progress into an honours degree you must maintain 60pc or higher throughout these programmes, it is a longer route but it caters for the a weaker Maths student.
Project Maths
The first exam for all students in Project Maths will be 2012, the differences from this exam to previous exams are Paper 2 will take on a new style while Paper 1 will be stay along the traditional lines
The major differences between the papers will be that on Paper 2 all questions must be attempted (unlike on Paper 1), and that on paper 2 up to half the marks will go for completely unseen problems, requiring methods drawn from the full length and breadth of the course.
As this is the first year for all students we have very few examples however we are selling excellent Revision notes which should help you prepare better for Project Maths see a sample here.
Section B:
Contexts and Applications questions will more than likely involve real-life situations which you have not analysed mathematically before. For each such question, your tool-box for solving the problem is the entire course on probability. Geometry and trigonometry. You will have to make huge decisions as to what constitutes a suitable approach.
To successfully deal with such questions, you first need to master the use of all the tools in the toolbox.
You need to understand the ideas behind them and their suitability for different problems and not just learning methods off by heart.
The key to Project Maths is understanding:
The more you understand the less you have to learn, and the better you will be prepared for Section B questions on Paper 2.
If students attempt the a and b parts of questions they should pass the paper , the c parts of questions generally determine who gets A's and B's.
Students should attempt all parts of questions as attempt marks are given and examiners are willing to give marks if a concerted effort is made. | 677.169 | 1 |
Each section has an interactive quiz. Algebra is often taught abstractly with little or no emphasis on what algebra is or how it can be used to solve real problems. Just as Refulations can be translated into other languages, word problems can be "translated" into the math language of algebra and easily solved. Real World Algebra explains this process in an easy to understand format using cartoons and drawings. This makes self-learning easy for both the student and 2004 nissan 350z performance teacher who never did quite understand algebra. Includes chapters on algebra and money, algebra and geometry, algebra and physics, algebra and levers and many more. Designed for children in grades 4-9 with higher math ability and interest but could be used by older industrial regulations 851 and adults as well.
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ISBN: 0801890128
Books : Misc. Educational : English
Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys 2nd Edition
Author: David Joyner
The Johns Hopkins University Press 2008-12-01
ISBN: 0801890128 328 pages PDF
This updated and revised edition of David Joyner's entertaining "hands-on" tour of group theory and
abstract algebra brings life, levity, and practicality to the topics through mathematical toys.
Joyner uses permutation puzzles such as the Rubik's Cube and its variants, the 15 puzzle, the Rainbow
Masterball, Merlin's Machine, the Pyraminx, and the Skewb to explain the basics of introductory algebra
and group theory. Subjects covered include the Cayley graphs, symmetries, isomorphisms, wreath
products, free groups, and finite fields of group theory, as well as algebraic matrices, combinatorics,
and permutations.
Featuring strategies for solving the puzzles and computations illustrated using the SAGE open-source
computer algebra system, the second edition of Adventures in Group Theory is perfect for mathematics
enthusiasts and for use as a supplementary textbook.
From the preface:
This book grew out of a combined fascination with games and mathematics, from a desire to marry 'play'
with 'work' in a sense. It pursues playing with mathematics and working with games. In particular,
abstract algebra is developed and used to study certain toys and games from a mathematical modeling
perspective. All the abstract algebra needed to understand the mathematics behind the Rubik's Cube,
Lights Out, and many other games is developed here | 677.169 | 1 |
Products - Overview of Math Academic Instructional Software Titles
proven, curriculum-based family of software titles for
Grades 6-12 math
. The series helps students build math skills, gain confidence, and boost achievement
both in the classroom and on standardized tests.
The Math Tutor series has three components - the CONCEPTS AND SKILLS SERIES,
the PRACTICE AND REVIEW SERIES, and the SPECIAL TOPICS SERIES.
The packages in each of the series feature self-paced, interactive math tutorials that help students learn
and master the concepts of math and build proficiency in math problem solving. Algebra 1 in the Concepts and Skills Series
is the
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in the series have achieved similar success.
Designed by math educators and in accordance with math curriculum standards, the titles in the
Math Tutor series provide dynamic and interactive math lessons that are suitable for students
at all levels of ability. The series provides students with the highest quality instructional
software anywhere.
The titles in the Math Tutor series have been acclaimed for their educational excellence and
flexibility. They can be used for classroom instruction, self-study, remediation, enhancement,
and for preparation for statewide assessments and the SAT and ACT exams.
CONCEPTS AND SKILLS SERIES
The Concepts and Skills Series features interactive, self-paced tutorials that allow students
to master the concepts of math and build proficiency in problem solving. The programs
in the series use graphics and animation to engage students in the learning experience.
The series is unparalleled in its ability to bring clarity to even the most challenging
concepts of math.
The Practice and Review Series helps students build problem solving skills through a wide variety
of problem solving exercises. In addition, students are able to review
concepts and principles, and see detailed problem explanations. Designed by educators,
the series is the most comprehensive drill and practice software available anywhere.
The Special Topics Series contains two programs covering special areas within the math
curriculum. One helps students gain proficiency in business and consumer
math. The other helps students prepare for the math section of the SAT I and ACT exams. | 677.169 | 1 |
Musical Artist & Twitter - Trigonometric Function Task
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Your favorite musical artist has just released a new single. If you haven't heard it yet, you will likely hear it more than enough times over the next several months. We all know that a hit song has the potential to end up on the radio dial for months at a time, but what effect does the popularity of a hit song have for an artist on a social media platform such as Twitter? Are the effects lasting, or are they as temporary as the popularity of the song itself? This mathematical task gives students the opportunity to further explore and understand trigonometric functions in a "real world" situation. Students will utilize 21st century skills as they incorporate tables and graphs from computer software programs into their official report. Students will have the ability to be creative and create an engaging presentation as they explain their results from the function they have developed. Students will be asked to analyze their information and support and defend their reasoning.
Great to use as a long term project or to spend a few days in the computer lab to get students accustomed to the computer tools in GeoGebra. This pdf file includes 1 title page, 2 pages of directions, 1 rubric for grading purposes, and 5 pages of a sample answer key. | 677.169 | 1 |
Pages mathematical | 677.169 | 1 |
In this work, we investigated first-year university students' skills in using the limit concept. They were expected to understand the relationship between the limit-value of a function at a point and the values of the function at nearby points. To this end, first-year students of a Turkish university were given two tests. The results showed that the students were able to compute the limit values by applying standard procedures but were unable to use the limit concept in solving related problems | 677.169 | 1 |
published:28 Apr 2017
views:10655published:29 Sep 2016
views:705619 Nov 2015
views:21252
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!Calculus 1 Lecture 1.1: An Introduction to Limits Calculus? (Mathematics)Calculus - Introduction to CalculusCalculus: What Is It? Michael Harrison
Michael Harrison received his BS in math from C...
published: 29 Sep 2016 19 Nov 2015 -- The foundation of modern science
This calculus review video tutorial provides an introduction / basic overview of the fundamental principles taught in an IB or AP calculus AB course. This video is also useful for students taking their first semester of college level calculus. It cover topics such as graphing parent functions with transformations, limits, continuity, derivatives, and integration. This video contains plenty of examples and practice problems.
Here is a list of topics:
1. Graphing Parent Functions With Transformations
2. Linear Functions, Quadratic & Cubic Functions
3. Rational Functions, Fractions, Square Roots, and Radical Functions
4. Exponential, Logarithmic and Trigonometric Functions
5. Limits - DirectSubstitution, Factoring, Multiplying by the conjugation Given Radicals and Multiplying by the...
published: 01 Jan 2017 geom...What is Calculus? (Mathematics)
What is calculus? In this video, we give you a quick overview of calculus and introduce the limit, derivative and integral.
We begin with the question "Who inv...Calculus - Introduction to Calculus
This video will give you a brief introduction to calculus. It does this by explaining that calculus is the mathematics of change. A couple of examples are pre...Calculus: What Is It?
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wonderi...
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here! jus... calculus? In this video, we give you a quick overview of calculus and introduce th...This video will give you a brief introduction to calculus. It does this by explaining tha...This video shows how calculus is both interesting and useful. Its history, practical uses,...9:05
Calculus I in 20 Minutes (The Original) by Thinkwell
Want to see the ENTIRE Calculus in 20 Minutes for FREE? Click on this...Burden Of Grief
The war is over The last battles are gone Swords laying broken My bloodwork is all done I sit down for calming My breath is lessening I�m starting to tremble My sight is clearing My head is weary A dreadful awakening What has driven me Into insanity Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger back on the shelf I look around As I raise from my rest Discover what I�ve done No life I have left My heart is in pieces My soul is laying bare Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegy Put my anger on the shelf Awaking from this dreadful tragedy I return to myself Beginning to dwell in this elegyelsey DeFrates is pursuing her passion for engineering in medicine at Rowan. Kelsey DeFrates lives for research ... DeFrates applied for the Goldwater in December, using research from Dr ... During her time at Audubon High School, DeFrates wanted to be a doctor, but quickly found her passion for the "hard science and math" behind the medicine in chemistry, physics and calculus, leading her to enroll as an engineering student in the Henry M ... ....
Europe's residents have genuine reason to be wary of the risk of terrorism in the streets, stadiums , markets and concert-halls . But how President Macron and the Philippe Cabinet respond to that risk is crucial to hard-won liberties and rights not just in France, but across the rest of the continent ... The first question to ask is whether France even needs a new counterterrorism law ... The political calculus is simple ... ....
Even as politicians argue over how to create or keep "good jobs" in the U.S., a recent National Federation of Independent Businesses survey reported that the percentage of small businesses saying that they get no or few qualified applicants for available jobs has hit a 17-year high ... Just to be considered for an apprenticeship, a student must have taken advanced mathematics courses (such as calculus) and pulled off A's and B's ... ....
The Chicago Bulls have reportedly agreed to trade Jimmy Butler and the No. 16 pick (JustinPatton) in the NBA draft to the Timberwolves in exchange for Zach LaVine, KrisDunn, and the No. 7 pick (Lauri Markkanen). With the draft day blockbuster in the books, The Crossover is grading both sides of the deal ... A ... NBA ... D ... There is some theoretical sense to the way the Bulls have changed shape, if not so much in the talent calculus of this deal ... ....
No matter what Pedro Martinez tweeted a few weeks back, David Ortiz is really retired. Further proof arrives Friday night, when the Red Sox will put his No ... 1. Miguel Cabrera's No ... I do think Justin Verlander has a strong Hall case, but, because of the specifics of pitcher usage in his era (i.e., guys don't rack up the wins and innings totals of eras past), that will require a change to the typical Cooperstowncalculus ... 2 ... 3 ... 4 ... 5 ... Louis ... 6 ... 7 At the suggestion of a school administrator, Chacha enrolled in several classes at Passaic County Community College, including calculus and engineering ... ....
American Software, Inc. (NASDAQ.AMSWA). Q4 2017 Earnings Conference Call. June 22, 2017 17.00 ET. Executives. Vincent Klinges - CFO... Analysts ... Operator ... I realize that your fairly strategic acquirers -- when that does happen, clearly you're sitting out of pretty nice cash position here; so I'm just wondering if that changes the calculus at all now or if you're actively evaluating things or really no -- increased motivation at this point? ... ....
Bay Area political events. Climate change, diplomacy Political events Climate change ... in San Francisco. Diplomacy talk ... A New Diplomatic Calculus?" The discussion is from 6.30 to 7.45 p.m ... Where money should go ... ....
Sharjah recently commenced construction of a science park that is to bring together investors, companies and scientists, as part of the emirate's drive to become a knowledge economy ...The answer is "no", for several reasons ... This is where the other reasons come in to play ... That's why scientists - especially the best ones - prefer inventing calculus to inventing a tablet PC, even if the resulting financial windfall is modest.... | 677.169 | 1 |
Similar faster courseThis textbook is an introduction to algebra via examples. The book moves from properties of integers, through other examples, to theAlgebra I For Dummies, 2nd Edition (9781119293576) was previously published as Algebra I For Dummies, 2nd Edition (9780470559642). While this version features a new Dummies cover and design, the content is the same as the prior release and should not be considered a new or updated product.
ThereIncludes revised and updated examples and practice problems Provides explanations and practical examples that mirror today's teaching methods Other titles by Sterling: Algebra II For Dummies and Algebra Workbook For Dummies
A solutions manual to accompany An Introduction to Numerical Methods and Analysis, Second Edition
An version System of the World by Isaac Newton. Sir Isaac Newton (1642–1727) was an English physicist and mathematician who is widely recognised as one of the most influential scientists of all time and as a key figure in the scientific revolution.
This great work supplied the momentum for the Scientific Revolution and dominated physics for over 200 years.
It was the ancient opinion of not a few, in the earliest ages of philosophy, that the fixed stars stood immoveable in the highest parts of the world; that, under the fixed stars the planets were carried about the sun; that the earth, us one of the planets, described an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.
This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.
Written by two pioneers of the concept of math anxiety and how to overcome it, Arithmetic and Algebra Again has helped tens of thousands of people conquer their irrational fear of math.
This revised and expanded second edition of the perennial bestseller:
Features the latest techniques for breaking through common anxieties about numbers Takes a real-world approach that lets mathphobes learn the math they need as they need it Covers all key math areas--from whole numbers and fractions to basic algebra Features a section on practical math for banking, mortgages, interest, and statistics and probability Includes a new section on the graphing calculator, a chapter on the metric system, a section on word problems, and all updated exercises statistics, data analysis, psychology, cognitive science, social sciences, clinical sciences, and consumer sciences in business.
Accessible, including the basics of essential concepts of probability and random samplingExamples with R programming language and JAGS softwareComprehensive coverage of all scenarios addressed by non-Bayesian textbooks: t-tests, analysis of variance (ANOVA) and comparisons in ANOVA, multiple regression, and chi-square (contingency table analysis)Coverage of experiment planningR and JAGS computer programming code on websiteExercises have explicit purposes and guidelines for accomplishment
Provides step-by-step instructions on how to conduct Bayesian data analyses in the popular and free software R and WinBugs courseA no-nonsense, practical guide to help you improve your algebra II skills with solid instruction and plenty of practice, practice, practice
Practice Makes Perfect: Algebra II presents thorough coverage of skills, such as handling decimals and fractions, functions, and linear and quadratic equations, as well as an introducing you to probability and trigonometry. Inside you will find the help you need for boosting your skills, preparing for an exam or re-introducing yourself to the subject. More than 500 exercises and answers covering all aspects of algebra will get you on your way to mastering algebra! badA is for Algebra-and that's the grade you'll pull when you use Bob Miller's simple guide to the math course every college-bound kid must take
With eight books and more than 30 years of hard-core classroom experience, Bob Miller is the frustrated student's best friend. He breaks down the complexities of every problem into easy-to-understand pieces that any math-phobe can understand-and this fully updated second edition of Bob Miller's Algebra for the Clueless covers everything a you need to know to excel in Algebra I and II systemsThe third book in Peterson's NEW series of guides for visual learners, this volume covers basic algebra topics that are essential forThis best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra.
The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
A coherent introduction to the techniques for modeling dynamic stochastic systems, this volume also offers a guide to the mathematical, numerical, and simulation tools of systems analysis. Suitable for advanced undergraduates and graduate-level industrial engineers and management science majors, it proposes modeling systems in terms of their simulation, regardless of whether simulation is employed for analysis. Beginning with a view of the conditions that permit a mathematical-numerical analysis, the text explores Poisson and renewal processes, Markov chains in discrete and continuous time, semi-Markov processes, and queuing processes. Each chapter opens with an illustrative case study, and comprehensive presentations include formulation of models, determination of parameters, analysis, and interpretation of results. Programming language–independent algorithms appear for all simulation and numerical procedures. asThis incredibly useful guide book to mathematics contains the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Now in its fifth updated edition, it is easy to understand, and convenient to use. Inside you'll find the information necessary to evaluate most problems which occur in concrete applications. In the newer editions emphasis was laid on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes. For the 5th edition, the chapters "Computer Algebra Systems" and "Dynamical Systems and Chaos" have been revised, updated and expanded Edition
N second The book covers major areas of modern algebra, which is a necessity for most mathematics students in sufficient breadth and depth.
In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian (LCA) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of LCA groups are proved in the opening chapters, in order to make the treatment relatively self-contained"This book covers many interesting topics not usually covered in a present day undergraduate course, as well as certain basic topics such as the development of the calculus and the solution of polynomial equations. The fact that the topics are introduced in their historical contexts will enable students to better appreciate and understand the mathematical ideas involved...If one constructs a list of topics central to a history course, then they would closely resemble those chosen here."
(David Parrott, Australian Mathematical Society)
"The book...is presented in a lively style without unnecessary detail. It is very stimulating and will be appreciated not only by students. Much attention is paid to problems and to the development of mathematics before the end of the nineteenth century... This book brings to the non-specialist interested in mathematics many interesting results. It can be recommended for seminars and will be enjoyed by the broad mathematical community."
(European Mathematical Society)
"Since Stillwell treats many topics, most mathematicians will learn a lot from this book as well as they will find pleasant and rather clear expositions of custom materials. The book is accessible to students that have already experienced calculus, algebra and geometry and will give them a good account of how the different branches of mathematics interact."
(Denis Bonheure, Bulletin of the Belgian Society)
This third edition includes new chapters on simple groups and combinatorics, and new sections on several topics, including the Poincare conjecture. The book has also been enriched by added exercises.
Due to the rapid expansion of the frontiers of physics and engineering, the demand for higher-level mathematics is increasing yearly. This book is designed to provide accessible knowledge of higher-level mathematics demanded in contemporary physics and engineering. Rigorous mathematical structures of important subjects in these fields are fully covered, which will be helpful for readers to become acquainted with certain abstract mathematical concepts. The selected topics are:
This book is essentially self-contained, and assumes only standard undergraduate preparation such as elementary calculus and linear algebra. It is thus well suited for graduate students in physics and engineering who are interested in theoretical backgrounds of their own fields. Further, it will also be useful for mathematics students who want to understand how certain abstract concepts in mathematics are applied in a practical situation. The readers will not only acquire basic knowledge toward higher-level mathematics, but also imbibe mathematical skills necessary for contemporary studies of their own fields. | 677.169 | 1 |
Function Machines
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This is a Do-it-yourself worksheet to help learners understand the concept of functions and get familiar with the different ways that we can use functions - function machines, as equations, tables or graphs. It also introduce terms like domain and range in a simple and easy way. | 677.169 | 1 |
College AlgebraJames Stewart, author of the worldwide, best-selling Calculus texts, along with two of his former Ph.D. students, Lothar Redlin and Saleem Watson, collaborated in writing this text to address a problem they frequently saw in their calculus courses: many students were not prepared to think mathematically but attempted instead to memorize facts and mimic examples. College Algebra was written specifically to help students learn to think mathematically and to develop true problem-solving skills. | 677.169 | 1 |
This activity is designed to help your Pre-Calculus Honors or College Algebra students evaluate sequences and series in an end-unit review for Discrete Mathematics. There are 24 task cards in the activity. Students will find recursive and explicit forms of sequences, find the sum of finite and infinite series, determine convergent and divergent series, find nth terms, partial sums, P(K+1) term for induction proofs, and more.
Conic Sections Cheat Sheet 2 Versions First version is a one page reference sheet for Parabola, Ellipse, Circle, and Hyperbola. This is great handout for worksheets and quizzes and great as a handout for upper level classes, such as Calculus. Use in Algebra 2 or PreCalculus. The second version is a Foldable version. Fold along dotted lines, cut for individual sections and paste into Interactive Notebooks. Organized in a way students can easily follow and learn. | 677.169 | 1 |
Year 9
Maths
Year 9 ALP Mathematics In Year 9 ALP Mathematics students extend their understanding of measurement, linear equations, straight-line graphs and their applications, probability and trigonometry and applications…
SUBJECT DESCRIPTION Year 9 Performing Arts is designed to give students an introduction to drama through practical group based and individual activities aimed to develop self confidence, effective communication…
COURSE DESCRIPTION In year 9 Music, students develop their listening, comprehension and compositional skills and refine their practice on an instrument. Therefore, those selecting this elective should have… | 677.169 | 1 |
Math algebra 1 help
Then you found the right place to get help. We have more than forty free, text-based algebra lessons listed on the left. If not, try the site search at the top of every page. Algebra 1 HelpClick your Algebra 1 textbook below for homework help. Our answers explain actual Algebra 1 textbook homework problems. Each answer shows how to solve a textbook problem, one step at a time
She was falling behind. Since subscribing to your program she has been 100% self-sufficient. After my son bombed the first quarter in Algebra 1, I decided to sign up. I wish I knew about you the last 2 years when my son was in Middle School. Your site is so user friendly and interactive. Everything is right there. Start learning. Modeling is an amazing world, full of challenges. In this topic, we will start to think about some general modeling concerns, before we dive into modeling situations with different kinds of functions and equa. | 677.169 | 1 |
Beginning Algebra (Expanded Edition)
by Stanley F. Schmidt Ph.D.
In five days of Fred's life, every aspect of beginning algebra pops up in our hero's life.
What it takes to get drafted into the army at age 6
New milkshake marketing techniques
About enjambment in the colonel's library
. . . and algebra
Older editions of Beginning Algebra had a companion book. This edition replaces both Life of Fred: Beginning Algebra and Fred's Home Companion: Beginning Algebra. All problems are completely worked out.
Zillions of Practice Problems Beginning Algebra
by Stanley F. Schmidt Ph.D.
Need a lot of practice or stuck on a particular kind of problem? This book has been requested by many readers. Keyed directly to the chapters and topics of Life of Fred: Beginning Algebra. Each problem worked out in complete detail. Eleven mixture word problems are each worked out step by step, often using a whole page of explanation for each problem. Thirteen quadratic equations solved by completing the square. Thirteen examples of two equations with two unknowns solved by the elimination method. This book is mandatory for those who need it. The Zillions of Practice Problems Slogan: If your cat can work through all the problems in this book, your cat can teach Beginning Algebra at any school in the nation.
This book replaces both Life of Fred: Advanced Algebra and Fred's Home Companion: Advanced Algebra. This book has all the problems completely worked out, which wasn't true in the old books. It costs less, too.
Zillions of Practice Problems Advanced Algebra
by Stanley F. Schmidt Ph.D.
Need a lot of practice or stuck on a particular kind of problem? This book has been requested by many readers. It is keyed directly to the chapters and topics of Life of Fred: Advanced Algebra. Each problem is worked out in complete detail.
Advanced Algebra (Fred's Home Companion)
by Stanley F. Schmidt Ph.D.
Multiple Uses!
Lesson plans for those studying Life of Fred: Advanced Algebra on their own. Each lesson offers you a "daily helping" of Fred.
Lecture notes for those teaching Life of Fred: Advanced Algebra. Outlines for each lecture. Problems to present at the blackboard that are not in the textbook. Additional insights to present in class. Quiz and test material.
Answer key for Life of Fred: Advanced Algebra.
Additional exercises for those who want more drill. All answers are included.
This book replaces both Life of Fred: Trigonometry and Fred's Home Companion: Trigonometry. This book has all the problems completely worked out, which wasn't true in the old books. It costs less, too. | 677.169 | 1 |
Mathematics is a compulsory subject for all Year 9-11 students. In each year we cover the 3 main strands of Number, Algebra, Measurement, Geometry and Statistics. The course is cyclic and each year we build on work from the previous years. In year 12 (level 2) and 13 (level 3) mathematics is optional but as many courses require level 2 or 3 as an entry qualification, it is wise to take mathematics are far as you are able to.
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Recently we have had a few students from Waitara High Compete in the maths spectacular '10 | 677.169 | 1 |
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A Guided Tour of Mathematical Methods
A Guided Tour of Mathematical Methods By:"Roel Snieder" Published on 2004-09-23 by Cambridge University Press
Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks, all the material is presented in the form of problems. Within these problems the basic mathematical theory and its physical applications are well integrated. The mathematical insights that the student acquires are therefore driven by their physical insight. Topics that are covered include vector calculus, linear algebra, Fourier analysis, scale analysis, complex integration, Green's functions, normal modes, tensor calculus and perturbation theory. The second edition contains new chapters on dimensional analysis, variational calculus, and the asymptotic evaluation of integrals. This book can be used by undergraduates and lower-level graduate students in the physical sciences. It can serve as a stand-alone text, or as a source of problems and examples to complement other textbooks. | 677.169 | 1 |
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This is a short outline of the objectives and keywords (or vocabulary) for an introductory unit on solving linear equations. This is the beginning level of solving with pretty simple equations using the addition and multiplication rules. Good for a teacher at any level to use when planning lessons. I have used these for students in grade school as well as those who are in remedial math at the community college level. | 677.169 | 1 |
Two power point presentations reviewing the Differential Calculus learnt in Year 11. The basics, power rule for differentiation, interpretation of the gradient function, equations of tangents and normal, decreasing and increasing functions. Examples modelling typical examination questions.
Converting between degrees and radians. Solving trig equations over a specified domain. General solutions to trig equations. Using zeros and sum of zeros commands. Ferris wheel application question with CAS. Graphs with display in terms of pi. A zipped file containing a word document and tns files.
Introduction to sketching parabolas in Year 9. Class activities to learn the basic shape of a parabola with simple transformations; determine the position of a vertex and axes intercepts. Simple assessment task included.
Fitting curves using direct and inverse variation. Modelling with triangular numbers resulting in a quadratic (using first and second differences to confirm) and with light intensity with a power function. Can be used as an assessment or as class activity. Use of technology is encouraged. Solutions provided.
Functional equations and functional identities with the range of functions. Using CAS and algebraically. Illustrated with past exam questions and typical examples. Notes pages to speed up answering MC questions. Followed by the test on Functions and Graphs. Solutions and MS included. | 677.169 | 1 |
Algebraic Expressions Worksheets
Many students find algebraic expressions difficult. At TuLyn, we created worksheets on algebraic expressions to help you better understand algebraic expressions. Our algebraic expressions worksheets are in pdf format and available for students, parents, and educators to print out. Are you looking for algebraic expressions printables?You no longer need to search for a source. We have tens of completely free worksheets on algebraic expressions and hundreds on other math topics.
Below is the list of all worksheets we have on algebraic expressions. We have designed each worksheet with different difficulty level. Check all the worksheets and print the ones you think suitable for your level. Our algebraic expressions worksheets are downloadable and printer friendly. And as long as for educational purposes, you can distribute them freely.
Algebraic Expressions Worksheets
Combining Like Terms Video Clip
This tutorial shows how to combine like terms in given polynomial, the variables, and the constants. You will learn to combine the coefficients of the same variable as well as pay attention to the signs of coefficients, for there are different rules for different signs and same signs.
Are you a math teacher looking for free algebraic expressions printables? We have tens of worksheets on algebraic expressions. If you are a teacher or parent, you can also request the solutions to the questions of the worksheet. You can also tell your students to visit TuLyn for more practice. | 677.169 | 1 |
Do you want easy access to the latest methods in scientific computing? This greatly expanded third edition of Numerical Recipes has it, with wider coverage than ever before, many new, expanded and updated sections, and two completely new chapters. The executable C++ code, now printed in colour for easy reading, adopts an object-oriented style particularly suited to scientific applications. Co-authored by four leading scientists from academia and industry, Numerical Recipes starts with basic mathematics and computer science and proceeds to complete, working routines. The whole book is presented in the informal, easy-to-read style that made earlier editions so popular. Highlights of the new material include: a new chapter on classification and inference, Gaussian mixture models, HMMs, hierarchical clustering, and SVMs; a new chapter on computational geometry, covering KD trees, quad- and octrees, Delaunay triangulation, and algorithms for lines, polygons, triangles, and spheres; interior point methods for linear programming; MCMC; an expanded treatment of ODEs with completely new routines; and many new statistical distributions. For support, or to subscribe to an online version, please visit Most comprehensive book available on scientific computing, now updated • New routines for classification and inference HMMs and SVMs, computational geometry, ODEs, interior point methods for linear programming, and MCMC • Over 600,000 Numerical Recipes products in print
Reviews
'This monumental and classic work is beautifully produced and of literary as well as mathematical quality. It is an essential component of any serious scientific or engineering library.' Computing Reviews '… an instant 'classic,' a book that should be purchased and read by anyone who uses numerical methods …' American Journal of Physics
'… replete with the standard spectrum of mathematically pretreated and coded/numerical routines for linear equations, matrices and arrays, curves, splines, polynomials, functions, roots, series, integrals, eigenvectors, FFT and other transforms, distributions, statistics, and on to ODE's and PDE's … delightful.' Physics in Canada
'… if you were to have only a single book on numerical methods, this is the one I would recommend.' EEE Computational Science & Engineering
'This encyclopedic book should be read (or at least owned) not only by those who must roll their own numerical methods, but by all who must use prepackaged programs.' New Scientist
'These books are a must for anyone doing scientific computing.' Journal of the American Chemical Society
'The authors are to be congratulated for providing the scientific community with a valuable resource.' The Scientist | 677.169 | 1 |
Geometric Analysis combines differential equations with differential geometry. An important aspect of geometric analysis is to approach geometric problems by studying differential equations. Besides some known linear differential operators such as the Laplace operator, many differential equations arising from differential geometry are nonlinear. A particularly important example is the Monge-Amperè equation. Applications to geometric problems have also motivated new methods and techniques in differential equations. The field of geometric analysis is broad and has had many striking applications. | 677.169 | 1 |
A student first learning physics usually faces two inter-related tasks that are difficult to unwind and see as separate fundamentals. Older students and practitioners rarely go back and address the assumptions they made at the very start, so the intuitive leaps they made rarely surface for examination.
Besides the early concepts of kinematics and Newton's forces, a student must also learn how to represent things of interest in the mathematical language they are expected to use. It is the issue of object representation that will be addressed by this paper. The objects to be represented are the typical concepts of physics. The reader is assumed to be knowledgeable about enough of them to follow the argument.
Early students are assumed to know the tools of algebra, geometry, calculus, and parts of linear algebra. After a time, the student usually picks up more knowledge concerning linear and matrix algebra, differential equations, complex numbers and various specialty functions and polynomials.
While each of the added mathematical tool sets greatly expands a student's capabilities for problem solving, they tend to get appended through the working of example problems to which they are powerfully adapted. It is rare for a student to face an unsolved problem to which they must find the appropriate tools until they are at the graduate level of their training. Even then, many never face this task.
This paper focuses upon how we use the mathematical tools we choose. It asks the questions 'How do we choose to represent that which we perceive?' and 'How do we know others are perceiving the same objects we are trying to address?' Which tools we use depends a great deal on our knowledge of them and the depth of our experience with them. This paper assumes the question of why we choose our tools has much to do with simplicity, power, and convention and pays it no further attention.
History
There is historical evidence of changes to the primary tool sets used for the oldest branches of physics. Newton crafted the contents of his Principia with geometry and his new calculus. Modern students use algebra, calculus, and vectors for the same material and usually find the Principia difficult to decode. Maxwell's description of Electric and Magnetic fields was radically different from what is taught today. With modern vectors and matrices, electromagnetism is hard to recognize for those who put forth the effort to read the older texts.
Whether Newton or the others would have used modern tools had they been available is not the question. The fact is, modern practitioners moved away from older approaches in favor of other means to represent much the same concepts. How they did so is worth some consideration because modern instructors rarely bother to teach or even mention the older approaches. Something important has occurred and the existence of these changes suggests it is worthwhile to consider the foundation of our approaches to object representation.
An Example
Imagine a sharp, straight stick on the ground. We wish to represent the length and orientation of the stick mathematically. There is more than one tool that can be used, but modern students will almost always reach for vectors once they have learned how to use them. Two will be shown here.
Vector Technique
Choose a reference frame. Pick an origin for it and then write down some information about the directions of your basis vectors relative to each other. You have quite a bit of freedom, but most people pick right-handed systems and make the basis vectors have a length of one unit and stand at right angles to each other.
Pick one of the basis vectors. Imagine a light bulb that projects a shadow of the sharp end of the stick onto a line that extends through your basis vector from the origin outward. Put a slit in front of the bulb. Arrange the slit so the light it allows out moves toward the stick parallel to the other basis vector you didn't choose. How long would your basis vector have to have been to be long enough to just reach the projected shadow? Write down that number.
Do the light trick again swapping your basis vectors and write down that number.
Multiply your first number by the first basis vector and add it to the product of the second number with the second basis vector. This vector points at the sharp end of the stick.
If you didn't pick your origin at the blunt end of the stick, do both projections and the addition trick again for the blunt end. This vector points at the blunt end of the stick.
The length and direction of the stick can be represented by the difference of the two vectors. Take the sharp end vector and subtract the blunt end vector and you are done.
Algebra Technique
Choose a coordinate system. Pick an origin for it and then write down some information about the directions and scale of your coordinate axes. You have quite a bit of freedom, but most people pick right-handed systems and make the axes stand at right angles to each other.
Draw out a coordinate grid using the axes you chose. Find the coordinates of both ends of your stick.
Find the straight line equation that goes through both of the points you found earlier.
The stick is represented by the straight line function over a domain and range limited to the span between the two coordinate points. The distance formula between two points gives the length while the domain and range spans start at the blunt end to give the orientation.
These two approaches may sound the same, but they are not. In the first, a vector represents the stick. In the second, a function represents the stick. Both of them have different descriptions for the reference frames used. There are some similarities, of course. This shouldn?t surprise anyone since both techniques are being used to try to describe the same properties of the stick
If the reader is still unsure that these two techniques and their supporting tools are different, imagine expanding the example in the following way. Suppose we also try to describe the size and orientation of the cross-sectional area of the same sharp stick. Vector proponents would be tempted to write another vector oriented normal to the cross-sectional area. Algebraic proponents might be tempted to do the same, but they could also represent the functional equivalent of the cylindrical or conical surface area surrounding the stick and use formulas to demonstrate the cross-sectional area. Both techniques work, so one cannot claim to be superior than the other.
The Postulates
For the purpose of further discussion, we propose three postulates to act as an abstraction of the representation techniques and intentions of the users. The postulates make no attempt to describe any physics. Instead, their purpose is to address how we recognize and represent physically interesting things for our physics theories to address. If the abstraction works, it should be possible for the reader to write examples similar to the one above and see their techniques described by the postulates.
1: Completion terms
Complete Objects are represented as a list of observables representing each of the independent properties of an Object.
2: Rendering of representations
Properties are rendered as observables through representations as combinations of components in a suitably complex mathematical tool.
3: Identity terms
Representations of an observable describe the same property of an Object if under an agreed upon family of passive transformations suitably defined for the mathematical tool, all Observers can passively transform their rendition into the renderings of the other Observers. This is not invariance.
Terminology Dictionary
Object
An Object is a physically interesting thing. It is the subject of theory and experiment. It is a part of reality singled out for further attention.
Property
A property is a part of an Object describing some definable portion we might observe and test through experimentation.
Observable
An observable is related to a property and is the direct result of a rendering of that property into an appropriate mathematical tool. The observable is what actually gets tested in experimentation because the experimenter must make assumptions about the property being tested. Those assumptions are embodied in the observable.
Independence
Independence is a concept useful for distinguishing properties of an Object from each other. An example of two properties that are not independent of each other might be color and the reflection coefficient as a function of energy for a body bathed in sunlight.
Component
A component of a math tool is defined relative to that tool. In geometry of two dimensions, line segments, area segments and points make up some basic components. In algebra, polynomials of order N make up an unending list of components. In a geometric algebra, the generators of an algebra and their products make up the available components.
Combination
A combination is a sum and products of components scaled by elements from a field such as real or complex numbers. If this definition were limited to sums, it would be equivalent to the standard meaning used for 'linear combination.'
Representation
A representation is a particular rendering of an observable within the mathematical tool of choice. There are often many ways to represent the same observable.
Transformation (passive)
A passive transformation is one that alters the representation of an observable without doing anything to the observable itself. These transformations are usually more a measure of what the Observer is doing that changes the representation than anything else.
Observer
An Observer is someone or some thing that can definitively answer questions about their representations of observables. The typical observers are the experimenters trying to understand an object.
Postulate Support and Explanation
Postulate one is largely about knowing how to decompose an object to parts that may be rendered and tested separately, but it will most often be used in the other direction. It is possible and quite likely that observers will initially fail to understand the full scope of an object. Without this understanding, initial representations are likely to be partial. A strategy for completing a representation, therefore, is in order.
The completion strategy is to form a list of properties known to be associated with an object. This is what is done today. The length of the list depends on the experience of the observer and in what properties they are interested. The appearance of the list depends on the tool to be used. Most practitioners simply write a natural language list of the representations of the expected observables. To many, initially including this author, this postulate is assumed and goes by without thinking.
Postulate two addresses the issue of how properties are rendered as observables. Each mathematical tool is different enough to make a general discussion of representations quite generic and of little interest here. The proponents of the vector technique in our sharp stick example earlier rendered the length and orientation during steps two through four. The correctly sized components were found in the first two steps while the combination was created in the last one. The proponents of the algebraic technique rendered the length and orientation during step three. The components were suitably scaled monomials (order one and order zero) while the combination was the sum that created the straight line function.
Postulate three forms the basis of a strategy that may be used by multiple observers who wish to know if their renderings describe the same object. The strategy is not fully deterministic. Probabilistic knowledge, therefore, can enter into the concept of object identity long before it enters into the physical theories using the representations.
Consider our earlier example of the sharp stick. Imagine further that many observers create representations of the length and orientation and share them with each other. How does Observer A know that the representation offered by Observer B addresses the same properties of the same stick? Do they simply trust each other? Is there a method for creating identity in the representation that may eliminate the need for trust? In practice, trust is used, but it tends to undermine the purpose of experimentation in the scientific method.
The notion of identity can be added to the representation through an agreed upon family of passive transformations shared by all observers. Ask an observer to render the observable again and a different outcome may occur. After all, the observers were free to choose a reference frame before finding a suitable linear combination for their vector. Different representations would occur for each choice of reference frame. If one observer is asked to create many different representations of the same observable, another observer could discover the passive transformation family used by the first observer through a careful study of the renderings. If that family is found, and there exists a transformation in that family that turns the representation written by observer B into one written by observer A, observer B may be reasonably confident both renderings refer to the same object.
Postulate three does not lead to a bulletproof concept of identity. If one observer produces a representation that fails the test, the other observers can be certain that particular observer rendered a different property or object. If all representations pass, though, all the observers can say is they are pretty sure the renderings refer to the same object. The higher number of representations that pass, the higher the odds are that they do refer to the same properties of the same object.
Invariance
Any observable that proves to be invariant under the family of passive transformations used by all observers obviously qualifies to represent an object seen by all observers. The only block to knowing absolutely that a given representation refers to an object is the possibility that the observer might have been looking at a similar object elsewhere. Are you and I looking at the same sharp stick? What if we were instead trying to describe an electron?
Issues of identity are rarely raised if each observer sticks to representations that are invariant under the agreed upon transformations. This suggests that invariance is a suitable stand-in for identity much of the time. It also suggests that observers will choose between two equally powerful tools in favor of the one where invariance is most easily found and demonstrated. This last statement depends largely on the desire of most practitioners to avoid unnecessary work.
Whether the identity of an object is fully agreed upon by all observers or not, the invariance of a representation can be determined with certainty by any one observer and then promised to the others. The mathematical tool offers this certainty, so no trust is needed.
This promise between observers is probably at the root of how we choose our representations. Any supporters of newly developed tools must be able to offer at least this much if they hope to have their work displace older techniques. To test this notion, we can ask if the newer tools offer an equally strong or stronger sense of identity to the objects Newton originally represented with geometry for Principia. If the answer is yes, this premise survives its first test and might be worth consideration by a larger audience.
To determine the times of the descent of a body falling from a given place A
Upon the diameter AS, the distance of the body from the centre at the beginning, describe the semicircle ADS, as likewise the semicircle OKH equal thereto, about the center S. From any place C of the body erect the ordinate CD. Join SD and make the sector OSK equal to the area ASD. It is evident (by Prop 35) that the body in falling will describe the space AC in the same time in which another body, uniformly revolving about the centre S, may describe the arc OK.
(Proposition 35 addresses the notion that equal areas are swept out in equal times on a gravitational orbit.)
Newton used proposition 36 in book 3 to show how the rate at which the Moon falls toward the Earth can be directly formulated from our knowledge of how masses accelerate here on the surface of the Earth.
Early physics students learn a similar lesson when they are taught about gravity. These students are taught to replace geometric line segments with distances calculated from a Cartesian or similar coordinate system. The areas swept out (sectors OSK and ASD) can be shown too, but often aren't. The algebraic formula for the force of gravity is used directly in modern texts and the approach is considered sufficient.
Each of the objects represented in Newton's proposition can be translated to modern renderings using algebra. In each case, care can be taken to use the typical coordinate systems taught in the texts and the identities of each object get preserved. The invariants are much the same as they were with the geometric description. Lengths and angles do not get altered between observers. Time ticks at the same rate for all, else proposition 35 would fail. Young students are taught not to violate those invariants whether they realize it or not. In their acceptance, then, the concept of object identity is likely to be preserved and different observers can be confident in their use of each other's renderings.
Anyone who knows special relativity well enough should be able to see the beginnings of how revolutionary the new concepts were. The two postulates of special relativity strike directly at the heart of the invariants used in Newton's work. The representations written in tools capable of supporting special relativity fundamentally alter what a property is, let alone how its observable behaves.
Summary
How we choose to represent our objects can be summarized by the creation of a trustable identity for the interesting properties. Invariance provides the root for trust between observers. The rest of the rules concerning representation are technique that varies from one tool to the other.
As long as the practitioner takes steps to represent complete properties for objects whose identity can be trusted by other observers, they have done what the community expects of them. Most everyone except the users of statistical mechanics and its related subjects largely ignore the probabilistic nature of the identification.
Since this is a part of a larger work, I'm wondering if you give any thought to useful models of explanation? What you have here is an attempt at rigorous and exhaustive descriptiveness, but pure description falls short of a thorough explanation (and vice versa). To say that an object responds in a particular manner in a situation does not say what that object means at that time, and often times that meaning cannot be interchanged as easily as the description of the object itself. "Morning star" and the "evening star" may very well be the same planet Venus if the use of those phrases have the same referant, but the context of either of those aspects would be lost if expressed in purely mathematical terms. This is often expressed as the difference between an object's sense (or meaning, context, intention, connotation) versus its reference (or naming, description, extension, denotation).
Quine says it better: "The general terms 'creature with a heart' and 'creature with a kidney,' e.g., are perhaps alike in extension but unlike in meaning."
As a physicist, I tend to shy away from explanation and stick to descriptive representations that feed predictive theories.
Explanation is rarely approachable by the scientific method. How do I know when I'm right or wrong?
I know that explanation is needed, though, to build intuitions. I will occasionally stretch a bit and write analogies that can be used temporarily as explanations. Be ready to throw out the analogies at the drop of a hat, though. I may use them, but I don't BELIEVE any of them.
that's why i ask. They are such tricky things. And yet, in many ways, that is the whole point of description models. Formulae don't show relevance, importance, or purpose. It may answer, "Who, what, when, where, how" but it neglects "Why" and "So what?"
OK, I read it once and here as some first thoughts, in no particular order.
I found it a bit hard to follow
to tell you the truth, mainly because it seems to me to jump around many topics,
making allusions but not going sufficiently deeply into things.
Of course, that's not surprising given that this is only a short article.
I found your terminology bothered me a bit. You mix computer science concepts (rendering, objects) with physics concepts (observable) and mathematics concepts (symmetry).
Since I know a little bit about these subjects, I found I was fairly defensive towards
the nontraditional use of these words. This probably accounted for some of the difficulties I had reading.
To elaborate a bit, you use the word component in its cs meaning (part of a system)
to apply to the building blocks of mathematical representations, and later on your
discussion of symmetries alludes to the mathematical meaning of components (of a vector) when introducing symmetries and invariance. So you've jumped from a very abstract
and simplistic description of systems through a modern algebraic approach to maths/physics back into a simpler vector analysis problem, where the word component has a very
concrete but imho different meaning.
Overall, it feels a bit like you're trying to write a computer algorithm for the
thought processes of a modern physicist, who has to weed through his full conceptual
framework to solve a simple problem (your illustrative example). But in doing so,
the algorithm wastes a lot of effort in deciding what to throw away for this concrete
problem. It also means that as a reader, I am constantly told about generalizations
which reduce to nothing in the concrete example provided. That makes it difficult to
follow I think.
I hear you on the terminology issues. I am straddling two or three camps and trying to pull good ideas from each one. The math and physics camps aren't all that far apart for those of us trained as theoretical types. The CS camp, though, is quite a reach.
The words I am using now aren't the ones I started using. I'm looking for a set that doesn't cause too much grief for anyone, so if you have suggestions, I'm all ears.
I like the CS terms quite a lot. They force me, as a physicist, to think about things I used to gloss over. Of course, that leads to algorithmic thinking which is counter to how I learned to solve physics problems. It's all useful for doing a bit of house cleaning and tidying, though. My problem solving approach has been shifting over the last couple years.
An algorithmic way to describe what a physicist does when setting up a problem may not be needed, but I think it will prove useful. I've seen some of the code that developers write for simulation applications. A lot of them write decent UI's and event models and then get bogged down in the physical model. The physics community isn't helping much with our basket of kludges and limited scope solution spaces. We can do better.
The math and physics camps aren't all that far apart for those of us trained as theoretical types. The CS camp, though, is quite a reach. ...
I like the CS terms quite a lot.
Ahem! My academic training is primarily math and CS with a little bit of physics thrown in. I wish [pedant warning!] people wouldn't use "CS" as a generic term for "things having to do with computers". CS comprises the formal underpinnings of the craft of "writing code". Words like "model" and "render" are part of that discipline. Concepts like the various complexity classes are the realm of CS. The confusion comes because, since it's quite a young field, there is a lot of overlap between the practitioners of the two. At various times I've been doing more of one than the other. (The "writing code" aspect certainly pays much better than the other, which is why these days that's what I do!)
Aside: I think it's because of the youth of the field that current literature (research) in CS is absurdly formal; the notation is far more obtuse than anything you'd find in, say, algebraic topology. It's like they are trying to out-notate and out-formal each other.
"Computer Science" has the further deadly drawback of using that S-word... any discipline that feels the need to tack it on is obviously not one! Cf. Political "Science". Instead of "CS" can we use something like "programming" or "engineering"?
I like this kind of article. Are you planning on discussing mathematical models: how they are constructed, what makes a good [effective] model, etc.? Talking to people I find that a lot of them think that "Science" is all about "absolute truth" and "reality". etc. I like to think that when a scientist says "this is how it works" what is really meant is "here's a very effective and useful model for that phenomenon".
Perhaps this subject is not of general interest, but the good and deep articles never are. I look forward to reading more.
CS comprises the formal underpinnings of the craft of "writing code". [sic] [1] Technological fields are theoretically subsidiary to theoretical fields.
You are also slightly linguistically misninformed as to the association of the word science with academic disciplines such as CS or Poli Sci. Science refers to scientific methodology and as such is correctly applied to the methodical study of computers, politics, the physical properties of the earth, and many other things.
I don't understand what you're trying to say here. My calling CS the "formal underpinnings of the craft of writing code" in no way implies any kind of temporal relationship about which came first. I could say that "statics is the formal underpinning of bridge-building". Does it imply that one came before the other? Does it imply that "statics" is "formal bridge-building"?
Methinks you should look in your own stable before naming my game.
Periods outside of quotation marks, please.
Is that a request or a scornful jibe?
Is that a practice you object to? If so, why?
You are also slightly linguistically misninformed as to the association of the word science with academic disciplines such as CS or Poli Sci.
For clearing up some confusing statements that I misread as arguments. To wit:
CS comprises the formal underpinnings of the craft of "writing code". [sic]
Instead of "CS" can we use something like "programming" or "engineering"?
The first reads as though CS is antecedent to writing code (see JonesBoy's post for a reading agreeing with mine.) The second reads as though you believe what the first implies.
As for placing periods outside of quote marks, that is a usage error. It's like putting periods (outside of parentheses). This orthography is a g**k affectation rather than Standard Written English. Hope this helps.
As for placing periods outside of quote marks, that is a usage error. It's like putting periods (outside of parentheses). This orthography is a g**k affectation rather than Standard Written English.
I'm afraid I'd have to take exception to this. As a kid I was taught that fullstops [period? why period?] went inside quotation marks. In the US, though, common usage puts them outside. I find myself vacillating back and forth on this issue, not having decided which is "correct". (Or which is "correct.") I tend to put the fullstop outside when I'm quoting just a word or a phrase; it goes inside when I'm quoting someone."Man," he said, "your spelling is all screwed up."
I have decided how to treat parentheses, though:
the fullstop goes inside the parentheses if it's a complete sentence; it goes outside if the parenthetical clause is at the end of the sentence. I consider this correct: blah blah (foobar bar foo.). However it looks ugly enough that I'd rewrite it.
I was taught to do it as you describe - period within enclosing quotes or parentheses. However, it makes it confusing as to what exactly is being quoted or parenthesized. Since computer people tend to be sensitive to minute details like this, there are a lot of people (myself included) who try to disambiguate this notation by only putting the period inside the quotes if you're quoting the period itself (for example, quoting a full sentence). I was unaware that others in the USA considered this to be standard usage; I knew that I was doing it differently and really didn't care, since I'm pretty sure I can justify my choice.
"CS comprises the formal underpinnings of the craft of 'writing code'".<GASP> I can't believe you even implied this. Computer science is all about the algorithms, describability, solvability, etc. Theory. Programming is merely a side effect of this knowledge. It is an implementation, but not the intent. There are a lot of CS people out there that are horrible coders, or do not code at all. Look at Alan Turing. He was doing CS before computers existed! There are also a lot of good CS programmers. Thats 'cause theory usually doesn't put food on the table.
Science is not necessarily about truth, but about abstraction. Engineering is about the implementation of scientific abstractions. Programmers that develop algorithms from theories may be classified as engineers, but most are merely technicians. I mean, calling a front end programmer a software engineer is like calling a garbage man a sanitation engineer - P.C. feelgood bullshit. Yes, far too many people lump Computer Science, Computer Engineering, Software Engineering, and Programming Technicians into CS, but martingale and adiffer were spot on.
CS is not new either. It has been a collegiate major for ~35 years, and the science it is teaching has been around even longer.
The camp I am reaching out toward is the one filled with people who write code and the languages in which the source is written. These are the people dealing with day-to-day issues of representation. They are also the most easily understood by a large group because their work is so transparent. I did not mean to misuse the term 'Computer Science' no matter how much people may differ on their definition of it.
Regarding future articles... yes. I am planning to write some more stuff and offer it here to K5 readers. It seems to pass the voting test even if it doesn't lead to a lot of discussion.
I don't know how good I'll be writing up comparisons of the various representation techniques, though. I know a couple of them well enough to write confidently, but others here have already found my limits elsewhere. I think what I'll do is write up stuff for the ones I know and then ask others to fill in the gaps.
Who knows... Maybe the book we were planning to write could contain a collection of articles along with it describing opposing systems.
Representation is an important issue (5.00 / 1) (#19) by sebpaquet on Tue May 07, 2002 at 12:36:59 PM EST
Nice, original article, although I must admit I didn't understand all of it. It would be nice to see examples/applications of your framework.
The way we represent things largely determines what we will be easy and hard to do with them. Most outstanding thinkers usually have several ways of seeing the same thing, and are able to shift from one to the other according to where they're trying to go. For instance, almost every problem in physics can be stated in several ways. If you use the right description the solution follows naturally; if you don't, you could be in for hard work.
You made an important point in stating that practitioners are often not aware of there being this additional "layer" of representation between themselves and objects. And I'd be inclined to add that they can become entraped in the particular layer they grew up in and learned by example (rather than by specification). At times someone will begin representing an object in a way that is fundamentally different and not strictly equivalent to previous ways of seeing it. The implications of the new view are not the same as those associated with the old. An example is the shift from geocentrism to heliocentrism, or from Newtonian mechanics to relativity theory. When the new view better matches reality, and if this comes to be widely recognized, a paradigm shift occurs. I recommend the excellent book "The Structure of Scientific Revolutions" by T.S. Kuhn for an in-depth look at how such important events are lived by scientific communities.
When you think about it, computer science is all about representations. The physicist or biologist typically thinks he's dealing with objects and so pays little attention to this issue. Knowledge about representations is somehow kept tacit. When the computer scientist comes into the picture, he is faced with the difficult task of unearthing tacit knowledge and making it explicit. And often has to make choices that the physicist never thought about.
All this borders on philosophy - a fascinating topic if you ask me. ---- Seb's Open Research - Pointers and thoughts on the evolution of knowledge sharing and scholarly communication.
I usually get sour looks from other physicists when I bring this topic up because it is so close to philosphy. I think it is a shame we have distanced ourselves so much from our roots as natural philosophers.
I also think a good house cleaning should be done by us regarding how we choose to represent stuff. The only physicist that interacts directly with the object is an experimentalist who then interprets their results in terms of the observables. The rest of us are playing games for which we should be able to write the rules. Rules lead to algorithmic thinking and suddenly (bang!) the theoreticians will have a powerful new tool to help them out.
The way we represent things largely determines what
we will be easy and hard to do with them.
The classic example of this is the old Newton/Leibniz flamewar
regarding derivatives. Newton wrote f', f" etc for the derivative
of f, while Leibniz invented df/dx etc. Same concept, but Leibniz'
notation was way more useful for hundreds of years. More geometric.
You could write df/dy = (df/dx)(dx/dy) which is really intuitive.
It had some problems with infinitesimals, but that's been solved.
Newton's notation only became interesting/useful last century with
the push into functional analysis.
Most
outstanding thinkers usually have several ways of
seeing the same thing, and are able to shift from
one to the other according to where they're trying
to go.
I agree completely with you on this one. This is perhaps the
most important lesson, but unfortunately it takes years to
assimilate, simply due to the sheer number of alternatives to be mastered.
But fortunately, people can still contribute even if they are
ignorant of some or most of the alternative viewpoints.
I'm afraid the concepts presented are not very clear, and the examples are lacking in clarity and validity.
The "vector" method describes the process of assigning coordinates by projection onto basis vectors, skipping the concept of a perpendicular space and ending up with (I hope) two points to describe a line segment.
The "algebraic" method assigns coordinates in an equivalent space, which apparently doesn't require projection or much of any process to find coordinates. The result is a line segment which is claimed to be described by "a function", but is really the image of a closed interval under a function. This image is itself a line segment, consisting of endpoints and every point in between, unlike the "vector" method.
Why these representations are different is more due to confusion than any real difference in representation.
Likewise the concepts of symmetry, invariance, conservation laws, and the like have been studied in detail by Physicists and Mathematicians for centuries. If the claim is that mathematically unsophisticated students have difficulty understanding the different descriptions, I have no quarrel with that, but I would suggest a bit of guidance from, say, a Mathematical Physicist.
My examples could use some improvement. I just made them up yesterday as a response to an earlier editorial comment. I'll bullet-proof them before anything goes in the book.
To the physicists, these representations are different. They hard to grasp for the beginning students largely because we don't talk much about how we do it. We just provide some examples and expect the students to imitate them long enough to get it right. Even the more experienced physicists rarely think about representation. The biggest hurdle the string theory folks face in getting a successful theory adopted by the rest of us is getting us to not look too closely at the space they concoct to get testable hypotheses that survive experimentation. I know plenty of physicists who reject the string theory approach simply because they can't accept the solution space being offered.
From a Mathematical standpoint (none / 0) (#28) by KWillets on Tue May 07, 2002 at 08:36:52 PM EST
Physicists are notorious abusers of notation. Part of what mathematicians do is to clean up and simplify the gobbledygook that physicists produce. If what physicists think up were not so significant, mathematicians would be happy to ignore them.
The reasons for this situation are many, but the main one is that experiment is more important in Physics than formal elegance. Making things up as one goes along is a useful practice, and you're correct that students are usually not aware of it.
Physics teaching is the art of compressing millenia of thought into a few months' time, and I'm afraid that much of the process of doing Physics is distilled out. I learned the same thing when I TA'd a Calculus class; this subject is one of the most random collections of Mathematical subfields I've run across. There's no structure to the topic except that each piece has proven useful in some area not related to Mathematics.
At the conferences I've attended, that is exactly what the mathemeticians thought of us. They work so hard coming up with elegant systems that actually work and we butcher them for the sake of practicality.
If you take the side of the mathemeticians, you will howl at my next installment. Just remember that I know and accept my place. 8)
bwa ha ha ha! yes! we destroy your beautiful mathematics! i laugh with glee every time i truncate a Taylor series approximation of a function at the 5th or 6th term because it's "close enough". inverse Laplace/Fourier transforms!? who cares! we've got tables you guys made for us! region of convergence!? bah! you guys can take your real and/or complex analysis and stick it where the sun don't shine! | 677.169 | 1 |
Precalculo II Advice
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This was a great continuation for MATE 3023. I learn a lot of things about Trigonometry. It is great for students looking for a rather different math than Algebra and Geometry. This gives you all. Geometry combines with Algebra in a different a good way. It helps you think and analyze.
Course highlights:
I gained lots of knowledge about angles, trigonometry functions, logarithms, exponential functions, trigonometry equations, logarithmic equations and lots of great things that helps specially in physics. Is a great base course for students that studied science.
Hours per week:
6-8 hours
Advice for students:
You should be revised your notes every week. Maybe do from three to five problems every time you took something new. It helps to remember for the next class. Is good to do at least 15 problems in the weekend, this helps to practice and if you have a doubt about a problem you know what to ask the professor. If you do this thing, one week before the exam you should be ready to take it. That week look for old exams in the course page and do them by yourself. Then, ask a tutor or teacher to revise it.
Course Term:Spring 2016
Professor:Aponte, José
Course Tags:Great Intro to the SubjectGo to Office HoursMany Small Assignments | 677.169 | 1 |
Excursions In Number Theory Dover Books On Mathematics
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Find a Worth, IL AlgebraAlgebra is a lot of rules and functions to remember to be successful. For my current students, we are constantly reviewing so they are prepared for the next lesson because algebra is just building skill on top of skill. One strategy I have used when solving problems using a mix of positive and negative numbers is to have my students draw a number line on their paper for their reference | 677.169 | 1 |
This handbook, written by an experienced math teacher, lets you quickly look up definitions, facts, and problem solving steps. It includes over 700 detailed examples and tips to help someone improve
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1.1 Aims and Objectives
This Lesson introduces some basic concepts in Set Theory, describing sets, elements, Venn diagrams and the union and intersection of sets.
1.2 Sets and elements
Sets of objects, numbers, departments, job descriptions, etc. are things that we all deal with every day of our lives. Mathematical Set Theory just puts a structure around this concept so that sets can be used or manipulated in a logical way. The type of notation used is a reasonable and simple one. For example, suppose a company manufactured 5 different products a, b, c, d, and e. Mathematically, we might identify the whole set of products as P, say, and write: P = (a,b,c,d,e) which is translated as 'the set of company products, P, consists of the members (or elements) a, b, c, d and e. The elements of a set are usually put within braces (curly brackets) and the elements separated by commas, as shown for set P above. A mathematical set is a collection of distinct objects, normally referred to as elements or members. Sets are usually denoted by a capital letter and the elements by small letters. Example 1 (Illustrations of sets)
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2 a) The employees of a company working in the purchase department could be written as: P = (Jones, Wilson, Gopan, Smith, Hari) b) The warehouse locations of a large supermarket chain could be written as: W = (Mumbai, Delhi, Bangalore, Chennai, Triandrum, Kochi)
1.3 Further set concepts
a) Subsets. A subset of some set A, say, is a set which contains some of the elements of A. For example, if A = (h,i,j,k,l), then: X = (i,j,l) is a subset of A Y = (h,1) is a subset of A Z = (i,j) is a subset of A and also a subset of X. b) The number of a set. The number of a set A, written as n[A], is defined as the number of elements that A contains. For example, if A = (a,b,c,d,e), then n[A] = 5 (since there are 5 elements in A); if D = (Sales, Purchasing, Inventory, Payroll), then n[D] = 4. c) Set equality. Two sets are equal only if they have identical elements. Thus, if A = (x, y, z) and B = (x, y, z), then A = B. d) The Universal Set. In some problems in involving sets, it is necessary to consider one or more sets under consideration as belonging to some larger set that contains them. For example, if we were considering the set of skilled workers (S, say) on a production line, it might be convenient to consider the universal set (U, say) as all of the workers on the line. In other words, where a universal set has been defined, all the sets under consideration must necessarily be subsets of it. e) The complement of a set. If A i s any set, with some universal set U defined, the complement of A, normally written as A', is defined as 'all those elements that are not contained in A but are contained in U'. For the example of the workers on the production line (given in d above), S was specified as the set of skilled workers within the universal set of all workers on the line. Therefore, S' would be all the workers that were not skilled. i.e. the set of unskilled workers.
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3
1.4 Venn Diagrams
. A Venn diagram is a simple pictorial representation of a set. For example, if M = (a,b,c,d,e,f,g) then we could represent this information in the form of a Venn diagram as in Figure 1.1 Figure 1.1 b a f c g e 6 Venn diagrams are useful for demonstrating general relationships between sets. For example, if a firm maintains a fleet of 7 cars, we might write A = (1,2,3,4,5,6,7) (each car being numbered for convenience). If also it was important to identify those cars of the fleet that were being used by the directors, we might have D = {3,5 ). i.e. Cars 3 and 5 are director's cars. This situation could be represented in Venn diagram form as in Figure 1.2. This diagram nicely demonstrates the fact that D is a subset of A, which normally means that n[D] < n[A]. In this case n[D] = 2 and n [A] = 7. d 2 5 M 1 D 3 7 4 Figure 1.2 A
1.5 Operations on Sets
In ordinary arithmetic and algebra there are four common operations that can be performed; namely, addition, subtraction, multiplication and division. With sets, however, just two operations are defined. These are set union and set intersection. Both of these operations are described, with examples, in the following sections.
The Union of two sets A and B is written as AÈB and defined as that set which contains all the elements lying within either A or B or both. For example, if A = (c,d,f,h,j) and B = (d,m,c,f,n,p), then the union of A and B is AÈB = (c,d,f,h,j,m,n,p), these being the elements that lie in either A or B. So that any element of A must be an element of AÈB; similarly any element of B must also be an element of AÈB. Set union for three or more sets is defined in an obvious way. That is, if A, B and C are any three sets, AÈBÈC is the set containing all the elements lying within (i) anyone of A, B or C, (ii) any two of them or (iii) all three. Example 2 (To demonstrate set union)
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4 If A = (m,n,o,p); B = (m,o,p,q); C = (m,p,r); and the universal set is defined as U = (k,l,m,n,o,p,q,r,s), then: a) AÈB = (m,n,o,p,q) b) AÈC = (m,n,o,p,r) c) BÈC = (m,o,p,q,r) d) AÈBÈC = (m,n,o,p,q,r) e) (AÈB)' = (k,l,r,s), which is describing all the elements that are not in AÈB but are in the universal set U.
1.6 Set Intersection
The intersection of two sets A and B is written as AÇ B and defined as that set which contains all the elements lying within both A or B. For example, if A = (a,b,c,d,f,g,) and B = (c,f,g,h,j), then the intersection of A and B is AÇ B = (c,f,g), since these are the elements that lie in both sets. The intersection of three or more sets is a natural extension of the above. If P, Q and R are any three sets then PÇ QÇ R is the set containing all the elements that lie in all three sets. Any combinations of union and intersection can be used with sets. For, example, if X and Y are the sets specified above and Z = (d,f,g,j). then: (XÇy) ÈZ = (c,f,g) È(d,f,g,j) =(c,d,f,g,j) which can be described in words as 'the set of elements that are in either both of X and Y or in Z'.
Example 3 (To demonstrate set intersection) If A=(m,n,o,p};B=(m,o,p.q);C=(n,q,r);with a universal set defined as (k,l,m,n,o,p,q,r,s). Then: a) AÇB = (m,o,p), since a1l these elements are in both sets. Similarly, b) AÇ e = (n) c) BÇC = (q). d) AÇ BÇ C has no elements, is sometimes called the empty set and can be written AÇ BÇC = {}. Note n[{}]=0. e) (AÇ B)' = (k,l,n,q,r,s) is the complement of AÇB and is the set of all elements that are NOT in both A and B. f) (AÈB)ÇC =(m,n,o,p,q)Ç(n,q,r)= (n.q) is the set of all elements that are in A o r B AND ALSO in C.
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5 Example 4 (The union and intersection of given sets) Question In a particular insurance life office, employees Smith, Jones, Williams and Brown have 'A' levels, with Smith and Brown also having a degree. Smith, Melville, Williams, Tyler, Moore and Knight are associate members of the Chartered Insurance Institute (ACII) with Tyler, and Moore having 'A' levels. Identifying set A as those employees with 'A' levels, set C as those employees who are ACII and set D as graduates: a) Specify the elements of sets A, C and D. b) Draw a Venn diagram representing sets A, C and D, together with their known elements. c) What special relationship exists between sets A and D? d) Specify the elements of the following sets and for each set, state in words what information is being conveyed. i. AÇC ii. DÈC iii. DÇC e) What would be a suitable universal set for this situation? Answer a) A = (Smith, Jones, Williams, Brown, Tyler, Moore); C = (Smith, Melville, Williams, Tyler, Moore, Knight); D = (Smith, Brown) b) The Venn diagram is shown in Figure 1.3.
A C Jones Moore D Brown Smith Williams Tyler Melville Knight
Figure 1.3
c) From the diagram, it can be seen that D is a subset of A. d) This information can be obtained either from the Venn diagram or from the sets listed in, a) above. i. AÇC = (Williams, Tyler, Smith). This set gives the employees who have both 'A' levels and are ACII. ii. DÈC = (Brown, Smith, Williams, Tyler, Melville, Knight). This set gives the employees who are either graduates or ACII.
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6 iii. DÇC = (Smith). This set gives the single employee who is both a graduate and ACII qualified. e) A suitable universal set for this situation would be the set of all the employees working in the Life office.
1.7 Let us Sum Up
This Lesson presented described about the set, set theory, Venn Diagrams, and its applications. A set is a collection of distinct objects, called elements, which are normally enclosed within brackets and separated by commas. Venn diagram is a pictorial representation of one or more sets. The Union and Intersection of sets were also discussed in detail. Some examples to understand the concept is also given in the Lesson.
2.1 Aims and OBjectives
For decision problems which use mathematical tools, the first requirement is to identify or formally define all significant interactions or relationships among primary factors (also called variables) relevant to the problem. These relationships usually are stated in the form of an equation (or set of equations) or inequations. Such type of simplified mathematical relations help the decision- maker in understanding (any) complex management problems. For example, the decision- maker knows that demand of an item is not only related to price of that item but also to the price of the substitutes. Thus if he can define specific mathematical relationship (also called model) that exists, then the demand of the item in the near future can be forecasted. The main objective of this unit is to study mathematical relationships (or functions) in the context of managerial problems.
2.2 Definitions
Variable A variable is something whose magnitude can vary or which can assume various values. The variables used in applied mathematics include: sale, price, profit, cost, etc. since magnitude of variables can vary, therefore these are represented by symbols (such as x,y,z etc) instead of a specific number. In applied mathematics a variable is represented by the first letter of its name, for example p for price or profit; q for quantity, c for cost; s for saving or sales; d for demand and so forth. When we write X = 5, the variable takes specific value. Variables can be classified in a number of ways. For example, a variable can be discrete (suspect to counting, e.g. 2 houses, 3 machines etc.), or continuous (suspect to measurement, e.g. temperature, height etc.).
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8 Constant and Parameter A quantity that remains fixed in the context of a given problem or situation is called a constant. An absolute (or numerical) constant such as 2, π, e, etc. retains the same value in all problems whereas an arbitrary (or parametric) constant or parameter retains the same value throughout any particular problem but may assume different values in different problems, such as wage rates of different category of labourers in an industrial unit. The Absolute or numerical value of a constant 'b' is denoted by |b| and means the magnitude of 'b' regardless of its algebraic sign. Thus |b| = |-b | or |+b|.
Functions We come across situations in which two or more variables are related to each other. For example, demand (D) of a commodity is related to its price (p). It can be mathematically expressed as D = f(p) (2-1) This relationship is read as "demand is function of price" or simply "f of p". it does not mean D equals f times p. This mathematical relationship has two variables, D and p. these are called variables because they can take on different numerical values. Let us now consider a mathematical relationship that contains three variables. Assume that the demand (D) of a commodity is related to the price (p) per unit of the commodity, and the level of advertising expenditure (A). then the general relationship among these variables can be expressed as D = f(p,A) (2-2) The functional notations of the type (2-1) and (2-2) are meant to give a general idea that certain variables are, somehow, related. However for making managerial decisions, we need a specific and explicit, not a general and implicit relationship among selected variables. For example, for the purpose of finding the value of demand (D), we make the general relationship (2-2) more specific as shown in (2-3). D = 4+3p-2pA+2A2 (2-3)
Now for any given values of p and A, the value of D can be calculated using the relationship (2-3). This means that the value of D depends on the values of p and A. Hence D is called the dependent variable and p and A are called independent variables. In this case, it may be noted that we have established a rule of correspondence between the dependent variable and independent variable (s). That is as soon as values are assigned to the independent variables (s), the corresponding unique value for the dependent variable is determined by the given specific relationship. That is why a function is sometimes defined as a rule of correspondence between variables. The set of values given to independent variable is called
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9 the domain of the function while the corresponding set of values of the dependent variable is called the range of the function. Other examples of functional relationships are as follows: i) ii) iii) iv) v) vi) the distance (d) covered is a function of time (T) and speed (s), i.e. d = f (T,s). Sales volume (v) of the commodity is a function of price (p), i.e. V = f(p). Total inventory cost (T) is a function of order quantity (Q), i.e. T = f(Q). The volume of the sphere (v) is a function of its radius ®, i.e. V = f® or V = 4/3 π r3 The extension (y) of a spring is proportional to the weight (m) (Hooke's law), i.e. Y m or Y = km. The net present value (y) of an investment is a function of net cash flows (Ct ) in different time periods, project's initial cash outlay (B), firm's cost of capital (P) and the life of the project (N), i.e. y = f(Ct , B,P,N).
The following example will illustrate the meaning of these terms. Example 1 Suppose an industrial worker gets Rs. 50 per day. If he works for 25 days in a particular month, then his total wage for this month is 50 x 25 = Rs. 1250. During some other month he may have worked a total of only 24 days, then he would have earned Rs. 1200. Thus, the total wages of the worker, assuming no overtime, can always be calculated as follows: Total wages = 25 X number of days worked Let, T = total wages D = number of days worked Then, T = 50 D. This represents the relationship between total wages and number of days worked. In general, the above relationship can also be written as: T = KD Where K is a constant for particular class of worker (s), to be assigned or determined in a specific situation. Since the value of K can vary for a specific situation, problem or context therefore it is called a parameter, whereas constants such as pi (denoted by π) which has approximate value of 3.1416 remains same from one problem context to another are called absolute constants. Quantities such as T and D which can assume various values in a given problem are called variables. Exercise 1. Find the domain and range of each of the following functions a) Y = 1/x-1 b) Y = x; y 0 c) Y = 4-x; y 0 2. Let 4p+6q = 60 be an equation containing variables p (price) and q (quantity). Identify the meaningful domain and range for the given function when price is considered as independent variable.
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10
2.3 Types of Functions
In this Section some different types of functions are introduced. Linear Functions: A linear function is one in which the power of independent variable is 1, the general expression of linear function having only one independent variable is: Y = f(x) = a + bx Where a and b are given real numbers and x is an independent variable taking all numerical values in an interval. A function with only one independent variable is also called single variable function. Further, a single- variable function can be linear and non-linear. For example, Y = 3+2x, (linear single-variable function) And Y = 2+3x-5x2 +x2 , ( non- linear single-variable function) A liner function with one variable can always be graphed in a two dimensional plane (or space). This graph can always be plotted by giving different values to x and calculating corresponding values of y. the graph of such functions is always a straight line. Example 2 Plot the graph of the function, y = 3+2x For plotting the graph of the given function, assigning various values to x and then calculating the corresponding values of y as shown in the table below:
X Y
0 3
1 5
2 7
3 9
4 11
5 13
… …
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11 The graph of the given function is shown in Figure 1.4 y 13 11 (4,11) y =3+2x 9(3,9)
7-
(2,7)
5-
(1,5)
30
(0,3) |
| 1
| 2
| 3
| 4
| 5
x
Figure 1.4 A function with more than one independent variable is defined, in general, form, as: Y=f(x1 ,x2 ,….,xn ) = a0 +a1 x+a2 x2 +…+an xn Where a0 ,a1 ,a2 ,…,an are given real numbers and x1 ,x2 ,…,xn are independent variables taking all numerical value in the given intervals. Such functions are also called multivariable functions. A multivariable function can be linear and non- liner, for example, Y = 2+3x1 +5x2 (linear multi- variable function) and Y = 3+4x1 +15x1 x2 +10x2 2 (non- linear multivariable function) Multivariable functions may not be graphed easily because these require threedimensional plane or more dimensional plane for plotting the graph. In general, a function with n variables will require (n+1) dimensional plane for plotting its graph. Polynomial Functions: A function of the form Y = f(x) = a1 xn-1 +…+an x0 (1-4) Where a1 's(I = 1,2,…,n) are real numbers, a1 0 and n is a positive integer is called a's polynomial of degree n. a) if n = 1, then the polynomial function is of degree 1 and is called a linear function. That is, for n = 1, function (1-4) cam be written as: y = a1 x1 +an x0 (a1 0) This is usually written as Y = a + bx (since x0 = 1) Where 'a' and 'b' symbolise an and a1 respectively.
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12
b) if n = 2, then the polynomial function is of degree 2 and is called a quadratic function. That is, for n = 2, the function (1-4) can be written as: c) y = a1 x2 +a2 x1 +an x0 (a1 0) This is usually written as Y = ax2 +bx+c where a1 = a, a2 = b and an = c Absolute Value Functions The functional relationship expressed by Y = |x| Is known as an absolute value function, where |x| is known as magnitude (or absolute value) of x. By absolute value we mean that whether x is positive or negative, its absolute value remains positive. For example |7|=7 and |-6|=6. Plotting of the graph of the function y=|x|, assigning various values to x and then calculating the corresponding values of y, is shown in the table below:
X Y
… …
-3 3
-2 2
-1 1
0 0
1 1
2 2
3 3
… …
The graph of the given function is shown in Figure 1.5
y y=|- x| 4(-3,3) 3- (3,3) y=|x|
(-2,2) -
2-
-(2,2)
(-1,1) - 1| | |
- (1,1)
| | |
-x
-3
-2
-1
0
1
2
3
+x
Figure 1.5
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13 Inverse Function Take the function y = f(x). Then the value of y, can be uniquely determined for given values of x as per the functional relationship. Sometimes, it is required to consider x as a function of y, so that for given values of y, the value of x can be uniquely determined as per the functional relationship. This is called the inverse function and is also denoted by x=f-1 (y). For example consider the linear function: Y = ax+b Expressing this in terms of x, we get X = y-b/a = y/a-b/a = cy + d where c = 1/a, and d = -b/a This is also a linear function and is denoted by x = f-1 (y) Step Function For different values of an independent variable x in an interval, the dependent variable y=f(x) takes a constant value, but takes different values in different intervals. In such cases the given function y = f(x) is called a step function. For example y1 , if 0 < x <50 y = f(x) y2 , if 51 < x < 100 y3 , if 101 x 150 The shape of the graph of this function looks as shown in Figure 1.6, for y3 < y2 <y1
y
y1Y2Y3| | |
x
50
100
150 Figure 1.6
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14 Algebraic and Transcendental Functions Functions can also be classified with respect to the mathematical operations (addition, subtraction, multiplication, division, powers and roots) involved in the functional relationship between dependent variable and independent variable (s). When only finite number of terms are involved in a functional relationship and variables are affected only by the mathematical operations, then the function is called an algebraic function, otherwise transcendental function. The following functions are algebraic functions of x. i) y = 2x3 +5x2 – 3x+9 ii) y = x+ 1/x2 iii) y = x3 - 1/ x +2 The sub-classes of transcendental functions are follows: a) Exponential function If the independent variable in any functional relationship appears as an exponent (or power), then that functional relationship is called exponential function, such as i) y = ax, a 1 ii) y = kax ,a 1 iii) y = kabx,a 1 iv) y = kex where a, b, e and k are constants with 'a' taking only a positive value. Such functions are useful for describing sharp increase or de crease in the value of dependent variable. For example, the exponential function y = kax curve rises to the right for a>1, k>0 and falls to the left a<1,k>0 as shown in the Figure 1.7 (a) and (b). Figure 1.7(a) Figure 1.7(b)
y y=kax (a>1,k>0) k 0 x k 0 x y=kax (a>1, k>0
b) Logarithmic functions A logarithmic function is expressed as Y=loga x Where a 1 and >0 is the base. It is read as 'y' is the log to the base a of x'. this can also be written as X=ay
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15 Thus from an exponential function y=ax , we may construct the logarithmic function x=ay by interchanging the variables. This shows that the inverse of an exponential function is a logarithmic function. The two most widely used bases for logarithms are '10' and 'e' (=2.7182). i) Common logarithm: It is the logarithm to the base 10 of a number x. it is written as log10 x. if y=log10 x, then x=10y ii) Natural logarithm: It is the logarithm to the base 'e' of a number x. it is written as loge x or In x. when no base is mentioned, it will be understood that the base is e. Some important properties of the logarithmic function y=loge x are as follows: i) ii) iii) iv) v) vi) log 1=0 loge=1 log (xy)=log x+log y log (x/y)=log x - log y log (xn ) = n log x loge 10 = 1/ log10 e
2.4 Solution of Functions
The value (s) of x at which the given function f(x) becomes equal to zero are called the roots (or zeros) of the function f(x). For the linear function Y = ax + b The roots are given by ax + b = 0 Or x = -b/a Thus if x = -b/a is substituted in the given linear function y = ax + b then it becomes equal to zero. In the case of quadratic function, Y = ax2 +bx+c,
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16 We have to solve the equation ax2 +bx+c=0; a 0 to fined the roots of y. The general value of x for which the given quadratic function will become zero is given by -b± (b2-4ac) X = 2a Thus, in general, there are two values of x for which y becomes zero. One value is -b+ (b2-4ac) X= 2a and other value is -b- (b2 -4ac) X= 2a It is very important to note that the number of roots of the given function are always equal to the highest power of the independent variable. Particular Cases: The expression b2 -4ac in the above formula is known as discriminant which determines the nature of the roots as discussed below: i) If b2 -4ac>0, then the two roots are real and unequal ii) If b2 -4ac=0 or b2 =4ac, then the two roots are equal and are equal to – b/2a iii) If b2 -4ac<0, then the two roots are imaginary (not real) because of the square root of a negative number. iv) The roots of a polynomial of the form: Y = (x-a) (x-b) (x-c) (x-d)… are a, b, c, d, …
2.5 Business Applications
In business applications, there are lot of situations to deal with supply and demand functions; cost functions; profit functions; revenue functions; production functions; utility functions; etc. in applied mathematics. In this section, a few examples are given by constructing such functions and obtaining their solutions: Example 3 (liner Functions) A company sells x units of an items each day at the rate of Rs. 50 per unit. The cost of manufacturing and selling these units is Rs. 35 per unit plus a fixed daily overhead cost of Rs. 1000. Determine the profit function. How would you interpret the situation if the company manufactures and sells 400 units of the items a day. Solution: The total revenue received by the company per day is given by: Total revenue( R ) = (price per unit) X (number of items sells) = 50.x The total cost of manufactured items per day is given by: Total cost (C)=(Variable cost per unit) X(no. of items manufactured)+(fixed daily overhead cost) = 35.x + 1000
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17 Thus, Total profit (p) = (Total revenue)-(Total cost) = 50.x - (35.x + 1000) = 15.x-1000 If 400 units of the item are manufactured and sold, then the profit is given by: P = 15X400-1000 = -400 The negative profit indicates loss. Thus if the company manufactures and sells 400 units of the item, it would incur a loss of Rs. 400 per day. Example 4 ( Quadratic Functions) Let the market supply function of an item be q = 160+8p, where q denotes the quantity supplied and p denotes the market price. The unit cost of production is Rs. 4. It is felt that the total profit should be Rs. 500. What market price has to be fixed for the item so as to achieve this profit? Solution: Total profit function can be constructed as follows: Total profit (p)= Total revenue – Total cost = (price per unit x Quantity supplied) – (unit cost x Quantity supplied) = p.q - c.q = (p-c).q Given that c = Rs. 4 and q = 160+8p. Then total profit function becomes P= (p-4) (160+8p) = 8p2 + 12 8p-640 If P = 500, then we have 500 = 8p2 + 12 8p-640 8p2 + 12 8p-1140 = 0 p = 6.36 or -22.37 since negative price has no economic meaning, therefore the required priced per unit should be Rs. 6.36. Exercise 1. Consider the quadratic equation x2 -8x+c = 0. For what value of c, the equation has i) real roots, ii) equal roots, and iii) imaginary roots? 2. A newsboy buys papers for p1 paise per paper and sells them at a price of p2 paise per paper (p1 >p1). The unsold papers at the end of the day are bought by a wastepaper dealer for p3 paise per paper (p3 <p1 ). i) Construct the profit function of the newsboy. ii) construct the opportunity loss function of the newsboy.
2.6 Let us Sum Up
The objective of this unit is to provide you exposure to functional relationship among decision variables. We started with the mathematical concept of function and defined terms such as constant, parameter, independent and dependent variable. Various examples of functional relationships are mentioned to see the concept in broad
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18 perspective. Various types of functions which are normally used in managerial decisionmaking are enumerated along with suitable examples, their graphs and solution procedure. Finally, the applications of functional relationships are demonstrated through several examples.
3.1 Aims and Objectives
Matrices have applications in management disciplines like finance, production, marketing etc. Also in quantitative methods like linear programming, game theory, input-output models and in many statistical applications matrix algebra is used as the theoretical base. Matrix algebra can be used to solve simultaneous linear equations.
3.2 Matrices : Definition and Notations
A matrix is a rectangular array or ordered numbers. The term ordered implies that the position of each number is significant and must be determined carefully to represent the information contained in the problem. These numbers (also called elements of the matrix) are arranged in rows and columns of the rectangular array and enclosed by either square brackets, []; or parantheses ( ), or by pair of double vertical line || ||. A matrix consisting of m rows and n columns is written in the following form
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20 A column
A11 A21 . . . Am1 am2 …….. amn a12 a22 ……… a1n
……. a2n
Where a 11 ,a12,… denote the numbers (or elements) of the matrix. The dimension (or order) of the matrix is determined by the number of rows and columns. Here, in the given matrix, there are m rows and n columns. Therefore, it is of the dimension m X n (read as m by n). In the dimension of the given matrix the number of rows is always specified first and then the number of columns. Boldface capital letters such as A,B,C…. are used to denote entire matrix. The matrix is also sometimes represented as A=[aij]m x n where aij denotes the ith row and the jth element of a. Some examples of the matrices are
-1 A= 2 3
2X2
1 ; B=
1 2
1 4
2 -3
2X3
5 ; C= 6 2
5 2 1
10 10 2
3X3
The matrix A is a 2X2 matrix because it has 2 rows and 2 columns. Similarly the matrix B is a 2X3 matrix while matrix C is a 3X3 matrix. Exercise Tick mark the correct alternative indicting the dimension of the matrix 2 6 3 i) 3x4 3 8 5 ii) 4x3 4 9 7 iii)None of these
3.3 Some special Matrices
a) Square matrix A matrix in which the number of rows equals the number of columns is called a square matrix. For example 2 3 4 3 5 3 7 2 1
3x3
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21 is a square matrix of dimension 3. The elements 2,5 and 1 in this matrix are called the diagonal elements and the diagonal is called the principal diagonal. b) Diagonal matrix A square matrix, in which all non-diagonal elements are zero whereas diagonal elements are non-zero, is called a diagonal matrix. For example 2 0 0 0 5 0 0 0 1
3x3
is a diagonal matrix of dimension 3. c) Scalar matrix A diagonal matrix in which all diagonal elements are equal is called a scalar matrix. For example
k 0 0
0 k 0
0 0 k
3x3
is a scalar matrix, where k is a real (or complex) number. d) Identity (or unit) matrix A Scalar matrix in which all diagonal elements are equal to one, is called an identity (or unit) matrix and is denoted by I. Following are two different identity matrices 1 I2 = 0 1
2x2
0
1 ; I3 = 0 0
0 1 0
0 0 1
3X3
An identity matrix of dimension n is denoted by In. It has n elements in its diagonal each equal to I and other elements are zero. d) The zero (or null) matrix A matrix is said to be the zero matrix if every element of it is zero. It is denoted as 0. Following are three different zero matrices
3.4 Matrix Representation of Data
Before discussing the operations on matrices, it is necessary for you to know a few situations in which data can be represented in matrix form. 1. Transportation Problem The unit cost of transportation of an item from each of the two factories to each of the three warehouses can be represented in a matrix as shown below: Warehouses
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22
W1 20 25 W2 15 20 W3 30 15
F1 Factory F2
Similarly, we can also construct a time matrix [tij], where tij=time of transportation of an item from factory I to warehouse j. Note that the time of transportation is independent of the amount shipped. 2. Distance Matrix The distance (in kms.) between given number of cities can be represented as matrix as shown below: City
A A B 1,470 C 2,158 D 1,732
City
B 1,470 --C 2,158 1,853 D 1,732 2,365
1,853 --1,635
2,385 1,635 ----
3. Diet matrix The vitamin content of two types of foods and two types of vitamins can be represented in a matrix as shown below:
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23 5. Pay – off Matrix Suppose two players A and B play a coin tossing game. If outcome (H,H) or (T,T) occurs, then player B loses Rs. 20 to player A, otherwise gains as shown in the matrix: Player B H T H 20 -20 Player A T -20 20
The minus sign with the pay off means that player A pays to B. 6. Brand Switching matrix The proportion of users in the population surveyed switching to brand j of an item in a period, given that they were using brand I can be represented as a matrix. To
Brand 1 From Brand 1 Brand 2 Brand 3 0.3 0.6 0.2 Brand2 0.6 0.3 0.5 Brand 3 0.1 0.1 0.3
Here the sum of the elements of each row is 1 because these are proportions.
3.5 Operations on Matrices
1. Addition and Subtraction of Matrices The sum of two matrices of same order is obtained by adding the corresponding elements of the given matrices. The difference of two matrices of same order is obtained by subtracting the corresponding elements of the given matrices. é 2 - 3ù é 1 3ù ê1 ú and B = ê 1 2ú , then, 0ú For example, if A = ê ê ú ê- 2 - 1ú ê- 1 0ú ë û ë û
é3 ´ 2 3 ´ 1 3 ´ -1ù ê ú 3A = ê3 ´ 3 3 ´ 1 3 ´ 5 ú ê3 ´ 2 3 ´ 0 3 ´ 1 ú ë û é6 3 - 3ù ê ú = ê9 3 15 ú ê6 0 3 ú ë û
3. Multiplication of Matrices If the number of columns in the first matrix is equal to the number of rows in the second matrix, then the matrices are compatible for multiplication. That is, if there are n columns in the first matrix then the number of rows in the second matrix must be n. Otherwise the matrices are said to be incompatible and their multiplication is not defined. The operation of multiplication a) The element of a row of the first matrix should be multiplied by the corresponding elements of a column of the second matrix.
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25 b) The products are then summed and the location of the resulting element in the new matrix determines the row from first matrix has to be multiplied with which column from second.
é2 1 ù é1 0 3ù ê ú Example 1. Let A= ê and B= ê1 0ú 2 1 5ú ë û ê 3 2ú ë û Since A is of order 2×3 and B is of order 3×2, the matrices are compatible for multiplication and the resultant matrix should 2×2.
In the first matrix R1 is [1 0 3] and R2 is [ 2 1 5] and é 2ù é1 ù ê1 ú and C is ê0ú Columns of the second matrix are C1 is ê ú 2 ê ú ê 3ú ê 2ú ë û ë û
Properties of multiplication 1. Matrix multiplication, in general, is not commutative. i.e, AB ¹ BA. 2. Matrix multiplication is associative. i.e., A(BC) =(AB)C 3. Matrix multiplication is distributive, i.e, A(B+C) = AB + AC 4. Transpose of a Matrix The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A and is denoted by A' or AT . Thus if A is an m×n matrix, then, AT will be an n×m matrix.
3.6 Determinant of a Square Matrix
The determinant of a square matrix is a scalar (i.e. a number). Determinants are possible only for square matrices. For more clarity, we shall be defining it in stages, starting with square matrix of order 1, then for matrix of order 2, etc. The determinant of a square matrix A is denoted either by |A| or det. A. i) Determinant of order 1. Let A = [a11 ] be a matrix of order 1. Then det A=a11 ii) Determinant of order 2. Let
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27
a11 A= a21
a12 a22
be a square matrix of order 2, then det. A is defined as a11 a12 A= = a11 a22 - a21a12 a21 a22 For example 3 4 det. A= = 3X2-1X4 = 2 1 2 to write the expansion of a determinant to matrices of order 3,4,…,let us first define two important terms: a) Minor: Let a be a square matrix of order m. Then minor of an element aij is the determinant of the residual matrix (or submatrix) obtained from a by deleting row I and column j containing the element aij. In the |A|, the minor of the element aij is denoted by Mij. Thus, in the determinant of order 3 a11 a12 a13 a21 a22 a23 a31 a32 a33 the minor of the element a11 is obtained by deleting first row and first column containing element a11 and is written as a22 a23 M11 = A32 a33 Similarly, minor of a12 is A21 M12 = A3 1 a33 a23
b) Cofactor. The cofactor cij of an element aij is defined as Cij=(-1)i+jMij Where Mij is the minor of an element aij. Now using the concept of minor and cofactor, you can write the expansion of a determinant of order 3 as shown below: a11 a12 a13 = a11 C11 +a12 C12+a13 C13 a21 a22 a23 1+1 1+2 M12-a13(-1)M13 =a11(-1) M11+a12 (-1) a31 a32 a33
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a11 =a11 a32
a22 -a12 a33
a21
a23 +a13
a21
a22
a31
a33
a31
a32
=a11 (a22 a33-a32 a23 ) – a12 (a21 a33-a31 a23) + a13 (a21 a32-a31 a22) The expansion of the given determinant can also be done by choosing elements in any row and column. In the above example expansion was done by using the elements of the first row. Example 2 Find the value of the determinant 1 18 72 det.A= 2 40 96 2 45 75 Solution: If you expand the determinant by using the elements of the first column, then you will get 1 2 2 18 40 45 72 96 75 40 =1 96 18 72 18 72
Properties of determinants Following are the useful properties of determinants of any order. These properties are very useful in expanding the determinants. 1. The value of a determinant remains unchanged. If rows are changed into column and columns into rows, i.e. |A| = |At | 2 If two rows (or columns) of a determinant are interchanged, then the value of the determinant so obtained is the negative of the original determinant. 3 If each element in any row or column of a determinant is multiplied by a constant number say K, then the determinant so obtained is K times the original determinant. 4 The value of a determinant in which two rows (or columns) are equal is zero. 5 If any row (or column) of a determinant is replaced by the sum of the row and a linear combination of other rows (or columns), then the value of the determinant so obtained is equal to the value of the original determinant. 6 The rows (or columns) of a determinant are said to be linearly dependent if |A|=0, otherwise independent. Example 3 Verify the following result
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Singular and Non-singular Matrices: A matrix A is said to be singular if |A| = 0; otherwise it is called non-singular.
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30 Exercise If a+b+c = 0, then verify the following result. a b c 0 a b = c(2ab-c2 ) b 0 a
3.7 Let us Sum Up
Matrices play an important role in quantitative analysis of managerial decision. They also provide very convenient and compact methods of writing a system of linear simultaneous equations and methods of solving them. These tools have also become very useful in all functional areas of management. Another distinct advantage of matrices is that once the system of equations can be set up in matrix form, they can be solved quickly using a computer. A number of basic matrix operations (such as matrix addition, subtraction, multiplication) were discussed in this Lesson. 3.8 Lesson – End Activities 1. Define matrix, square matrix, diagonal matrix, scalar matrix. 2. Mention the properties of transpose of a matrix. 3. List the properties of determinants.
3.9 References
P.R. Vittal – Business Mathematics and Statistics.
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4.1 Aims and Objectives
In the last Lesson, Matrix algebra, matrix operations and applications of matrix theory, etc., were discussed in details. This Lesson exclusively describes inverse of matrix, which is another important operation of matrix algebra.
4.2 Inverse of a Matrix
If for a given square matrix A, another square matrix B of the same order is obtained such that AB = BA = 1 Then matrix B is called the inverse of A and is denoted by B=A-1 Before start discussing the procedure of finding the inverse of a matrix, it is important to know the following results: 1. The matrix B=A-1 is said to be the inverse of matrix A if and only if AA-1 =A-1 A=I. 2. That is, if the inverse of a square matrix multiplied by the original matrix, then result is an identity matrix. The inverse A-1 does not mean I/A or I/A. This is simply a notation to denote the inverse of A 3. Every square matrix may not have an inverse. For example, zero matrix has no inverse. Because, inverse of square matrix exists only if the value of its determinant is non-zero, i.e. A-1 exists if and only if |A| 0. For example, let B be the inverse of the matrix A, then AB=BA=I Or |AB|=I Or |A|B|=1(|I|=1) Hence |A| 0. 4. If a square matrix A has an inverse, then it is unique. It can also be proved by letting two inverse B and C of A. We then have AB = BA = I …(i) And AC = CA = I …(ii) Pre-multiplying (i) by C, we get
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32 CAB = CI IB = CI or B = C (CA = I) This implies that the inverse of a square matrix is unique. Singular Matrix A matrix is said to be singular if its determinant is equal to zero; Otherwise non-singular. Properties of the inverse i) The inverse of the inverse is the original matrix, i.e. (A-1 )-1 =A. ii) The inverse of the transpose of a matrix is the transpose of its inverse, i.e. (At )-1 =(A-1 )t iii) The identity matrix is its own inverse, i.e. I-1 =I iv) The inverse of the product of two non-singular matrices is equal to the products of two inverse in the reverse order, i.e.(AB)-1 =B-1 A-1 Methods of finding inverse of a matrix The procedure of finding inverse of a square matrix A=[aij] of order n can be summarized in the following steps: 3. Construct the matrix of co- factors of each element aij in |A| as follows: C11 C21 . . . Cm1 C12 ….CIn C22 …. C2n
Cm2 ……
Cmn
In this case cofactors are the elements of the matrix 2. Take the transpose of the matrix of cofactors constructed in step 1. It is called adjoint of A and is denoted by Adj. A. 3. Find the value of |A| 4. Apply the following formula to calculate the inverse of A A-1= Adj A , |A| 0 |A| Example 1 Find the inverse of the matrix 1 -2 1 3 3 1 0 3 4
A
Solution The determinant of matrix A is expanded with respect to the elements of first row:
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4.3 Let us Sum Up
Subsequent to the last Lesson, a discussion on matrix inversion and procedure for finding matrix inverse was discussed in this Lesson. Examples were also given in support of the inverse of a matrix. The inverse of matrix finds applications in most of the problems in matrix algebra like inn business applications while solving linear equations.
4.4 Lesson – End Activities
1. How to find the Inverse of a Matix?
4.5 Reference
Navaneethan, P. – Business Mathematics.
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5.1 Aims and Objectives
Matrix theory was discussed in detail in the previous Lessons. In business applications there are several occasions in which mathematical solution are to be made using simultaneous equations. Matrix algebra is useful in solving a set of linear simultaneous equations involving more than two variables. Now the procedure for getting the solution will be demonstrated in this Lesson.
5.2 Solution of Linear Simultaneous Equations
Consider the set of linear simultaneous equations x-y+z=4 2x + 5y-2x = 3 These equations can also be solved by using ordinary algebra. However, to demonstrate the use of matrix algebra, the first step is to write the given system of equations to matrix form as follows: 1 2 or 1 5 1 -2 X 4 Y = Z 3
AX=B where 1 1 1 A= 2 5 -2 Is known as the coefficient matrix in which coefficients of x are written in first column, coefficients of y in second column and the coefficients of z in the third column. X X= Y Z
Is the matrix of unknown variables x,y and z, and
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b1 b2 B= . mX1 . . bm Classification of linear Equations If matrix B is zero matrix, i.e. B=0, then the system AX=0 is said to be homogeneous system. Otherwise, the system is said to be non- homogeneous. Homogeneous Linear Equations When the system is homogenous, i.e. b1 =b2 = … =bm=0, the only possible solution is X=0 or X1 =X2 =…Xn =0. it is called a trivial solution. Any other solution if it exists is called non-trivial solution of the homogenous linear equations. In order to solve the equation Ax=0, we perform such an elementary operations or transformations on the given coefficient matrix A which does not change the order of the matrix. An elementary operation is of any one of the following three types: i) The interchange of any two rows (or columns)
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37 ii) The multiplication (or division) of the elements of any row (or column) by any nonzero number, e.g. the Ri(row i) can be replaced by KRi (K 0). iii) The addition of the elements of any row (or column) to the corresponding elements of any other row (or column) multiplied by any number, e.g.Ri (row i) can be replaced by Ri+KRj where Rj is the row j and K 0. The elementary operation is called row operation if it applies to rows, and column operation if it applies to column. For the purpose of applying these elementary operations, we form another matrix called augmented matrix as shown below: A11 a12 …..a1n . b1 [A:B]= a21 a22 …. A2n . b2 …………………… am1 am2……amn . bm Solution Method We shall apply Gauss-Jordon Method (also called Triangular form Reduction Method) to solve homogeneous linear equations. In this method the given system of linear equations is reduced to an equivalent simpler system (i.e. system having the same solution as the given one). The new system looks like: X1 +b1 X2 +C1 X3 = d1 X2 + C2X3 = d2 X3 = d3 Solution The given system of equation in matrix form is: 1 3 -2 X1 0 2 -1 4 X2 = 0 or AX=0 1 -11 14 X3 0 The augmented matrix becomes 1 [A:0]+ 2 1 3 -1 -11 -2 4 14 :0 :0 :0
Applying elementary row operations R2 R2 – 2R1 R3 R3 - R1
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The equations equivalent to the given system of equations obtained by elementary row operations are: X1 +3X2-2X3=0 -7X2+8X3=0 or X2 -(8/7)X3=0 0=0 The last equation, though true, is redundant and the system is equivalent to X1 +3X2-2X3=0 X2 -(8/7)X3=0 This is not in triangular form because the number of equations being less than the number of unknowns. This system can be solved in terms of X3 by assigning an arbitrary constant value, k to it. The general solution to the given system is given by X3 = k X2 = (8/7)k X1 +3X2 = 2k3 or X1 = -3(8/7)k+2k = (-10/7)k Exercise Solve the following system of equations using Gauss-Jordon Method i) 4X1+X2=0 -8X1+2X2=0 ii) X1 -2X2+3X3=0 2X1+5X2+6X3=0 Non-homogeneous Linear Equations: The non-homogeneous linear equations can be solved by any of the following methods 1 Matrix Inverse Method 2 Cramer's Method 3 Gauss-Jordon Method Again, for the purpose of demonstrating above solution methods, we shall consider three equations with three unknowns. 1. Matrix Inverse Method Let AX = B Be the given system of linear equations, and also A-1 be the inverse of a. Pre-multiplying both sides of the equation by A-1 , A-1 (AX) = A-1B
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39 (A-1 A)X = A-1 B IX = A-1 B X = A-1 B Where I is the identity matrix. The value of X gives the general solution to the given set of simultaneous equations. This solution is thus obtained by (i) first finding A-1 , and (ii) post multiplying A-1 by B. When the system has a solution, it is said to be consistent, otherwise inconsistent. A consistent system has either just one solution or infinitely many solutions. Example 1 The daily cost, C of operating a hospital, is a linear function of the number of in patients I, and out-patients, P, plus a fixed cost a, i.e, C = a+b P+dI. Given the following data for three days, find the values of a, b, and d by setting up a linear system of equations and using the matrix inverse. Day Cost No.of No.of (in Rs.) in–patients, I out-patients, P 1 6,950 40 10 2 6,725 35 9 3 7,100 40 12
Solution: Based on the given daily cost equation, the system of equations for three days cost can be written as: a+10b+40d = 6,950 a+9b+35d = 6,725 a+12b+40d = 7,100 This system can be written in the matrix form as follows: 1 1 1 10 9 12 40 35 40 a b d 6,950 6,725 7,100
or a = 5000,b = 75 and d = 30 Exercise A salesman has the following record of sales during three months for three items A,B and C, which have different rates of commission.
Months January February March A 90 150 60
Sales of Units B C 100 20 50 40 100 30
Total Commission drawn (inRs.) 800 900 850
Find out the rates of commission on items A,B and C. 2. Cramer's Method When the number of equations is equal to the number of unknowns and the determinant of the coefficients has non-zero value, then the system has a unique solution which can be found by using Cramer's formula.
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41 Xj = Dj, j= 1,2,…,n D Where D=|aij| and determinant Dj is obtained from D by replacing column j by the column of constant terms (i.e. matrix B). Example 2 An automobile company uses three types of steel, S1 ,S2 and S3 for producing three different types of cars C1 ,C2 and C3 . Steel requirements (intones) for each type of car and total available steel of all the three types is summarized in the following table. Types of steel Type of car Total steel available C11 C2 C3 S1 2 3 4 29 S2 1 1 2 13 S3 3 2 1 16 Determine the number of cars of each type which can be produced. Solution: Let X1 ,X2 and X3 be the number of cars of the type C1 ,C2 and C3 respectively which can be produced. Then system of three linear equations is: 2X1+3X2+4X3 = 29 x1+x2+2X3 = 13 3X1+2X2+X3 = 16 These equations can also be represented in matrix form as shown below: 2 1 3 3 1 2 4 2 1 x1 29 x2 = 13 x3 16
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42 Applying Cramer's Method 29 13 16 2 1 3 3 1 2 29 13 16 4 2 1 4 2 1
x1 = D1 = 1 D 5
=2
x2 = D2 = 1 D 5
=3
x3 = D3 = 1 D 5
2 1 3
3 1 2
29 13 = 4 16
Hence, the number of cars of type C1,C2 and C3 which can be produced are 2,3 and 4 respectively. Total time available is 80 hours and 60 hours in department I and II respectively. Determine the number of units of product A and B which should be produced. 3. Gauss – Jordan Method We can solve a system of linear equations by transform the augmented matrix [A:B] into a triangular form. Example 3 : Solve x + 2y = 3 2x + 5y = 2
é1 2 ù é x ù é 3 ù Solution: The system of equations can be written as ê ú ê ú= ê ú ë2 5û ë y û ë 2û é1 2 ù That is, A = ê ú and B = ë2 5û é 3ù ê 2ú ë û
é1 2 3 ù The augmented matrix is [A:B] = ê ú ë 2 5 2û
Apply transformation, R2 = R2 – 2R1
é1 2 3 ù = ê ú ë0 1 - 4 û
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5.3 Let us Sum Up
The methods for solving linear equations using matrix theory is described in this Lesson. The three important methods of solving the equations using Cramer's rule, matrix inverse method and Gauss – Jordon method, are described in detail in this Lesson. Number of examples were given in support of the above said operations.
5.4 Lesson – End Activities
1. How to solve a system of linear equation by Gauss – Jordan Method?
6.1 Aims and Objectives
This Lesson deals with the concepts and applications of sequence and series. A clear understanding about sequence and series is provided. Applications of series like Arithmetic Progression and Geometric Progression and practical applications in business are also dealt with in this Lesson.
6.2 Sequence
If for every positive integer n, there corresponds a number an such that an is related to n by some rule, then the terms a1 , a2 ,….an …. are said to form a sequence. A sequence is denoted by bracketing its nth term, i.e. (an ) or {an }. Example of a few sequences are: i) If an = n2 , then sequence {an }is 1,4,9,16….an ,… ii) If an = 1/n, then sequence {an } is 1,1/2,1/3,1/4…1/n… iii) If an = n2 /n+1, then sequence {an }is ½, 4/3, 9/4,…n2 /n+1,…. The concept of sequence is very useful in finance. Some of the major areas where it plays a vital role are: "instalment buying'; simple and compound interest problems'; 'annuities and their present values', mortgage payments and so on
6.3 Series
A series is formed by connecting the terms of a sequences with plus or minus sign. Thus if an is the nth term of a sequence, then a1 + a2 + … + an is the given series of n terms.
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45
6.4 Arithmetic Progression (AP)
A progression is a sequence whose successive terms indicate the growth or progress of some characteristics. An arithmetic progression is a sequence whose term increases or decreases by a constant number called common difference of an A.P. and is denoted by d. In other words, each term of the arithmetic progression after the fist is obtained by adding a constant d to the preceding term. The standard form of an A.P. is written as a, a+d, a+2d, a+3d,… where 'a' is called the first term. Thus the corresponding standard form of an arithmetic series becomes a+(a+d)+(a+2d)+(a+3d)+…. Example 1 Suppose we invest Rs. 100 at a simple interest of 15% per annum for 5 years. The amount at the end of each year is given by 115,130,145,160,175 This forms an arithmetic progression The nth Term of an A.P. The nth term of an A.P. is also called the general term of the standard A.P. it is given by. Tn = a+(n-1)d; n=1,2,3,… Sum of the First n terms of an A.P Consider the first n terms of an A.P. a, a+d, a+2d, a+3d,…., a+(n-1)d The sum, Sn of the these terms is given by Sn = a+(a+d) + (a+2d) +(a+3d) + …+ a+(n-1)d = (a+a+…+a) + d(1+2+(n-1) 3+….+) = n.a + d n(n-1) 2
(using formula for the sum of first (n-1) natural numbers)
= n/2 {2a+(n-1)d} Example 2 Suppose Mr. Anil repays a loan of Rs. 3250 by paying Rs. 20 in the first month and then increases the payment by Rs. 15 every month. How long will be take to clear his loan? Solution Since Mr. Anil increases the monthly payment by a constant amount, Rs.15 every month, therefore d = 15 and first month instalment is, a = Rs. 20. This forms an A.P. Now if the entire amount be paid in n monthly instalments, then we have Sn = n/2 {2a+(n-1)d} Or 3250 = n/2{2X20+(n-1)15} 6500 = n{25+15n} 15n2 +25n-6500 = 0
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= -25 ± 625 = 20 or -21.66 30 The value, n = -21.66 is meaningless as n is positive integer. Hence Mr. Anil will pay the entire amount in 20 months. Exercise 1. Find the 15th term of an A.P. whose first term is 12 and common difference is 2. 2. A firm produces 1500 RV sets during its first year. The total production of the firm at the end of the 15th year is 8300 TV sets, then a) estimate by how many units, production has increased each year. b) based on estimate of the annual increment in production, forecast the amount of production for the 10th year.
6.5 Geometric Progression (GP)
A geometric progression (GP) is a sequence whose each term increases or decreases by a constant ratio called common ratio of G.P. and is denoted by r. In other words, each term of G.P. is obtained after the first by multiplying the preceding term by a constant r. The standard from of a G.P. is written as : a, ar, ar2 ,…. Where 'a' is called the first term. Thus the corresponding geometric series in standard form becomes a + ar + ar2 + …. Example 3 Suppose we invest Rs. 100 at a compound interest of 12% per annum for three years. The amount at the end of each year is calculated as follows: i) Interest at the end of first year Amount at the end of first year = 100X12/100 = Rs. 12 = Principal + Interest = 100 + 100(12/100) =100 (1+12/100)
This shows that the principal of Rs. 100 becomes Rs. 100 (1+12/100) at the end of first year. ii) Amount at the end of second year = = (Principal at the beginning of second year) {1+12/100} = 100{1+12/100} {1+12/100} = 100{1+12/100}2 iii) Amount at the end of third year ==100{1+12/100}2 {1+12/100}
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47 =100{1+12/100}3 Thus, the progression giving the amount at the end of each year is 100{1+12/100}2 ; 100{1+12/100}2 ; 100{1+12/100}3 ;….. This is a G.P with common ratio r = (1+12/100) In general, if P is the principal and I is the compound interest rate per annum, then the amount at the end of first year becomes P(1+i). Also the amount at the end of successive years forms a G.P. is P(1+i/100); P(1+i/100)2 :… with r = (1+i/100) th The n Term of G.P. The nth term of G.P. is also called the general term of the standard G.P. It is given by Tn =arn-1 , n=1,2,3,… It may be noted here that the power of r is one less than the index of Tn , which denotes the rank of this term in the progression. Sum of the First in Terms in G.P. Consider the first n terms of the standard form of G.P. a, ar, ar2 ,….arn-1 The sum, Sn of these terms is given by Sn = a + ar + ar2 +….+ arn-2 + arn-1 Multiplying both sides by r, we get rSn = ar + ar2 + ar3 + … + arn-1 + arn Subtracting the aboce equations, we have Sn - r Sn = a-arn Sn (1-r) = a(1-rn ) or Sn = a(1-rn ) ; r 1 and<1 (1-r) Changing the signs of the numerator and denominator, we have Sn = a(rn-1) , r 1and>1 (r-1) a) If r = 1, G.P. becomes a, a, a….so that Sn in this case is Sn = n.a. b) If number of terms in a G.P. are infinite, the Sn = a/1-r , r<1 = a/r-1, r>1 Example 4 A car is purchased for Rs. 80,000. Depreciation at 5% per annum for the first 3 years and 10% per annum for the next 3 years. Find the money value of the car after a period of 6 years. Solution: i) Depreciation for the first year = 80,000 x 5/100. Thus the depreciated value of the car at the end of first year is: = (80,000-80,000x5/100)
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48 = 80,000(1-5/100) ii) Depreciation for the second year = (Depreciated value at the end of first year)X (Rate of depreciation for the second year) = 80,000(1-5/100) 5/100) Thus the depreciated value at the end of the second year is = (Depreciated value after first year)- ( Depreciation for the second year) = 80,000(1-5/100) – 80,000(1-5/100) (5/100) = 80,000(1-5/100) (1-5/100) = 80,000(1-5/100)2 Calculating in the same way, the depreciated value at the end of three years is iii) Depreciation for fourth year = 80,000(1-5/100)3 (10/100) Thus the depreciated value at the end of the fourth year is = (Depreciated value after three year)X(Depreciation for fourth year) =80,000(1-5/100)3 – 80,000(1-5/100)3 (10/100) = 80,000(1-5/100)3 (1-10/100) Calculating in the same way, the depreciated value at the end of six years becomes = 80,000(1-5/100)3 (1-10/100)3 = Rs. 49,989.24 Exercise 1. Determine the common ratio of the G.P. 49,7,1/7,1/49,…. a) Find the sum to first 20 terms of G.P. b) Find the sum to infinity of the terms of G.P. 2. The population of a country in 1985 was 50 crore. Calculate the population in the year 2000 if the compounded annual rate of increase is (a) 1% (b) 2%
6.6 Let us Sum Up
Attention is then directed to defining the Arithmetic and Geometric Progressions and subsequently to their applications. An Arithmetic Progression is a sequence whose terms increases or decreases by a constant number. Geometric Progression is a sequence whose terms increases or decreases by a constant ration. Applications of these series find in sales, calculating interest in business, population studies, other socio-economic applications. Some of the examples shown in this Lesson brings out the importance of the study.
7.1 Aims and Objectives
So far we have discussed about various mathematical functions and theories. This Lesson deals with the applications of such theories in Finance. In financial management, lot of calculations are involved in the case of interest, depreciation values, and so on.
7.2 Some terms used in business calculations
a) Principal amount (P). This is the amount of money that is initially being considered. It might be an amount about to be invested or loaned or it may refer to the initial value or cost of plant or machinery. Thus if a company was considering a bank loan value or cost of plant or machinery. Thus if a company was considering a bank loan of say Rs.20000, this would be referred to as the principal amount to be borrowed. b) Accrued amount (A). This term is applied generally to a principal amount after some time has elapsed for which interest has been calculated and added. It is quite common to qualify a precisely according to time elapsed. Thus A1 , A2 , etc would mean the amount accrued at the end of the first and second years and so on. The company referred to in (a) above might owe, say, an accrued amount of Rs.22000 at the end of the first year and Rs. 24200 at the end of the second year (if no repayments had been made prior to this time). c) Rate of interest (i). Interest is the name given to a proportionate amount of money which is added to some principal amount (invested or borrowed). It is normally denoted by symbol i and expressed as a percentage rate per annum. For example if Rs. 100 is invested at interest rate 5% per annum (pa), it will accrue to Rs. 100 + (5% of Rs. 100) = Rs 100 + Rs.5 = Rs.105 at the end of one year. Note however, that for calculation purposes, a percentage rate is best written as a proportion. Thus, an interest rate of 10% would be written as i = 0.1 and 12.5% as i = 0.125 and so on.
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50 d) Number of time periods (n). The number of time periods over which amounts of money are being invested or borrowed is normally denoted by the symbol n. although n is usually a number of years, it could represent other time periods, such as a number of quarters or months.
7.3 Difference between Simple and Compound Interest
When an amount of money is invested over a number of years, the interest earned can be dealt with in two ways. a) Simple interest. This is where any interest earned is NOT added back to the principal amount invested. For example, suppose that Rs. 200 is invested at 10% simple interest per annum. The following table shows the state of the investment, year by year. Amount on which interest is calculated Rs. 200 Rs. 200 Rs. 200 Interest earned 10% of Rs. 200 = Rs. 20 10% of Rs. 200 = Rs. 20 10% of Rs. 200 = Rs. 20 Cumulative amount accrued Rs. 220 Rs. 240 Rs. 260 … etc.
Year 1 2 3
b) Compound interest. This is where interest earned is added back to the pervious amount accrued. For example, suppose that Rs. 200 is invested at 10% compound interest. The following table shows the state of the investment, year by year: Cumulative amount accrued 10% of Rs. 200 = Rs.20 Rs. 220 10% of Rs. 220 = Rs. 22 Rs. 242 10% of Rs. 242 = Rs. 24. 20 Rs. 266.20 … …etc The difference between the two methods can easily be seen by comparing the above two tables. Notice that the amount on which simple interest is calculated is always the same, namely, the original principal. Note that whether a principal amount is being invested [(as in a) and b)] or borrowed makes no difference to the considerations for interest. Amount on which interest is calculated Rs. 200 Rs. 220 Rs. 242 Interest earned
Year 1 2 3
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7.4 Formula for Amount Accrued (simple interest)
If Rs. 1000 is invested at 5% simple interest (pa) then 5% of Rs. 1000 = Rs.50 will be earned every year. In general terms, if amount P is invested at 100 i % simple interest (per time period), then the amount of interest earned per time period is given by P.i. For the data given, P = 1000 (Rs.) and 100i% = 5% (i.e.i = 0.05). Therefore, the interest earned per year = P.i = 1000(0.05) = Rs. 50. Also, if Rs. 3000 (P=3000) is invested at a (simple) interest rate of 9.5% (i=0.095), the interest earned in any year is P.i = 3000(0.095) = Rs. 285. Over a period of, say, 5 years, the accrued amount of the investment would be given by: Rs.[2000+5(285)] = Rs. 4425. that is, the original principal plus 5 year's-worth of interest. In general terms, if amount P is invested at 100i% simple interest, then the amount accrued over n years is given by the following formula. Accrued amount formula (simple interest) : An = P (1+i.n) Where: An = accrued amount at end of n-th year P = Principal amount i = (proportional) interest rate per year n = number of years. Generally, simple interest is of no great practical value in modern business and commercial situations, since in practice interest is always compounded.
7.5 Formula for Amount Accrued (compound interest)
Section 2.2.3 gave details of the year-by-year state of an investment of Rs. 200 at 10% compounded per annum (sometimes called an investment schedule). The following derives, in general terms, a formula to give the accrued amount of a principal at the end of any time period. If principal P is invested at a (compound) interest rate of 100% over n time periods, then: A1 = P + i i.e. the total amount accrued at the end of the first time period is the amount of the original principal plus the interest earned (i) for this period. So, A1 = P + P.i the interest earned in one time period is P.i i.e. A1 = P(1+i) factorizing A2 = P(1+i)+I amount accrued at end of second time period is the amount at the beginning of the period plus the interest earned (I) i.e. A1 = P(1+i)+P(1+i)i interest earned is accrued amount (at beginning) times i = P(1+i)(1+i) factorizing i.e. A2 = P(1+i)2 Similarly, A3 = P(1+i)3 , A4 = P(1+i)4 …. and so on.
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52 For example, if Rs. 5000 (P=5000) is invested at 9% p.a (i=0.09), then: Amount accrued after 1 year: A1 = P(1+i) = 5000(1.09) = Rs. 5450. Similarly: A2 = P(1+i)2 = 5000(1.09)2 = Rs. 5940.50. and A3 = P(1+i)3 = 5000(1.09)3 = Rs. 6475.15. … and so on. In general terms, if amount P is invested at 100i% compound interest, then the amount accrued over n years is given by the following formula. Accrued amount formula (simple interest) : An = P(1+i)n An =accrued amount at end of n-th year P= Principal amount i=(proportional) interest rate per year n=number of years.
Where:
7.6 Notes on previous formula
a) The above formula can be transposed to make P the subject as follows: P= A (1+i)n So that, given an interest rate and a time period, a principal can be found if accrued amount is known. For example, if some principal amount is invested at 12% (i=0.12) and amounts to Rs. 4917.25 (=A) after 3 years (n=3), then P (the principal amount) can be found using the above formula as: P= 4917.25 = 4917.25 = Rs. 3,500 1.123 1.4049 b) The standard time period for the calculation of interest is usually a year. Hence, from this point on, the value of n (the number of time periods) will be assumed to be a number of years unless stated otherwise. Example 1 (using accrued amount formula to solve simple problems) a) What will be the value of Rs. 450 compounded at 12% for 3 years? Here, P = 450, i=12/100 = 0.12 and n=3 Therefore A= P(1+i)n = 450(1+0.12)3 = 450 (1.12)3 = Rs. 632.22 c) A principal amount accrues to Rs. 8500 if it is compounded at 14.5% over 6 years. Find the value of this original amount. Here it is necessary to use the inverted formula, since P needs to be found. With A = 8500, i=0.145 and n=6 (and using the formula in section 11(a)): We have: P= A = 8500 = n (1+i) (1+0.145)6 8500 = 3772.12 (1.145)6
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53 That is, Rs. 37772.12 needs to be invested (at 14.5% over 6 years) in order to accrue to Rs. 8500.
7.7 Formula for calculating APR
Formula to calculate APR : Given a nominal annual rate of interest, the effective rate or actual percentage rate (APR) can be calculated as: APR = (1+i/n)n -1 Where: i= given nominal rate (as a proportion) n= number of equal compounding periods in one year.
Thus, for example a) 10% nominal, compounded quarterly, has APR = (1.025)4 – 1=0.1038=10.38% b) 24% nominal, compounded monthly, has APR = (1.02)12-1=0.2682=26.82% Example 7 (To calculate principal amount and APR) A company will have to spend Rs. 300,000 on new plant in two years from now. Currently investment rates are at a nominal 10%. a) What single sum should now be invested, if compounding is six- monthly? b) What is the APR? Answer a) Since compounding is six- monthly, the investment (P, say) must accrue to a value of Rs. 300,000 after four six- monthly periods. Note also that the interest rate for each sixmonth period is (10/2)% = 5%. Using, the compounding (accrued amount) formula, 300000=P(1+0.05)4 And re-arranging gives: 300000 = 246810.75 P= 1.054 That is, the amount to be invested is Rs. 246810.75 b) Using the previous APR formula: APR = (1+0.05)2 -1=(1.05)2 -1=0.1025=10.25%
7.8 Depreciation
Depreciation is an allowance made in estimates, valuations or balance sheets, normally for 'wear and tear'. It is normal accounting practice to depreciate the values of certain assets. There are several different techniques available for calculating depreciation, two of which are: a) Straight line (or Equal Instalment) depreciation and b) Reducing balance depreciation. These two methods can be thought of as the converse of the interest techniques dealt with so far in the Lesson. That is, instead of adding value to some original principal amount (as with interest), value is taken away in order to reduce the original amount. Straight
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54 line depreciation is the converse of simple interest with amounts being subtracted (rather than added), while the reducing balance method is the converse of compound interest. Straight line depreciation of the value of a machine Figure 2.1 3000 _ Initial value= Rs. 2500 Value(Rs.) 2000 -
1000 - Value at end of
Year 2 = Rs.1700
Final value=Rs.500
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7.8.1 Straight line depreciation
Given that the value of an asset must be depreciated, this technique involves subtracting the same amount from the original book value each year. Thus, for example, if the value of a machine is to depreciate from Rs. 2500 to Rs. 500 over a period of five years, then the annual depreciation would be Rs (2500-500)/5=Rs. 400. The term 'straight line' comes from the fact that 2500 can be plotted against year 0 and 500 against year 5 on a graph, the two points joined by a straight line and then values for intermediate years can be read from the line. See Figure 2.1
7.8.2 Reducing balance depreciation
We already know that to increase some value P by 100%, we need to calculate the quantity P(1+i), and if this is done n times, successively, the accrued value is given by P(1+i)n (from section 10). A similar argument is applied if we need to decrease (or equivalently depreciate) some value B say by 100%. Here, we need to calculate B(1- i). Notice that the multiplier is now 1-i, rather than 1+i. If the depreciation is carried out successively, n times, the accrued value will be given as B(1- i)n . For example, Rs. 2550 depreciated by 15% is Rs. 25500(1-0.15) = Rs. 2550(0.85)= Rs. 2167.50. Also, if Rs 2550 was successively depreciated over four time periods by 15%, the final depreciated value=Rs. 2550(1-0.15)4 =Rs. 2550(0.85)4 = Rs 1331.12. Reducing balance depreciation is the name given to the technique of depreciating the book value of an asset by a constant percentage.
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7.9 Formula for Reducing Balance Depreciation
Reducing balance depreciated value formula : If book value B is subject to reducing balance deprecation at rate 100i% over n time periods, the depreciated value at the end of the n-th time period is given by: D = B(1-i)n Where: D= depreciated value at the end of the n-th time period B= original book value i= depreciation rate (as a proportion) n= number of time periods (normally years). Note that by re-arranging the above formula, any one of the variables B, i and n could be found, given the other three. For example: B = D (giving B in terms of D, i and n) (1-i)n Thus if the depreciated value (D) of an asset was Rs. 5378.91 after three years depreciation at 25%, the original book value can be calculated as: B = Rs. 5378.91 0.753 = Rs. 12750. Also, since: (1- i)n = D/B Then: 1- i=n D/B (giving 1- i in terms of D,V and n) Example 8 (Reducing balance and straight line depreciation) A mainframe computer whose cost is Rs. 220,000 will depreciate to a scrap value of Rs. 12000 in 5 years. a) If the reducing balance method of depreciation is used, find the depreciation rate. b) What is the book value of the computer at the end of the third year? c) How much more would the book value be at the end of the third year if the straight line method of depreciation had been used ? Answer Variables given are: D=12000, B=220000, n=5 and D/B = 12,000 = 0.0545 220,000 a) Now 1-i = 5 0.0545 Therefore, 1- i = 0.5589 and so i=0.4411. Thus the depreciation rate is 44.11% b)The book value at the end of year 3 is given by: B(1- i)3 = 220000(0.5589)3 = Rs. 38408. c) Using the straight line method, the annual depreciation is: Rs. (220000-12000) 5
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56 = Rs. 41600 Thus, after three years, the book value would be : Rs[220000-3(41600)]=Rs 95200. So that the book value, using this method, would be Rs[95200-38408]=Rs. 56792 more than using the reducing balance method (at the end of the third year).
7.10 Let us Sum Up
Let us sum up the Lesson with the understanding of certain mathematical applications in Financial management. Usual applications involves calculations of different types of interest on capital, like simple interest, compound interest, accrued interest, calculation of depreciation, etc. With suitable examples and with simple illustration, the concept has been well presented in this Lesson for better understanding.
7.11 Lesson – End Activities
1. Mention the differences between simple interest and compound interest. 2. What is the need for giving depreciation?
8.1 Aims and Objectives
You may be aware of the fact that prior to the industrial revolution individual business was small and production was carried out on a very small scale mainly to cater to the local needs. The management of such business enterprises was very different from the present management of large scale business. The decisions was much less extensive that at present. Thus they used to make decisions based upon his past experience and intuition only. Some of the reasons for this were: 1. The marketing of the product was not a problem because customers were, for the large part, personally known to the owner of the business. There was hardly any competition in the business. 2. Test marketing of the product was not needed because the owner used to know the choice and requirement of the customers just by personal interaction. 3. The manager (also the owner) also used to work with his workers at the shopfloor. He knew all of them personally as the number was small. This reduced the need for keeping personal data. 4. The progress of the work was being made daily at the work centre itself. Thus production records were not needed. 5. Any facts the owner needed could be learnt direct from observation and most of what he required was known to him. Now, in the face of increasing complexity in business and industry, intuition alone has no place in decision- making because basing a decision on intuition becomes highly questionable when the decision involves the choice among several courses of action each of which can achieve several management objectives simultaneously. Hence there is a need for training people who can manage a system both efficiently and creatively. Quantitative techniques have made valuable contribution towards arriving at an effective decision in various functional areas of management- marketing, finance, production and
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58 personnel. Today, these techniques are also widely used in regional planning, transportation, public health, communication, military, agriculture, etc. Quantitative techniques are being used extensively as an aid in business decision- making due to following reasons: 1. Complexity of today's managerial activities which involve constant analysis of existing situation, setting objectives, seeking alternatives, implementing, coordinating, controlling and evaluating the decision made. 2. Availability of different types of tools for quantitative analysis of complex managerial problems. 3. Availability of high speed computers to apply quantitative techniques ( or models) to real life problems in all types of organisations such as business, industry, military, health, and so on. Computers have played an important role in arriving at the optimal solution of complex managerial problems. In spite of these reasons, the quantitative approach, however, does not totally eliminate the scope of qualitative or judgment ability of the decision- maker. Of course these techniques complement the experience and knowledge of decision- maker in decisionmaking.
8.2 Meaning of Quantitative Techniques
Quantitative techniques refer to the group of statistical, and operations research (or programming) techniques as shown in the following chart.
Quantitative-Techniques
Operations research (or Programming) Techniques The quantitative approach in decision-making requires that, problems be defined, analysed and solved in a conscious, rational, systematic and scientific manner based on data, facts, information, and logic and not on mere whims and guesses. In other words, quantitative techniques ( tools or methods) provide the decision – maker a scientific method based on quantitative data in identifying a course of action among the given list of courses of action to achieve the optimal value of the predetermined objective or goal. One common characteristic of all types of quantitative techniques is that numbers, symbols or mathematical formulae ( or expressions) are used to represent the models of reality.
Statistical Techniques
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8.3 Statistics
Statistics The word statistics can be used in a number of ways. Commonly it is described in two senses namely: 1. Plural Sense ( Statistical Data)
The plural sense of statistics means some sort of statistical data. When it means statistical data, it refers to numerical description of quantitative aspects of things. These descriptions may take the form of counts or measurements. For example, statistics of students of a college include count of the number of students, and separate counts of number of various kinds as such, male and females, married and unmarried, or undergraduates and post-graduates. They may also include such measurements as their heights and weights.
2.
Singular Sense ( Statistical Methods)
The large volume of numerical information ( or data) gives rise to the need for systematic methods which can be used to collect, organise or classify, present, analyse and interpret the information effectively for the purpose of making wise decisions. Statistical methods include all those devices of analysis and synthesis by means of which statistical data are systematically collected and used to explain or describe a given phenomena. The above mentioned five functions of statistical methods are also called phases of a statistical investigation. Methods used in analysing the presented data are numerous and contain simple to sophisticated mathematical techniques. As an illustration, let us suppose that we are interested in knowing the income level of the people living in a certain city. For this we may adopt the following procedures: a) Data Collection: The following data is required for the given purpose: § Population of the city § Number of individuals who are getting income § Daily income of each earning individual b) Organise ( or Condense) the data: the data so obtained should now be organised in different income groups. This will reduce the bulk of the data. c) Presentation: the organised data may now be presented by means of various types of graphs or other visual aids. Data presented in an orderly manner facilitates statistical analysis. d) Analysis: on the basis of systematic presentation (tabular form or graphical form) determine the average income of an individual and extent of disparities that exist. This information will help to get an understanding of the phenomenon ( i.e. income of individuals.)
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60 e) Interpretation: All the above steps may now lead to drawing conclusions which will aid in decision-making-a policy decision for improvement of the existing situation. Characteristics of data It is probably more common to refer to data in quantitative form as statistical data. It is probably more common to refer to data in quantitative form as statistical data. But not all numerical data is statistical. In order that numerical description may be called statistics they must possess the following characteristics: i) They must be aggregate of facts, for example, single unconnected figures cannot be used to study the characteristics of the phenomenon. ii) They should be affected to a marked extent by multiplicity of causes, for example, in social services the observations recorded are affected by a number of factors ( controllable and uncontrollable) iii) They must be enumerated or estimated according to reasonable standard of accuracy, for example, in the measurement of height one may measure correct upto 0.01 of a cm; the quality of the product is estimated by certain tests on small samples drawn from a big lot of products. iv) They must have been collected in a systematic manner for a pre-determined purpose. Facts collected in a haphazard manner, and without a complete awareness of the object, will be confusing and cannot be made the basis of valid conclusions. For example collected data on price serve no purpose unless one knows whether he wants to collect data on wholesale or retail prices and what are the relevant commodities in view. v) They must be placed in relation to each other. That is, data collected should be comparable; otherwise these cannot be placed in relation to each other, e.g. statistics on the yield of crop and quality of soil are related byt these yields cannot have any relation with the statistics on the health of the people. vi) They must be numerically expressed. That is, any facts to be called statistics must be numerically or quantitatively expressed. Qualitative characteristics such as beauty, intelligence, etc. cannot be included in statistics unless they are quantified.
8.4 Types of Statistical Data
An effective managerial decision concerning a problem on hand depends on the availability and reliability of statistical data. Statistical data can be broadly grouped into two categories: 1) Secondary ( or published) data 2) Primary (or unpublished) data The Secondary data are those which have already been collected by another organisation and are available in the published form. You must first check whether any such data is available on the subject matter of interest and make use of it, since it will save considerable time and money. But the data must be scrutinised properly since it was
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61 originally collected perhaps for another p8urpose. The data must also be checked for reliability, relevance and accuracy. A great deal of data is regularly collected and disseminated by international bodies such as: World Bank, Asian Development Bank, International Labour Organisation, Secretariat of United Nations, etc., Government and its many agencies: Reserve Bank of India, Census Commission, Ministries-Ministry of Economics Affairs, Commerce Ministry; Private Research Organisations, Trade Associations etc. When secondary data is not available or it is not reliable, you would need to collect original data to suit your objectives. Original data collected specifically for a current research are known as primary data. Primary data can be collected from customers, retailers, distributors, manufacturers or other information sources, primary data may be collected through any of the three methods: observation, survey, and experimentation. Data are also classified as micro and macro. Micro data relate to a particular unit region whereas macro data relate to the entire industry, region or economy.
8.5 Classification of Statistical Methods
The filed of statistics provides the methods for collecting, presenting and meaningfully interpreting the given data. Statistical Methods broadly fall into three categories as shown in the following chart. Statistical Methods
Descriptive Statistics
Inductive Statistics
Statistical Decision Theory
Data Collection Presentation
Statistical Inference Estimation
Analysis of Business Decision
Descriptive Statistics There are statistical methods which are used for re-arranging, grouping and summarising sets of data to obtain better information of facts and thereby better description of the situation that can be made. For example, changes in the price- index. Yield by wheat etc. are frequently illustrated using the different types of charts and graphs. These devices summarise large quantities of numerical data for easy understanding. Various types of averages, can also reduce a large mass of data to a single descriptive number. The
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62 descriptive statistics include the methods of collection and presentation of data, measure of Central tendency and dispersion, trends, index numbers, etc. Inductive Statistics It is concerned with the development of some criteria which can be used to derive information about the nature of the members of entire groups ( also called population or universe) from the nature of the small portion (also called sample) of the given group. The specific values of the population members are called 'parameters' and that of sample are called 'Statistics'. Thus, inductive statistics is concerned with estimating population parameters from the sample statistics and deriving a statistical inference. Samples are drawn instead of a complete enumeration for the following reasons: i) the number of units in the population may not be known ii) the population units may be too many in number and/or widely dispersed. Thus complete enumeration is extremely time consuming and at the end of a full enumeration so much time is lost that the data becomes obsolete by that time. iii) It may be too expensive to include each population item. Inductive statistics, includes the methods like: probability and probability distributions; sampling and sampling distribution; various methods of testing hypothesis; correlation, regression, factor analysis; time series analysis. Statistical Decision Theory Statistical decision theory deals with analysing complex business problems with alternative course of action ( or strategies) and possible consequences. Basically,. It is to provide more concrete information concerning these consequences, so that best course of action can be identified from alternative courses of action. Statistical decision theory relies heavily not only upon the nature of the problem on hand, but also upon the decision environment. Basically there are four different states of decision environment as given below: State of decision Certainty Risk Uncertainty Conflict Consequences Deterministic Probabilistic Unknown Influenced by an opponent
Since statistical decision theory also uses probabilities (subjective or prior) in analysis, therefore it is also called a subjectivist approach. It is also known as Bayesian approach because Baye's theorem, is used to revise prior probabilities in the light of additional information.
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8.6 Various Statistical Techniques
A brief comment on certain standard techniques of statistics which can be helpful to a decision- maker in solving problems is given below. i) Measures of Central Tendency: Obviously for proper understanding of quantitative data, they should be classified and converted into a frequency distribution ( number of times or frequency with which a particular data occurs in the given mass of data.). This type of condensation of data reduces their bulk and gives a clear picture of their structure. If you want to know any specific characteristics of the given data or if frequency distribution of one set of data is to be compared with another, then it is necessary that the frequency distribution help us to make useful inferences about the data and also provide yardstick for comparing different sets of data. Measures of average or central tendency provide one such yardstick. Different methods of measuring central tendency, provide us with different kinds of averages. The main three types of averages commonly used are: a) Mean: the mean is the common arithmetic average. It is computed by dividing the sum of the values of the observations by the number of items observed. b)Median: the median is that item which lies exactly half-way between the lowest and highest value when the data is arranged in an ascending or descending order. It is not affected by the value of the observation but by the number of observations. Suppose you have the data on monthly income of households in a particular area. The median value would give you that monthly income which divides the number of households into two equal parts. Fifty per cent of all the households have a monthly income above the median value and fifty per cent of households have a monthly income below the median income. c) Mode: the mode is the central value (or item) that occurs most frequently. When the data organised as a frequency distribution the mode is that category which has the maximum number of observations. For example, a shopkeeper ordering fresh stock of shoes for the season would make use of the mode to determine the size which is most frequently sold. The advantages of mode are that (a) it is easy to compute, (b) is not affected by extreme values in the frequency distribution, and (c) is representative if the observations are clustered at one particular value or class. ii) Measures of Dispersion: the measures of central tendency measure the most typical value around which most values in the distribution tend to coverage. However, there are always extreme values in each distribution. These extreme values indicate the spread or the dispersion of the distribution. The measures of this spread are called 'measures of dispersion' or 'variation' or 'spread'. Measures of dispersion would tell you the number of values which are substantially different from the mean, median or mode. The commonly used measures of dispersion are range, mean deviation and standard deviation. The data may spread around the central tendency in a symmetrical or an asymmetrical pattern. The measures of the direction and degree of symmetry are called measures of the skewness. Another characteristic of the frequency distribution is the shape of the peak, when it is plotted on a graph paper. The measures of the peakedness are called measures of Kurtosis.
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iii) Correlation: Correlation coefficient measures the degree to which the charge in one variable ( the dependent variable) is associated with change in the other variable (independent one). For example, as a marketing manager, you would like to know if there is any relation between the amount of money you spend on advertising and the sales you achieve. Here, sales is the dependent variable and advertising budget is the independent variable. Correlation coefficient, in this case, would tell you the extent or relationship between these two variables,' whether the relationship is directly proportional (i.e. increase or decrease in advertising is associated with decrease in sales) or it is an inverse relationship (i.e. increasing advertising is associated with decrease in sales and vice- versa) or there is no relationship between the two variables. However, it is important to note that correlation coefficient does not indicate a casual relationship, Sales is not a direct result of advertising alone, there are many other factors which affect sales. Correlation only indicates that there is some kind of association-whether it is casual or causal can be determined only after further investigation. Your may find a correlation between the height of your salesmen and the sales, but obviously it is of no significance. iv) Regression Analysis: For determining causal relationship between two variables you may use regression analysis. Using this technique you can predict the dependent variables on the basis of the independent variables. In 1970, NCAER ( National Council of Applied and Economic Research) predicted the annual stock of scooters using a regression model in which real personal disposable income and relative weighted price index of scooters were used as independent variable. The correlation and regression analysis are suitable techniques to find relationship between two variables only. But in reality you would rarely find a one-to-one causal relationship, rather you would find that the dependent variables are affected by a number of independent variables. For example, sales affected by the advertising budget, the media plan, the content of the advertisements, number of salesmen, price of the product, efficiency of the distribution network and a host of other variables. For determining causal relationship involving two or more variables, multi- variable statistical techniques are applicable. The most important of these are the multiple regression analysis deiscriminant analysis and factor analysis. v) Time Series Analysis : A time series consists of a set of data ( arranged in some desired manner) recorded either at successive points in time or over successive periods of time. The changes in such type of data from time to time are considered as the resultant of the combined impact of a force that is constantly at work. This force has four components: (i) Editing time series data, (ii) secular trend, (iii) periodic changes, cyclical changes and seasonal variations, and (iv) irregular or random variations. With time series analysis, you can isolate and measure the separate effects of these forces on the variables. Examples of these changes can be seen, if you start measuring increase in cost of living, increase of population over a period of time, growth of agricultural food production in India over the last fifteen years, seasonal requirement of items, impact of floods, strikes, wars and so on.
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65 vii) Index Numbers: Index number is a relative number that is used to represent the net result of change in a group of related variables that has some over a period of time. Index numbers are stated in the form of percentages. For example, if we say that the index of prices is 105, it means that prices have gone up by 5% as compared to a point of reference, called the base year. If the prices of the year 1985 are compared with those of 1975, the year 1985 would be called "given or current year" and the year 1975 would be termed as the "base year". Index numbers are also used in comparing production, sales price, volume employment, etc. changes over period of time, relative to a base. viii) Sampling and Statistical Inference: In many cases due to shortage of time, cost or non-availability of data, only limited part or section of the universe (or population) is examined to (i) get information about the universe as clearly and precisely as possible, and (ii) determine the reliability of the estimates. This small part or section selected from the universe is called the sample, and the process of selection such a section (or past) is called sampling. Schemes of drawing samples from the population can be classified into two broad categories: a) Random sampling schemes: In these schemes drawing of elements from the population is random and selection of an element is made in such a way that every element has equal change ( probability) of being selected. b) Non-random sampling schemes: in these schemes, drawing of elements for the population is based on the choice or purpose of selector. The sampling analysis through the use of various 'tests' namely Z-normal distribution, student's 't' distribution; F-distribution and x2 –distribution make possible to derive inferences about population parameters with specified level of significance and given degree of freedom.
8.7 Advantages of Quantitative Approach to Management
Executives at all levels in business and industry come across the problem of making decision at every stage in their day-to-day activities. Quantitative techniques provide the executive with scientific basis for decision- making and enhance his ability to make longrange plans and to solve every day problems of running a business and industry with greater efficiency and confidence. Some of the advantages of the study of statistics are: 1. Definiteness: the study of statistics helps us in presenting general statements in a precise and a definite form. Statements of facts conveyed numerically are more precise and convincing than those stated qualitatively. For example, the statement that "literacy rate as per 1981 census was 36% compared to 29% for 1971 census" is more convincing than stating simply that "literacy in our country has increased".
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66 2. Condensation: The new data is often unwieldy and complex. The purpose of statistical methods is to simplify large mass of data and to present a meaningful information from them. For example, it is difficult to form a precise idea about the income position of the people of India from the data of individual income in the country. The data will be easy to understand and more precisely if it can be expressed in the form of per capita income. 3. Comparison: According to Bodding, the object of statistics is to enable comparisons between past and present results with a view to ascending the reasons for change which have taken place and the effect of such changes in the future. Thus, if one wants to appreciate the significance of figures, then he must compare them with other of the same kind. For example, the statement "per capita income has increased considerably" shall not be meaningful unless some comparison of figures of past is made. This will help in drawing conclusions as to whether the standard of living of people of India is improving. 4. Formulation of policies: Statistics provides that basic material for framing policies not only in business but in other fields also. For example, data on birth and mortality rate not only help is assessing future growth in population but also provide necessary data fro framing a scheme of family planning. 5. Formulating and testing hypothesis: statistical methods are useful in formulating and testing hypothesis or assumption or statement and to develop new theories. For example, the hypothesis: "whether a student has benefited from a particular media of instruction", can be tested by using appropriate statistical method. 6. Prediction: For framing suitable policies or plans, and then for implementation it is necessary to have the knowledge of future trends. Statistical methods are highly useful for forecasting future events. For example, for a businessman to decide how many units of an item should be produced in the current year, it is necessary for him to analyse the sales data of the past years.
8.8 Applications of Quantitative Techniques in Business and Management
Some of the areas where statistics can be used are as follows:
Economics
§ Measurement of gross national product and input-output analysis § Determination of business cycle, long-term growth and seasonal fluctuations § Comparison of market prices, cost and profits of individual firms § Analysis of population, land economics and economic geography Operational studies of public utilities Formulation of appropriate economic policies and evaluation of their effect
§ §
Research and Development
§ § § Development of new product lines Optimal use of resources Evaluation of existing products
Natural Science
§ Diagnosing the disease based on data like temperature, pulse rate, blood pressure etc. § Judging the efficacy of particular drug for curing a certain disease § Study of plant life Exercises 1. Comment on the following statements: a) "Statistics are numerical statement of facts but all facts numerically stated are not statistics" b) "Statistics is the science of averages". 2. What is the type of the following models? a) Frequency curves in statistics. b) Motion films. c) Flow chart in production control, and c) Family of equations describing the structure of an atom.
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68 3. List at least two applications of statistics in each, functional area of management. 4. What factors in modern society contribute to the increasing importance of quantitative approach to management? 5. Describe the major phases of statistics. Formulate a business problem and analyse it by applying these phases. 6. Explain the distinction between: a) Static and dynamic models b) Analytical and simulation models c) Descriptive and prescriptive models. 7. Describe the main features of the quantitative approach to management.
8.9 Let us Sum Up
We have so for learned the quantitative techniques and quantitative approach to management with its characteristics.
8.10 Lesson – End Activities
1. What are the different types of statistical data available. 2. Mention the advantages of quantitative approach to management.
8.11 References
1. Gupta. S.P. – Statistical Methods.
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9.1 Aims and Objectives
The successful use of the data collected depends to a great extent upon the manner in which it is arranged, displayed and summarized. This Lesson mainly deals with the presentation of data. Presentation of data can be displayed either in tabular form or through charts. In the tabular form, it is necessary to classify the data before the data tabulated. Therefore, this unit is divided into two section, viz., (a) classification of data and (b) charting of data.
9.2 Classification of Data
After the data has been systematically collected and edited, the first step in presentation of data is classification. Classification is the process of arranging the data according to the points of similarities and dissimilarities. It is like the process of sorting the mail in a post office where the mail for different destinations is placed in different compartments after it has been carefully sorted out from the huge heap.
9.3 Objectives of Classification
The principal objectives of classifying data are: i) to condense the mass of data in such a way that salient features can be readily noticed ii) to facilitate comparisons between attributes of variables iii) to prepare data which can be presented in tabular form iv) to highlight the significant features of the data at a glance
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9.4 Types of Classification
Some common types of classification are: · Geographical i.e., according to area or region · Chronological, i.e., according to occurrence of an event in time. · Qualitative, i.e., according to attributes. · Quantitative, i.e., according to magnitudes. Geographical Classification: In this type of classification, data is classified according to area or region. For example, when we consider production of wheat State wise, this would be called geographical classification. The listing of individual entries are generally done in an alphabetical order or according to size to emphasise the importance of a particular area or region. Chronological Classification: when the data is classified according to the time of the occurrence, it is known as chronological classification. For example, sales figure of a company for last six years are given below: Year Sales (Rs. Lakhs) 175 220 350 Year Sales (Rs. Lakhs) 485 565 620
1982-83 1983-84 1984-85
1985-86 1986-87 1987-88
Qualitative Classification: When the data is classified according to some attributes(distinct categories) which are not capable of measurement is known as qualitative classification. In a simple (or dichotomous) classification, as attribute is divided into two classes, one possessing the attribute and the other not possessing it. For example, we may classify population on the basis of employment, i.e., the employed and the unemployed. Similarly we can have manifold classification when an attribute is divided so as to form several classes. For example, the attribute education can have different classes such as primary, middle, higher secondary, university, etc. Quantitative Classification: when the day is classified according to some characteristics that can be measured, it is called quantitative classification. For example, the employees of a company may be classified according to their monthly salaries. Since quantitative data is characterized by different numerical values, the data represents the values of a variable. Quantitative data may be further classified into one or two types: discrete or continuous. The term discrete data refers to quantitative data that is limited to certain numerical values of a variable. For example, the number of employees in an organisation or the number of machines in a factory are examples of discrete data. Continuous data can take all values of the variable. For example, the data relating to weight, distance, and volume are examples of continuous data. The quantitative classification becomes the basis for frequency distribution.
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When the data is arranged into groups or categories according to conveniently established divisions of the range of the observations, such an arrangement in tabular form is called a frequency distribution. In a frequency distribution, raw data is represented by distinct groups which are known as classes. The number of observations that fall into each of the classes is known as frequency. Thus, a frequency distribution has two parts, on its left there are classes and on its right are frequencies. When data is described by a continuous variable it is called continuous data and when it is described by a discrete variables, it is called discrete data. The following are the two examples of discrete and continuous frequency distributions.
No.of Employees 110 120 130 140 150 160
No.of companies 25 35 70 100 18 12
Age (years) 20-25 25-30 30-35 35-40 40-45 45-50
No.of workers 15 22 38 47 18 10
Discrete frequency distribution
Continuous frequency distribution
9.5 Construction of a Discrete Frequency Distribution
The process of preparing a frequency distribution is very simple. In the case of discrete data, place all possible values of the variable in ascending order in one column, and then prepare another column of 'Tally' mark to count the number of times a particular value of the variable is repeated. To facilitate counting, block of five 'Tally' marks are prepared and some space is left in between the blocks. The frequency column refers to the number of 'Tally' marks, a particular class will contain. To illustrate the construction of a discrete frequency distribution, consider a sample study in which 50 families were surveyed to find the number of children per family. The data obtained are: 3 2 2 1 3 4 2 1 3 4 5 0 2 1 2 3 3 2 1 1 2 3 0 3 2 1 4 3 5 5 4 3 6 5 4 3 1 0 6 4 3 1 2 0 1 2 3 4 5 To condense this data into a discrete frequency distribution, we shall take the help of 'Tally' marks as shown below:
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9.6 Construction of a Continuous Frequency Distribution
In constructing the frequency distribution for continuous data, it is necessary to clarify some of the important terms that are frequently used. Class Limits: Class limits denote the lowest and highest value that cab be include in the class. The two boundaries (i.e., lowest and highest) of a class are known as the lower limit and the upper limit of the class. For example, in the class 60-69, 60 is the lower limit and 69 is the upper limit or we can say that there can be no value in that class which is less than 60 and more than 69. Class Intervals: The class interval represents the width (span or size) of a class. The width may be determined by subtracting the lower limit of one class from the lower limit of the following class (alternatively successive upper limits may be used). For example, if the two classes are 10-20 and 20-30, the width of the class interval would be the difference between the two successive lower limit of the same class, i.e., 20-10=10. Class Frequency: The number of observations falling within a particular class is called its class frequency or simply frequency. Total frequency (sum of all the frequencies) indicate the total number of observations considered in a given frequency distribution. Class Mid-point: Mid-point of a class is defined as the sum of two successive lower limits divided by two. Therefore, it is the value lying halfway between the lower and upper class limits. In the example taken above the mid-point would be(10+20)/2=15 corresponding to the class 10-20 and 25 corresponding to the class 20-30. Types of Class Interval: There are different ways in which limits of class intervals can be shown such as: i) Exclusive and Inclusive method, and ii) Open-end Exclusive Method: The class intervals are so arranged that the upper limit of one class is the lower limit of the next class. The following example illustrates this point.
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Sales (Rs. Thousands) 20-25 25-30 30-35
No. of firms 20 28 35
Sales (Rs. Thousands) 35-40 40-45 45-50
No. of firms 27 12 8
In the above example there are 20 firms whose sales are between Rs. 20,000 and Rs. 24,999. A firm with sales of exactly Rs. 25 thousand would be included in the next class viz. 25-30. Therefore in the exclusive method, it is always presumed that upper limit is excluded.
In this example, there are 20 firms whose sales are between Rs. 20,000 and Rs. 24,999. A firm whose sales are exactly Rs. 25,000 would be included in the next class. Therefore in the inclusive method, it is presumed that upper limit is included. It may be observed that both the methods give the same class frequencies, although the class intervals look different. Whenever inclusive method is used for equal class intervals, the width of class intervals can be obtained by taking the difference between the two lower limits (or upper limits). Open-End: In an open-end distribution, the lower limit of the very fist class and upper limit of the last class is not given. In distribution where there is a big gap between minimum and maximum values, the open-end distribution can be used such as in income distributions. The income disparities, of residents of a region may vary between Rs. 800 to Rs. 50,000 per month. In such a case, we can form classes like: Less than Rs. 1,000 1,000 - 2,000 2,000 - 5,000 5,000 - 10,000 10,000 - 25,000 25,000 and above Remark: To ensure continuity and to get correct class intervals, we shall adopt exclusive method. However, if inclusive method is suggested then it is necessary to make an
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74 adjustment to determine the class interval. This can be done by taking the average value of the difference between the lower limit of the succeeding class and the upper limit of the class. In terms of formula:
Correction factor = Lower Limit of second class-Upper Limit of the first class 2
This value so obtained is deducted from all lower limits and added to all upper limits. For instance, the example discussed for inclusive method can easily be converted into exclusive case. Take the difference between 25 and 24,999 and divide it by 2. Thus correction factor becomes (25-24,999)/2=0.0005. Deduct this value from lower limits and add it to upper limits. The new frequency distribution will take the following.
Sales (Rs. Thousands) 19.9995-24.9995 24.9995-29.9995 29.9995-34.9995
No.of firms 20 28 35
Sales (Rs. Thousands) 34.9995-39.9995 39.9995-44.9995 44.9995-49.9995
No.of firms 27 12 8
9.7 Guidelines for Choosing the Classes
The following guidelines are useful in choosing the class intervals. 1. The number of classes should not be too small or too large. Preferably, the number of classes should be between 5 and 15. However, there is no hard and fast rule about it. If the number of observations is smaller, the number of classes formed should be towards the lower side of this towards the upper side of the limit. 2. If possible, the widths of the intervals should be numerically simple like 5,10,25 etc. Values like 3,7,19 etc. should be avoided. 3. It is desirable to have classes of equal width. However, in case of distributions having wide gap between the minimum and maximum values, classes with unequal class interval can be formed like income distribution. 4. The starting point of a class should begin with 0,5,10 or multiplies thereof. For example, if the minimum value is 3 and we are taking a class interval of 10, the first class should be 0-10 and not 3-13. 5. The class interval should be determined after taking into consideration the minimum and maximum values and the number of classes to be formed. For example, if the income of 20 employees in a company varies between Rs. 1100 and Rs.5900 and we want to form 5 classes, the class interval should be 1000 (5900-1100) 1000
= 4.8 or 5.
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9.8 Cumulative and Relative Frequencies
It is often useful to express class frequencies in different ways. Rather than listing the actual frequency opposite each class, it may be appropriate to list either cumulative frequencies or relative frequencies or both. Cumulative Frequencies: As its name indicates, it cumulates the frequencies, starting at either the lower or highest value. The cumulative frequency of a given class interval thus represents the total of all the previous class frequencies including the class against which it is written. To illustrate the concept of cumulative frequencies consider the following example Monthly salary (Rs.) 1000-1200 1200-1400 1400-1600 1600-1800 1800-2000 No.of employees 5 14 23 50 52 Monthly Salary (Rs.) 2000-2200 2200-2400 2400-2600 2600-2800 No.of employees 25 22 7 2
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76 If we keep on adding the successive frequency of each class starting from the frequency of the very first class, we shall get cumulative frequencies as shown below: Monthly Salary(Rs.) 1000-1200 1200-1400 1400-1600 1600-1800 1800-2000 2000-2200 2200-2400 2400-2600 2600-2800 No. of employees 5 14 23 50 52 25 22 7 2 Total 200 Relative Frequencies: Very often, the frequencies in a frequency distribution are converted to relative frequencies to show the percentage for each class. If the frequency of each class is divided by the total number of observations (total frequency), then this proportion is referred to as relative frequency. To get the percentage of each class, multiply the relative frequency by 100. For the above example, the values computed for relative for relative frequency and percentage are shown below: Monthly Salary (Rs.) 1000-1200 1200-1400 1400-1600 1600-1800 1800-2000 2000-2200 2200 -2400 2400-2600 2600-2800 No. of employees 5 14 23 50 52 25 22 7 2 200 Relative frequency 0.025 0.070 0.115 0.250 0.260 0.125 0.110 0.035 0.010 1.000 percentage Cumulative frequency 5 19 42 92 144 169 191 198 200
2.5 7.0 11.5 25.0 26.0 12.5 11.0 3.5 1.0 100%
There are two important advantages in looking at relative frequencies (percentages) instead of absolute frequencies in a frequency distribution. 1. Relative frequencies facilitate the comparisons of two or more than two sets of data. 2. Relative frequencies constitute the basis of understanding the concept of probability.
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9.9 Charting of Data
Charts of frequency distributions which cover both diagrams and graphs are useful because they enable a quick interpretation of the data. A frequency distribution can be presented by a variety of methods. In this section, the following four popular methods of charting frequency distribution are discussed in detail. i) Bar Diagram ii) Histogram iii) Frequency Polygon iv) Ogive or Cumulative Frequency Curve Bar Diagram: Bar diagrams are most popular. One can see numerous such diagrams in newspapers, journals, exhibitions, and even on television to depict different characteristics of data. For example, population, per capita income, sales and profits of a company can be shown easily through bar diagrams. It may be noted that a bar is thick line whose width is shown to attract the viewer. A bar diagram may be either vertical or horizontal. In order to draw a bar diagram, we take the characteristic (or attribute) under consideration on the X-axis and the corresponding value on the Y-axis. It is desirable to mention the value depicted by the bar on the top of the bar. To explain the procedure of drawing a bar diagram, we have taken the population figures (in millions) of India which are given below: Bar Diagram
800700600-
514.18
500 468.16
391.01
400 318.66 300 200100 Year 1931 1941 1951 278.98
429.23
1961
1971
1981
Year
Take the years on the X-axis and the population figure on the Y-axis and draw a bar to show the population figure for the particular year. This is shown above:
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78 As can be seen from the diagram, the gap between one bar and the other bar is kept equal. Also the width of different bars is same. The only difference is in the length of the bars and that is why this type of diagram is also known as one dimensional. Histogram: One of the most commonly used and easily understood methods for graphic presentation of frequency distribution is histogram. A histogram is a series of rectangles having areas that are in the same proportion as the frequencies of a frequency distribution. To construct a histogram, on the horizontal axis or X-axis, we take the class limits of the variable and on the vertical axis or Y-axis, we take the frequencies of the class intervals shown on the horizontal axis. If the class intervals are of equal width, then the vertical bars in the histogram are also of equal width. On the other hand, if the class intervals are unequal, then the frequencies have to be adjusted according to the width of the class interval. To illustrate a histogram when class intervals are equal, let us consider the following example.
Daily Sales (Rs. Thousand) 10-20 20-30 30-40 40-50
No. of companies 15 22 35 30
Daily Sales (Rs. Thousand) 50-60 60-70 70-80 80-90
No. of companies 25 20 16 7
In this example, we may observe that class intervals are of equal width. Let us take class intervals on the X-axis and their corresponding frequencies on the Y-axis. On each class interval (as base), erect a rectangle with height equal to the frequency of that class. In this manner we get a series of rectangles each having a class interval as its width and the frequency as its height as shown below :
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79 Histogram with Equal Class Intervals
35 35 30 30
Number of Companies
25 22 20
25 20 15 15 10 10 7 5
0
10
20
30
40
50
60
70
80
90
100
Daily Sales (In Thousand Rupees)
It should be noted that the area of the histogram represents the total frequency as distributed throughout the different classes. When the width of the class intervals are not equal, then the frequencies must be adjusted before constructing the histogram. The following example will illustrate the procedure Income (Rs.) 1000-1500 1500-2000 2000-2500 2500-3500 No. of employees 5 12 15 18 Income(Rs.) 3500-5000 5000-7000 7000-8000 No. of employees 12 8 2
As can be seen, in the above example, the class intervals are of unequal width and hence we have to find out the adjusted frequency of each class by taking the class with the lowest class interval as the basis of adjustment. For example, in the class 2500-3500, the class interval is 1000 which is twice the size of the lowest class interval, i.e., 500 and therefore the frequency of this class would be divided by two, i.e., it would be 18/2=9. In a similar manner, the other frequencies would be obtained. The adjusted frequencies for various classes are given below:
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The histogram of the above distribution is shown below: Histogram with Unequal Class Intervals 15 15 12
Number of Employees10
9
5 5 4 2 1
1000 Income (In Rupees) ›
2000
3000
4000
5000
6000
7000
8000
It may be noted that a histogram and a bar diagram look very much alike but have distinct features. For example, in a histogram, the rectangles are adjoining and can be of different width whereas in bar diagram it is not possible. Frequency Polygon: The frequency polygon is a graphical presentation of frequency distribution. A polygon is a many sided figure. A frequency polygon is
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81 Frequency Polygon
35 35 30 30
Number of Companies 25
25 22 20 15 15 10 5 10
Daily Sales (In Rupees)
20 16
7
20
30
40 50
60
70 80
90 100
Constructed by taking the mid-points of the upper horizontal side of each rectangle on the histogram and connecting these mid-points by straight lines. In order to close the polygon, an additional class is assumed at each end, having a zero frequency. To illustrate the frequency polygon of this distribution is shown above. If we draw a smooth curve over these points in such a way that the area included under the curve is approximately the same as that of the polygon, then such a curve is known as frequency curve. The following figure shows the same data smoothed out to form a frequency curve, which is another form of presenting the same data.
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82 Frequency Curve
35 30
Number of Companies
25 20 15 10 5 10 20 30 40 50 60 70 80 90 100
Daily Sales (In Rupees)
Remark: The histogram is usually associated with discrete data and a frequency polygon is appropriate for continuous data. But this distinction is not always followed in practice and many factors may influence the choice of graph. The frequency polygon and frequency curve have a special advantage over the histogram particularly when we want to compare two or more frequency distributions. Ogives or Cumulative frequency Curve: An ogive is the graphical presentation of a cumulative frequency distribution and therefore when the graph of such a distribution is drawn, it is called cumulative frequency curve or ogive. There are two methods of constructing ogive, viz., i) ii) Less than ogive More than ogive
Less than Ogive: In this method, the upper limit of the various classes are taken on the X-axis and the frequencies obtained by the process of cumulating the preceding frequencies on the Y-axis. By joining these points we get less than ogive. Consider the example relating to daily sales discussed earlier.
Daily sales (Rs. Thousand) 10-20
No. of companies 15
Daily sales (Rs. Thousand) Less than 20
No.of Companies 15
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83 20-30 30-40 40-50 50-60 60-70 70-80 80-90 22 35 30 25 20 16 7 Less than 30 Less than 40 Less than 50 Less than 60 Less than 70 Less than 80 Less than 90 37 72 102 127 147 163 170
More than Ogive: Similarly more than ogive or cumulative frequency curve can be drawn by taking the lower limits on X-axis and cumulative frequencies on the Y-axis. By joining these points, we get more than ogive. The table and the curve for this case is shown below: Daily sales (Rs. Thousand) 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 No. of companies 15 22 35 30 25 20 16 7 Daily sales (Rs. Thousand) More than 10 More than 20 More than 30 More than 40 More than 50 More than 60 More than 70 More than 80 No.of Companies 170 155 133 98 68 43 23 7
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84
The more than ogive curve is shown below:
210 180
(10,170) (20,155) (30,133)
150
Number of Companies (40,98)
120 90
(50,68) (60,43) (70,23)
60 30
(80,7)
10
20
30
40
50
60
70
80
90
100
Daily Sales (In Rupees)
The shape of less than ogive curve would be a rising one whereas the shape of more than ogive curve should be falling one. The concept of ogive is useful in answering questions such as : How many companies are having sales less than Rs. 52,000 per day or more than Rs. 24,000 per day or between Rs. 24,000 and Rs. 52,000?
Exercises
1. Explain the purpose and methods of classification of data giving suitable examples. 2. What are the general guidelines of forming a frequency distribution with particular reference to the choice of class intervals and number of classes? 3. Explain the various diagrams and graphs that can be used for charting a frequency distribution. 4. What are ogives? Point out the role. Discuss the method of constructing ogives with the help of an example. 5. The following data relate to the number of family members in 30 families of a village. 4 3 2 3 4 5 5 7 3 2 3 4 2 1 1 6 3 4 5 4 2 7 3 4 5 6 2 1 5 3 Classify the above data in the form of a discrete frequency distribution.
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Draw less than and more than ogives. Determine the number of companies whose sales are (i) less than Rs. 13 lakhs (ii) more than 36 lakhs and (iii) between Rs. 13 lakhs and Rs. 36 lakhs.
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86
9.10 Let us Sum Up
This Lesson illustrated the Presentation of data through tables and charts which is essential for a management student to understand. A frequency distribution is the principal tabular Let us Sum Up of either discrete or continuous data. The frequency distribution may show actual, relative or cumulative frequencies. Actual and relative frequencies may be charted as either histogram (a bar chart) or a frequency polygon. Two graphs of cumulative frequencies are: less than ogive or more than ogive. These aspects discussed in this Lesson find major applications while presenting any data with a managerial perspective.
9.11 Lesson – End Activities
1. How the data is classified? 2. What are the guidelines for choosing the classes?
10.1 Aims and Objectives
This Lesson deals with the statistical methods for summarizing and describing numerical methods for summarizing and describing numerical data. The objective here is to find one representative value, which can be used to locate and summarise the entire set of varying values. This one value can be used to make many decisions concerning the entire set. We can define measures of central tendency (or location) to find some central value around which the data tend to cluster. Needless to say the content of this Lesson is important for a manager in taking decisions and also while communicating the decisions.
10.2 Significance of Measures of Central Tendency
Measures of central tendency i.e condensing the mass of data in one single value, enable us to get an idea of the entire data. For example, it is impossible to remember the individual incomes of millions of earning people of India. But if the average income is obtained, we get one single value that represents the entire population. Measures of central tendency also enable us to compare two or more sets of data to facilitate comparison. For example, the average sales figures of April may be compared with the sales figures of previous months.
10.3 Properties of a Good Measure of Central Tendency
A good measure of central tendency should posses, as far as possible, the following properties. i) It should be easy to understand.
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88 ii) It should be simple to compute. iii) It should be based on all observations. iv) It should be uniquely defined. v) It should be capable of further algebraic treatment. vi) It should not be unduly affected by extreme values. Following are some of the important measures of central tendency which are commonly used in business and industry. Arithmetic Mean Weighted Arithmetic Mean Median Quantiles Mode Geometric Mean Harmonic Mean
10.4 Arithmetic Mean
The arithmetic mean ( or mean or average) is the most commonly used and readily understood measure of central tendency. In statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the numerical values of each and every observation divided by the total number of observations. Symbolically, it can be represented as:
X =å X
N
Where X indicates the sum of the values of all the observations, and N is the total number of observations. For example, let us consider the monthly salary (Rs.) of 10 employees of a firm : 2500, 2700, 2400, 2300, 2550, 2650, 2750, 2450, 2600, 2400 If we compute the arithmetic mean, then X = 2500+ 2700+ 2400+ 2300+ 2550+ 2650+ 2750+ 2450+ 2600+ 2400 10 25300 = Rs. 2530 10 Therefore, the average monthly salary is Rs. 2530.
=
We have seen how to compute the arithmetic mean for ungrouped data. Now let us consider what modifications are necessary for grouped data. When the observations are classified into a frequency distribution, the midpoint of the class interval would be treated as the representative average value of that class. Therefore, for grouped data, the arithmetic mean is defined as X =å fX
N
Where X is midpoint of various classes, f is the frequency for corresponding class and N is the total frequency. i.e. N= f.
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89 This method is illustrated for the following data which relate to the monthly sales of 200 firms.
200 Hence the average monthly sales are Rs. 510. To simplify calculations, the following formula for arithmetic mean may be more convenient to use. X = A + å fd X i
N
Where A is an arbitrary point, d= X-A , and i=size of the equal class interval. i REMARK: A justification of this formula is as follows. When d= X-A , then X=A+i d. Taking summation on both sides and dividing by N, we get ì X = A + å fd X i
N
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=525-15=510 or Rs. 510 It may be observed that this formula is much faster than the previous one and the value of arithmetic mean remains the same. Properties of AM 1. The algebraic sum of deviations of a set of values from their AM is zero. 2. Sum of squares of deviations of a set of values is minimum when deviations taken about AM.
10.5 Combined Mean of Two Groups
Let x 1 and x 2 be the means of two groups. Let there be n1 observations in the first group and n2 observations in the second group. Then x , the mean of the combined group can be obtained as
x=
n1 x1 + n 2 x 2 n1 + n 2
Example : Average daily wage of 60 male workers in a firm is Rs. 120 and that of 40 females is Rs.100. Find the mean wage of all the workers. Solution: Here n1 = 60,
x1 = 120
and n2 = 40,
x2 = 100
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91
Combined Mean =
60 ´ 120 + 40 ´ 100 60 + 40 = 112
10.6 Weighted AM
When calculating AM we assume that all the observations have equal importance. If some items are more important than others, proper weightage should be given in accordance with their importance. Let w1 , w2 , …, wn be the weights attached to the items x1 , x2 , …, xn , then the weighted AM is defined as Weighted mean =
10.7 Median
The median of a set of observations is a value that divides the set of observations in half, so that the observations in one half are less than or equal to the median and the observations in the other half are greater than or equal to the median value. In finding the median of a set of data it is often convenient to put the observations in ascending or descending order. If the number of observations is odd, the median is the middle observation. For example, if the values are 52, 55, 61, 67, and 72, the median is 61. If there were 4 values instead of 5, say 52, 55, 61, and 67, there would not be a middle value. Here any number between 55 and 61 could serve as a median; but it is desirable to use a specific number for the median and we usually take the AM of two middle values, i.e, (55+61)/2 = 58.
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92 Median is the primary measure of location for variables measured on ordinal scale because it indicates which observation is central without attention to how far above or below the median the other observations fall. Example: Find the median of 10, 2, 4, 8, 5, 1, 7 Solution: Observations in ascending order of magnitude are 1, 2, 4, 5, 7, 8, 10 Here there are 7 observations, so median is the 4th observation. That is, median = 5
10.8 Median for a grouped frequency distribution
In a grouped frequency distribution, we do not know the exact values falling in each class. So, the median can be approximated by interpolation. Let the total number of observations be N. for calculating median we assume that the observations in the median class are uniformly distributed. Median class is the class in which the (N/2) th observation belongs. Also assume that median is the (N/2)th observation. Here the frequency table must be continuous. If it is not, convert it into continuous table. Prepare a less than cumulative frequency table and find the median class. Let 'l' be the lower limit of the median class, 'f' the frequency of the median class, and 'c' is the class width of the median class. By the assumption of uniform distribution, the 'f' c 2c fc observations in the median class are l + , l + , …, l + . Let 'm' be the cumulative f f f N frequency of the class above the median class. Then the median will be the ( - m) th 2 observation in the median class. N c That is, median = l + ( - m) 2 f Example : Calculate the median of the following data: class 0 10 20 31 40 50 60 71 frequency 4 12 24 36 20 16 8 5
10 20 30 40 50 60 70 80
Solution: Since the frequency table is of inclusive, convert it into exclusive by subtracting 0.5 from the lower limits and adding 0.5 to the upper limits.
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Property of Median: The sum of absolute deviations of a set values is minimum when the deviations are taken from median.
10.9 Mode
The mode of a categorical or a discrete numerical variable is that category or value which occurs with the greatest frequency. Example : The mode of the data 2, 5, 4, 4, 7, 8, 3, 4, 6, 4, 3 is 4 because 4 repeated the greatest number of times.
10.10 Mode of a grouped frequency distribution
In a grouped frequency distribution, to find the mode, first locate the modal class. Modal class is that class with maximum frequency. Let l be the lower limit of the modal class, 'c' be the class interval, f1 be the frequency of the modal class, f0 be the frequency of the class preceding and f2 be the frequency of the class succeeding the modal class. c(f1 - f 2 ) Then, Mode = l + 2f 1 - f 0 - f 2 Example : Find the mode of the distribution given below class frequency
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2. Calculate the mean, median and mode of the following data: Class: 10 –20 20 - 30 30 – 40 40 – 50 50 - 60 Frequency: 25 52 73 40 10 3. From the following data of income distribution, calculate the AM. It is given that i) the total income of persons in the highest group is Rs. 435, and ii) none is earning less than Rs. 20. Income ( Rs) Below 30 " 40 " 50 " 60 " 70 " 80 80 and above No. of persons 16 36 61 76 87 95 5
4. Mean of 20 values is 45. If one of these values is to be taken 64 instead of 46. Find the correct mean. 5. The mean yearly salary of employees of a company was Rs. 20,000. The mean yearly salaries of male and female employees were Rs. 20,800 and Rs. 16,800 respectively. Find out the percentage of males employed.
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95
6. The average wage of 100 male workers is Rs. 80 and that 50 female workers is 75. Find the mean wage of workers in the company.
10.11 Let us Sum Up
The importance of measures of central tendency is described in this Lesson followed with different terms like mean, median, mode, etc. Measures of central tendency give one of the very important characteristics of data. Any one of the various measures of central tendency may be chosen as the most representative or typical measure. The AM is widely used and understood as a measure of central tendency. The concepts of weighted arithmetic mean, geometric mean and harmonic mean, are useful for specific types of applications. The median is a more representative measure for open-end distribution and highly skewed distribution. The mode should be used when the most demanded or customary value is needed. The examples shown in the Lesson clearly brings out the probable applications and the solution for specific problems.
11.1 Aims and Objectives
In the previous Lesson, we have discussed about the common measures of central tendency which are widely used in statistics. Median, as has been indicated, is a locational average, which divides the frequency distribution into two equal parts. Quartiles, deciles and percentiles are not averages. They are the partition values, which divides the distribution into certain equal parts. Quartiles Quartiles are the values, which divides a frequency distribution into four equal parts so that 25% of the data fall below the first quartile (Q1 ), 50% below the second quartile (Q2 ), and 75% below the third quartile (Q3 ). The values of Q1 and Q3 can be found out as in the case of Q2 (Median). For a raw data, Q1 is the (n/4)th observation and Q3 is the (3n/4)th observation. For a grouped table, Q1 = l1 + (
N c1 - m1 ) 4 f1
Where N is the total frequency, l1 is the lower limit of the first quartile class ( class in which (N/4)th observation belongs), m1 is the cumulative frequency of the class above the first quartile class, f1 is the frequency of the first quartile class and c1 is the width of the first quartile class. Q3 = l3 + (
C 3N - m3 ) 3 f3 4
Where l3 is the lower limit of the third quartile class ( class in which (3N/4)th observation belongs), m3 is the cumulative frequency of the class above the third quartile class, f3 is the frequency of the third quartile class and C3 is the width of the third quartile class.
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97 Deciles and Percentiles Deciles are nine in number and divide the frequency distribution into 10 equal parts. Percentiles are 99 in number and divide the frequency distribution into 100 equal parts. Selecting the Most Appropriate Measure of Central Tendency Generally speaking, in analyzing the distribution of a variable only one of the possible measures of central tendency would be used. Its selection is largely a matter of judgment based upon the kind of data, the aspect of the data to be examined, and the research question. Some of the points that must be considered are following. Central tendency for interval data is generally represented by the A.M., which takes into account the available information about distances between scores. For ranked (ordinal) data, the median is generally most appropriate, and for nominal data, the mode. If the distribution is badly skewed, one may prefer the median to the mean, because the example, the median income of people is usually reported rather than the A.M. If one is interested in prediction, the mode is the best value to predict if an exact score in a group has to be picked.
11.2 Measures of Dispersion
So far we have discussed averages as sample values used to represent data. But the average cannot describe the data completely. Consider two sets of data : 5, 10, 15, 20, 25 15, 15, 15, 15, 15 Here we observe that both the sets are with the same mean 15. But in the set I, the observations are more scattered about the mean. This shows that, even though they have the same mean, the two sets differ. This reveals the necessity to introduce measures of dispersion. A measure of dispersion is defined as a mean of the scatter of observations from an average. Commonly used measures of dispersion are Range, Mean deviation, Standard deviation, and quartile deviation. 11.2.1 Range Range of a set of observations is the difference between the largest and the smallest observations. In the case of grouped frequency table, range is the difference between the upper bound of last class and the lower bound of the first class. Example : The range of the set of data 9, 12, 25, 42, 45, 62, 65 is 65 – 9 = 56
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98
Range is the simplest measure of dispersion but its demerit is that it depends only on the extreme values. 11.2.2 Mean Deviation about the Mean You have seen that range is a measure of dispersion, which does not depend on all observations. Let us think about another measure of dispersion, which will depend on all observations. One measure of dispersion that you may suggest now is the sum of the deviations of observations from mean. But we know that the sum of deviations of observations from the A.M is always zero. So we cannot take the sum of deviations of observations from the mean as a measure. One method to overcome this is to take the sum of absolute values of these deviations. But if we have two sets with different numbers of observations this cannot be justified. To make it meaningful we will take the average of the absolute deviations. Thus mean deviation (MD) about the mean is the mean of the absolute deviations of observations from arithmetic mean. 1 n If x1 , x2 , …, xn are n observations, then, MD = å | xi - x | n i =1 Example : Find the MD for the following data 12, 15, 21, 24, 28 Solution: 12 + 15 + 21 + 24 + 28 = = 20 X 5 x 12 15 21 24 28 Total MD = | xi - x | 8 5 1 4 8 26
26 = 5.2 5 Mean deviation about mean for a frequency table
Let x1 , x2 , …, xn be the values and f1 , f2 , …, fn are the corresponding frequencies. Let N 1 n be the sum of the frequencies. Then, MD = å | xi - x | fi N i =1 In the case of a grouped frequency table, take the mid-values as x-values and use the same method given above.
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Short-cut method to find standard deviation If the values of x are very large, the calculation of SD becomes time consuming. Let the mid-values of k classes be x1 , x2 , …, xk and f1 , f2 , …, fk be the corresponding xi - A frequencies. We use the transformation of the form ui = for i = 1,2, …, k. C Here A and C can be any two numbers. But it is better to take A as a number among the middle part of the mid-values. If all the classes are of equal width, C can be taken as the class width. 1 Variance of ui's , Var(u) = S fi ui2 - u 2 N Then variance of xi 's, Var(x) = C2 ´ Var(u) That is, SD(x) = C ´ SD(u) Example : Consider the problem in example 5, let us find out the SD using short-cut method. Solution: class 0 – 10 10 – 20 20 - 30 30 – 40 40 - 50 mid-value (x) 5 15 25 35 45 Total ui =
xi - 25 10 -2 -1 0 1 2
frequency (f) 3 4 6 10 7 30
fu -6 -4 0 10 14 14
fu2 12 4 0 10 28 54
u =
å fu
N
=
14 = 0.467, S fi ui2 = 54, N = 30 30
Variance(u) =
54 - (0.467)2 30 = 1.8 – 0.21809
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Combined Variance If there are two sets of data consisting of n1 and n2 observations with s1 2 and s2 2 as their respective variances, then the variance of the combined set consisting of n1 +n2 observations is : S2 = [n1 (s1 2 + d12 ) + n2 (s2 2 + d2 2 )] / (n1 + n2 ) Where d1 and d2 are the differences of the means, x1 and x2 , from the combined mean x respectively. Example : Find the combined standard deviation of two series A and B Series A 50 5 100 Series B 40 6 150
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104
11.4 Relative Measures
The absolute measures of dispersion discussed above do not facilitate comparison of two or more data sets in terms of their variability. If the units of measurement of two or more sets of data are same, comparison between such sets of data is possible directly in terms of absolute measures. But conditions of direct comparison are not met, the desired comparison can be made in terms of the relative measures. Coefficient of Variation is a relative measure of dispersion which express standard deviation( s ) as percent of the mean. That is Coefficient of variation, C.V = ( s / x )100. Another relative measure in terms of quartile deviations is Coefficient of quartile Q3 - Q1 deviation and is defined as Qr = ´ 100 . Q3 + Q1 Example: An analysis of the monthly wages paid to workers in two firms A and B, belonging to the same industry, gives the following results:
Number of workers Average monthly wage Standard deviation
Firm A 586 52.5 10
Firm B 648 47.5 11
In which firm, A or B, is there greater variability in individual wages? Solution: Coefficient of variation for firm A =
10 ´ 100 52.5 = 19% 11 Coefficient of variation for firm B = ´ 100 47.5 = 23% There is greater variability in wages in firm B.
11.5 Skewness and Kurtosis
Skewness Very often it becomes necessary to have a measure that reveals the direction of dispersion about the center of the distribution. Measures of dispersion indicate only the extent to which individual values are scattered about an average. These do not give information about the direction of scatter. Skewness refers to the direction of dispersion leading departures from symmetry, or lack of symmetry in a direction. If the frequency curve of a distribution has longer tail to the right of the center of the distribution, then the distribution is said to be positively skewed. On the other hand, if the
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105 distribution has a longer tail to the left of the center of the distribution, then distribution is said to be negatively skewed. Measures of skewness indicate the magnitude as well as the direction of skewness in a distribution.
Empirical Relationship between Mean, Median and Mode The relationship between these three measures depends on the shape of the frequency distribution. In a symmetrical distribution the value of the mean, median and the mode is the same. But as the distribution deviates from symmetry and tends to become skewed, the extreme values in the data start affecting the mean. In a positively skewed distribution, the presence of exceptionally high values affects the mean more than those of the median and the mode. Consequently the mean is highest, followed, in a descending order, by the median and the mode. That is, for a positively skewed distribution, Mean > Median> Mode. In a negatively skewed distribution, on the other hand, the presence of exceptionally low values makes the values of the mean the least, followed, in an ascending order, by the median and the mode. That is, for a negatively skewed distribution, Mean < Median < Mode. Empirically, if the number of observations in any set of data is large enough to make its frequency distribution smooth and moderately skewed, then, Mean – Mode = 3(Mean – Median) Measures of Skewness 3. Karl Pearson's measure of skewness: Prof. Karl Pearson has been developed this measure from the fact that when a distribution drifts away from symmetry, its mean, median and mode tend to deviate from each other. Mean - Mode Karl Pearson's measure of skewness is defined as, SkP = SD 4. Bowley's measure of skewness: developed by Prof. Bowley, this measure of skewness is derived from quartile values. Q3 + Q1 - 2Q2 It is defined as SkB = Q3 - Q1 5. Moment measure of skewness: If x1 , x2 , …, xn are n observations, then the rth moment about mean is defined as 1 n mr = å (xi - x )r n i =1 The moment measure of skewness is defined as b 1 = m3 /(SD)3 In a perfectly symmetrical distribution b 1 =0, and a greater or smaller value of b 1 results in a greater or smaller degree of skewness.
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106 Kurtosis Kurtosis refers to the degree of peakedness, or flatness of the frequency Curve. If the curve is more peaked than the normal curve, the curve is said to be lepto kurtic. If the curve is more flat than the normal curve, the curve is said to be platy kurtic. The normal m4 curve is also called meso kurtic. The moment measure of kurtosis is b 2 = . The m2 2 value of b 2 =3, if the distribution is normal; more than 3, if the distribution is lepto kurtic; and less than 3, if the distribution is platy kurtic. Example : Given m2 (variance) = 40, m3 = -100. Find a measure of skewness. Solution: Moment measure of skewness, b 1 = m3 /(SD)3 - 100 = = - 0.4 ( 40 ) 3 Hence, there is negative skewness Example : The first four moments of a distribution about mean are 0, 2.5, 0.7, and 18.75. Comment on the Kurtosis of the distribution Moment measure of kurtosis is, b 2 =
m4 . m2 2 18.75 = =3 2.5 2
So, the curve is normal.
Exercises
1. Find the standard deviation of the values: 11, 18, 9, 17, 7, 6, 15, 6, 4, 1 2. Daily sales of a retail shop are given below: Daily sales(Rs): 102 106 110 114 118 122 126 No. of days: 3 9 25 35 17 10 1 Calculate the mean and standard deviation of the above data and explain what they indicate about the distribution of daily sales? 3. Goals scored by two teams A and B in a foot ball season were as follows: No. of goals scored: 0 1 2 3 4 No. of matches A: 2 9 8 5 4 B: 1 7 6 5 3 Find which team may be considered more consistent? 4. The mean of two samples of sizes 50 and 100 respectively are 54.1 and 50.3 and the standard deviations are 19 and 8. Find the mean and the standard deviation of the combined sample.
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11.6 Let us Sum Up
In this Lesson we have discussed about how the concepts of measures of variation and skewness are important. Measures of variation considered were the range, average deviation, quartile deviation and standard deviation. The concept of coefficient of variation was used to compare relative variations of different data. The skewness was used in relation to lack of symmetry. Some example problems were also shown solved for a better understanding.
11.7 Lesson – End Activities
1. Define Quartile deviation. 2. Give the necessity for finding the skewness of the data.
11.8 References
R.S.N. Pillai and Mrs. Bhagavathi – Statistics.
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12.1 Introduction
Sample statistics form the basis of all inferences drawn about populations. If we know the probability distribution of the sample statistic, then we can calculate the probability of that the sample statistic assumes a particular value or has a value in a given interval. This ability to calculate the probability that the simple statistic lies in a particular interval is the most important factor in all statistical inferences. Such aspects are covered in this Lesson. Examples are shown for better understanding of the subject.
12.2 Sampling Distributions
Suppose we wish to draw conclusions about a characteristic of a population. We draw a random sample of size n and take measurements about the characteristic, which we interested to study. Let the sample values be x1, x2 , x3 , …, xn . Then any quantity which can be determined as a function of the sample values x1 , x2 , x3 , …, xn is called a statistic. Since the sample values are the results of random selections, a statistic is a random variable. Therefore, a statistic has a probability distribution. It is known as sampling distribution. The standard deviation of the sampling distribution is called standard error. The process of inferring certain facts about a population based on a sample is known as statistical inference. Sample statistics and their distributions are the basis of all inferences drawn about the population.
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109
12.3 Sampling Distribution of the Sample Mean
Suppose we have a sample of size n from a population. Let x1 , x2 , x3 , …, xn be the values of the characteristic under study corresponding to the selected units. Then the sample __ __ x + x2 + x3 +¼+ xn mean X is defined as X = 1 . n If we draw another sample of size n from the same population, we may end up with a different set of sample values and so a different sample mean. Thus the value of the sample mean is determined by chance causes. The distribution of the sample mean is called sampling distribution of the sample mean.
12.4 Distribution of Sample mean
12.4.1 Distribution of sample mean of sample taken from any infinite population If x1 , x2 , x3 , …, xn constitute a random sample from an infinite population having the mean m and variance s2 , then the distribution of sample mean will be normal with mean m 2 and variance s , when n is large. n 12.4.2 Distribution of sample mean of sample taken from the normal population
__
If X is the mean of a random sample of size n from a normal population with the mean m and variance s2 , its sampling distribution is a normal distribution with the mean m and 2 variance s . n Example 1: a random sample of size 100 is taken from a normal population with s = 25. What is the probability that the mean of the sample will greater from the mean of the population by atleast 3.
__
Solution: Let m be the population mean and x be the sample mean. Given that n = 100, s=25.
__
12.5 Some Uses of Sampling distribution of Mean
1. 2. 3. 4. To test the mean of a normal population when population standard deviation is known To test the mean of any population when sample size is large ( usually n >30) To test the equality of means of two populations when sample sizes large. To test the equality of means of two normal populations when population standard deviations are known. 5. To find out the confidence interval for population mean; difference of population means of two populations. ( both cases sample sizes are large).
12.6 The Chi- Square Distribution
If a random variable X has the standard normal distribution, then the distribution X2 is called chi-square (c2 ) distribution with one degree of freedom. This distribution would be quite different from a normal distribution because X2 , being a square term, can assume only non-negative values. The probability curve of c2 will be higher near 0, because most of the x-values are close to 0 in a standard normal distribution. If X1 , X2 , …, Xn are independent standard normal variables, then X1 +X2 +… + Xn has the c2 distribution with n degrees of freedom. Here 'n' is the only one parameter. c2 – table Since c2 -distribution arises in many important applications, especially in statistical inference, integrals of its density has been tabulated. The table gives the value of c2 a,n such that probability that c2 is greater than c2 a,n is equal to a for a = 0.005, 0.01, 0.025, 0.05 etc. and n = 1, 2, 3, … . That is, the table gives P(c2 >c2 a,n) = a
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111
a c2 a,n Some Uses of Chi – Square Distribution 1. To test the variance of a normal population. 2. To test the independence of two attributes. 3. To test the homogeneity of two attributes. 4. To find the confidence interval for the variance of a normal population.
12.7 The Student – t Distribution
If X and Y are two independent random variables, X has the standard normal distribution and Y has a chi-square distribution with 'n' degrees of freedom, then the distribution of the statistic t = X is called Student 't' distribution. The t-distribution was first obtained
Y
n
by by W.S. Gosset, who is known under the pen name 'Student'.
-
x- m An example of a t-statistic is t = n , which follows t-distribution with (n-1) degrees of s
__
freedom, where x and s are mean and standard deviation of a random sample of size n from a normal population with mean m and variance s2 .
12.8 Student 't' table
Student 't' table has many applications in statistical inference. The t-table gives the values ta,n for a = 0.25, 0.125, 0.10, 0.05 etc. and n = 1, 2, 3, …, where ta,n is such that the area to its right under the curve of the t-distribution with 'n' degrees of freedom is equal to a. That is, ta,n is such that P(t > ta,n) = a. Also note that the t-distribution is a symmetric distribution.
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112
a
. ta,n
Some Uses of t-distribution 1. To test the mean of a normal population when the sample size is small and population variance is unknown. 2. To test the equality of means of two normal populations when the sample sizes are small and population variances are unknown but same. 3. To test the correlation coefficient is zero. 4. To find the confidence interval of mean of normal population when sample size is small and population variance is unknown.
12.9 The F- Distribution
If U and V are independent random variables having chi-square distribution with m and n U degrees of freedom, then the distribution of m is called the F-distribution with m and n V n degrees of freedom. For example, if S1 2 and S2 2 are the variances of independent random samples of sizes m and n from normal populations with variances s1 2 and s2 2 , then, 2 2 S1 s 2 F= 2 2 has an F-distribution with m-1 and n-1 degrees of freedom. S2 s 1 Table of F-distribution The table of F-distribution gives the values Fa;m,n for a=0.05 and 0.01 for various values of m and n where Fa;m,n is such that the area to the right under the curve of F-distribution with m, n degrees of freedom is equal to a.
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113 That is Fa;m,n is such that P(F> Fa;m,n) = a
a
Fa;m,n Some Uses of F-distribution 1. To test the equality of variances of two normal populations. 2. F-distribution is used in analysis of variance.
12.10 Estimation of Parameters
The problem of estimation is of finding out a value for unknown population parameters, which we cannot directly observe, as precisely as possible. Managers deal this problem most frequently. They make quick estimates too. Since our estimates are based only on a sample, the estimates are not likely to be exactly equal to the value we are looking for. Still we will be able to obtain estimates whose possible values are around the true, but unknown value. The difference between the true value and the estimate is the error in estimation. There are two types of estimates 1. Point Estimate 2. Interval Estimate If an estimate of a population parameter is given by a single value, then the estimate is called point estimate of the parameter. But if two distinct numbers give an estimate of a population parameter between which the parameter may be considered to lie, then the estimate is called an interval estimate of the parameter. A function, T, used for estimating a parameter q, is called an estimator and its value given a sample is known as estimate. Required Properties of an Estimator 1. Unbiasedness: An estimator must be an unbiased estimator of the parameter. That is an estimator T is said to be unbiased for a parameter q if E(T) = q. 2. Efficiency: Efficiency refers to the size of the standard error of the estimator. That is, an estimator T1 is said to be more efficient than another estimator T2 if standard error of T1 is less than the standard error of T2 . 3. Consistency: As the sample size increases the value of the estimator must get close to the parameter. 4. Sufficiency: An estimator T is said to be sufficient for a parameter q if T contains all information which the sample contains and furnishes about q.
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114 Some Point Estimators
__
1. The sample mean X is a point estimator of the population mean m 2. The sample proportion is a point estimate of the population proportion. 3. The sample variance is a point estimator of population variance.
12.11 Testing Hypotheses
Statistical testing or testing hypotheses, is one of the most important aspects of the theory of decision- making. Testing hypotheses consists of decision rules required for drawing probabilistic inferences about the population parameters. Definition: A Statistical Hypothesis is a statement concerning a probability distribution or population parameters and a process by which a decision is arrived at, whether or not a hypothesis is true is Testing Hypothesis. For example, the statement, mean of a normal population is 30, the variance of a population is greater than 12 are statistical hypotheses. Null Hypothesis and Alternate Hypothesis The hypothesis under test is known as the null hypothesis and the hypothesis that will be accepted when the null hypothesis is rejected is known as the alternate hypothesis. The null hypothesis is usually denoted by H0 and the alternate hypothesis by H1 . For example, if the population mean is represented by m, we can set up our hypothesis as follows: H0 : m £ 30; H1 : m > 30. The following are the steps in testing a statistical hypothesis. We draw a sample from the concerned population. Then choose the appropriate test statistic. A test statistic is a statistic, based on the value of it we decide either to reject or accept a hypothesis. Divide the sample space of the test statistic into two regions, namely, rejection region and acceptance region. (The set of sample points, which lead to the rejection of the null hypothesis, is called the Critical Region or Rejection Region). Calculate the value of the test statistic for our sampled data. If this value falls in the rejection region, reject the hypothesis; otherwise accept it. Type I Error and Type II Error Since we have to depend on the sample there is no way to know, which of the two hypotheses is actually true. The test procedure is to fix the rejection region, in which the value of test statistic observed, the null hypothesis would be rejected. The null hypothesis may be true, but the test procedure may reject the null hypothesis. This error is known as the first kind of error. It is also possible that the null hypothesis is actually false but the test accepts it. This error is known as the second kind of error. Thus, the error committed in rejecting a true null hypothesis is called type I error and the error in accepting a false null hypothesis is called the type II error.
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115 Significance Level The probabilities of two errors cannot be simultaneously reduced, since is we increase the rejection region the probability of type I error will increase whereas the reduction in rejection region will increase type II error. The procedure usually adopted is to keep the probability of type I error below a pre-assigned number and subject to this condition minimize the type II error. A pre-assigned number a between 0 and 1 chosen as an upper bound of type I error is called the level of significance. Two-tailed and One-tailed Tests A test where the critical region is found to lie under one tail of the distribution of the test statistic is called One-tailed test. In two-tailed tests the critical region lies under both the tails of the distribution of the test statistic. Example: Let m be the mean of a population. Then, 1. H0 : m = 30; H1 : m ¹ 30 is a two tailed test 2. H0 : m = 30; H1 : m > 30 is a single tailed test. Exercise 3. A population is normally distributed with mean 90. A sample of size 10 is taken at random from the population. Find the probability that the population mean is greater than 85. 4. In the above problem, suppose we have to test whether the population mean is equal to 85. Formulate the null hypothesis and alternate hypothesis.
12.12 Let us Sum Up
The concept of sampling distribution is introduced in this Lesson. Some of the commonly used sampling distributions used in statistics and some of the applications are also shown. The sampling distribution is very important in statistical calculations and inferences.
13.1 Introduction
There are situations where data appears as pairs of figures relating to two variables. A correlation problem considers the joint variation of two measurements neither of which is restricted by the experimenter. The regression problem discussed in this Lesson considers the frequency distribution of one variable (called the dependent variable) when another (independent variable) is held fixed at each of several levels. Examples of correlation problems are found in the study of the relationship between IQ and aggregate percentage of marks obtained by a person in the SSC examination, blood pressure and metabolism or the relation between height and weight of individuals. In these examples both variables are observed as they naturally occur, since neither variable is fixed at predetermined levels. Examples of regression problems can be found in the study of the yields of crops grown with different amount of fertilizer, the length of life of certain animals exposed to different levels of radiation, and so on. In these problems the variation in one measurement is studied for particular levels of the other variable selected by the experimenter.
13.2 Correlation
Correlation measures the degree of linear relation between the variables. The existence of correlation between variables does not necessarily mean that one is the cause of the change in the other. It should noted that the correlation analysis merely helps in
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117 determining the degree of association between two variables, but it does not tell any thing about the cause and effect relationship. While interpreting the correlation coefficient, it is necessary to see whether there is any cause and effect relationship between variables under study. If there is no such relationship, the observed is meaningless. In correlation analysis, all variables are assumed to be random variables.
13.3 The Scatter Diagram
The first step in correlation and regression analysis is to visualize the relationship between the variables. A scatter diagram is obtained by plotting the points (x1 , y 1 ), (x2 , y 2 ), …, (x n ,yn ) on a two-dimensional plane. If the points are scattered around a straight line , we may infer that there exist a linear relationship between the variables. If the points are clustered around a straight line with negative slope, then there exist negative correlation or the variables are inversely related ( i.e, when x increases y decreases and vice versa. ). If the points are clustered around a straight line with positive slope, then there exist positive correlation or the variables are directly related ( i.e, when x increases y also increases and vice versa. ). For example, we may have figures on advertisement expenditure (X) and Sales (Y) of a firm for the last ten years, as shown in Table 1. When this data is plotted on a graph as in Figure 1 we obtain a scatter diagram. A scatter diagram gives two very useful types of information. First, we can observe patterns between variables that indicate whether the variables are related. Secondly, if the variables are related we can get an idea of what kind of relationship (linear or non- linear) would describe the relationship.
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118
Correlation examines the first Question of determining whether an association exists between the two variables, and if it does, to what extent. Regression examines the second question of establishing an appropriate relation between the variables.
X The scatter diagram may exhibit different kinds of patterns. Some typical patterns indicating different correlations between two variables are shown in Figure 2. Figure 2: Different Types of Association Between Variables
r>0
Y
X
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119
(a) r>0
Positive Correlation
Y
X r=0
(b) Negative Correlation
Y
X
( c ) No Correlation
Y
X (d) Non-linear Association
13.4 The Correlation Coefficient
Definition and Interpretation The correlation coefficient measure the degree of association between two variables X and Y. Pearson's formula for correlation coefficient is given as 1 r = å ( X - X ) (Y - Y ) n sxsy Where r is the correlation coefficient between X and Y, sxandsy are the standard deviation of X and Y respectively and n is the number of values of the pair of variables X
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120
1 å ( X - X ) ( X - Y ) is known as the covariance n between X and Y. Here r is also called the Pearson's product moment correlation coefficient. You should note that r is a dimensionless number whose numerical value lies between +1 and -1. Positive values of r indicate positive (or direct) correlation between the two variables X and Y i.e. as X increase Y will also increase or as X decreases Y will also decrease. Negative values of r indicate negative (or inverse) correlation, thereby meaning that an increase in one variable results in a decrease in the value of the other variable. A zero correlation means that there is an o association between the two variables. Figure II shown a number of scatter plots with corresponding values for the correlation coefficient r. The following form for carrying out computations of the correlation coefficient is perhaps more convenient :
and Y in the given data. The expression
r =
å xy åX 2 åy
2
where ……..(18.2) x = X - X = deviation of a particular X value from the mean- X y= Y - Y = deviation of a particular Y value from the mean Y Equation (18.2) can be derived from equation (18.1) by substituting for sxandsy as follows:
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121
S xy S xy Sy Sx
S xy
2 2
The geometric mean of byx and bxy is
bxy b yx =
SxSy = r, the correlation coefficient.
=
Also note that the sign of both the regression coefficients will be same, so the sign of correlation coefficient is same as the sign of regression coefficient.
13.7 Coefficient of Determination
Coefficient of determination is the square of correlation coefficient and which gives the proportion of variation in y explained by x. That is, coefficient of determination is the ratio of explained variance to the total variance. For example, r2 = 0.879 means that 87.9% of the total variances in y are explained by x. When r2 = 1, it means that all the points on the scatter diagram fall on the regression line and the entire variations are explained by the straight line. On the other hand, if r2 = 0 it means that none of the points on scatter diagram falls on the regression line, meaning thereby that there is no linear relationship between the variables. Example: Consider the following data: X: 15 16 17 18 19 20 Y: 80 75 60 40 30 20 1. Fit both regression lines 2. Find the correlation coefficient 3. Verify the correlation coefficient is the geometric mean of the regression coefficients 4. Find the value of y when x = 17.5
13.8 Spearman's Rank Correlation Coefficient
Sometimes the characteristics whose possible correlation is being investigated, cannot be measured but individuals can only be ranked on the basis of the characteristics to be measured. We then have two sets of ranks available for working out the correlation coefficient. Sometimes tha data on one variable may be in the form of ranks while the data on the other variable are in the form of measurements which can be converted into ranks. Thus, when both the variables are ordinal or when the data are available in the ordinal form irrespective of the type variable, we use the rank correlation coefficient.
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124
6S d i The Spearman's rank correlation coefficient is defined as , r = 1 n(n 2 - 1) Example: Ten competitors in a beauty contest were ranked by two judges in the following orders:
2
13.9 Tied Ranks
Sometimes where there is more than one item with the same value a common rank is given to such items. This rank is the average of the ranks which these items would have got had they differed slightly from each other. When this is done, the coefficient of rank correlation needs some correction, because the above formula is based on the supposition that the ranks of various items are different. If in a series, 'mi ' be the frequency of ith tied ranks, 1 2 6[Sd i + S (m 3 - m)] 12 Then, r = 1 n(n 2 - 1) Example: Calculate the rank correlation coefficient from the sales and expenses of 10 firms are below: Sales(X): 50 50 55 60 65 65 65 60 60 50 Expenses(Y): 11 13 14 16 16 15 15 14 13 13
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Exercises
1. A company selling household appliances wants to determine if there is any relationship between advertising expenditures and sales. The following data was compiled for 6 major sales regions. The expenditure is in thousands of rupees and the sales are in millions of rupees. Region : 1 Expenditure(X): 40 Sales (Y): 25 2 45 30 3 80 45 4 20 20 5 15 20 6 50 40
a) Compute the line of regression to predict sales b) Compute the expected sales for a region where Rs.72000 is being spent on advertising 2. The following data represents the scores in the final exam., of 10 students, in the subjects of Economics and Finance. Economics: 61 78 77 97 65 95 30 74 55 Finance: 84 70 93 93 77 99 43 80 67 a) Compute the correlation coefficient? 3. Calculate the rank correlation coefficient from the sales and expenses of 9
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126 firms are below: Sales(X): 42 40 Expenses(Y): 10 18
54 18
62 17
55 17
65 14
65 13
66 10
62 13
13.10 Regression
In industry and business today, large amounts of data are continuously being generated. This may be data pertaining, for instance, to a company's annual production, annual sales, capacity utilisation, turnover, profits, ,manpower levels, absenteeism or some other variable of direct interest to management. Or there might be technical data regarding a process such as temperature or pressure at certain crucial points, concentration of a certain chemical in the product or the braking strength of the sample produced or one of a large number of quality attributes. The accumulated data may be used to gain information about the system (as for instance what happens to the output of the plant when temperature is reduced by half) or to visually depict the past pattern of behaviours (as often happens in company's annual meetings where records of company progress are projected) or simply used for control purposes to check if the process or system is operating as designed (as for instance in quality control). Our interest in regression is primarily for the first purpose, mainly to extract the main features of the relationships hidden in or implied by the mass of data. What is Regression? Suppose we consider the height and weight of adult males for some given population. If we plot the pair (X1 X2 )=(height, weight), a diagram like figure I will result. Such a diagram, you would recall from the previous Lesson, is conventionally called a scatter diagram. Note that for any given height there is a range of observed weights and vice- versa. This variation will be partially due to measurement errors but primarily due to variations between individuals. Thus no unique relationship between actual height and weight can be expected. But we can note that average observed weight for a given observed height increases as height increases. The locus of average observed weight for given observed height (as height varies) is called the regression curve of weight on height. Let us denote it by X2 =f(X1 ). There also exists a regression curve of height on weight similarly defined which we can denote by X1 =g(X2 ). Let us assume that these two "curves" are both straight lines (which in general they may not be). In general these two curves are not the same as indicated by the two lines in Figure 3.
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127
Figure 3: Height and Weight of thirty Adult Males
x 90 Weight in kg (X2) 80 x
x x
x
X1 =g(X2 ) x x x X2 =f(X1 )
x x x x x x x x x
-
x x x x x 60 x x x 50 | | | | | | 164 168 172
70 -
| | 176
| | | 180
| 184
188
Height in cms (X1)
A pair of random variables such as (height, weight) follows some sort of bivariate probability distribution. When we are concerned with the dependence of a random variable Y on quantity X, which is variable but not a random variable, an equation that relates Y to X is usually called a regression equation. Simply when more than one independent variable is involved, we may wish to examine the way in which a response Y depends on variables X1 X2 …Xk . We determine a regression equation from data which cover certain areas of the X-space as Y=f(X1 ,X2 …Xk )
13.11 Linear Regression
Regression analysis is a set of statistical techniques for analyzing the relationship between two numerical variables. One variable is viewed as the dependent variable and the other as the independent variable. The purpose of regression analysis is to understand the direction and extent to which values of dependent variable can be predicted by the corresponding values of the independent variable. The regression gives the nature of relationship between the variables. Often the relationship between two variable x and y is not an exact mathematical relationship, but rather several y values corresponding to a given x value scatter about a value that depends on the x value. For example, although not all persons of the same height have exactly the same weight, their weights bear some relation to that height. On the average, people who are 6 feet tall are heavier than those who are 5 feet tall; the mean weight in the population of 6- footers exceeds the mean weight in the population of 5footers.
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128 This relationship is modeled statistically as follows: For every value of x there is a corresponding population of y values. The population mean of y for a particular value of x is denoted by f(x). As a function of x it is called the regression function. If this regression function is linear it may be written as f(x) = a + bx. The quantities a and b are parameters that define the relationship between x and f(x) In conducting a regression analysis, we use a sample of data to estimate the values of these parameters. The population of y values at a particular x value also has a variance; the usual assumption is that the variance is the same for all values of x. Principle of Least Squares Principle of least squares is used to estimate the parameters of a linear regression. The principle states that the best estimates of the parameters are those values of the parameters, which minimize the sum of squares of residual errors. The residual error is the difference between the actual value of the dependent variable and the estimated value of the dependent variable. Fitting of Regression Line y = a + bx By the principle of least squares, the best estimates of a and b are S xy b = 2 and a = y - b x Sx Where Sxy is the covariance between x and y and is defined as Sxy = And Sx 2 is the variance of x, that is, Sx 2 =
1 Sxi2 – ( x )2 n - 1 Sxi yi - x y n
and a = y - b x = 6.33 – 0.252´13.9 = 2.8272 Therefore, the straight line is y = 2.8272 + 0.252 x Two Regression Lines There are two regression lines; regression line of y on x and regression line of x on y. In the regression line of y on x, y is the dependent variable and x is the independent variable and it is used to predict the value of y for a given value of x. But in the regression line of x on y, x is the dependent variable and y is the independent variable and it is used to predict the value of x for a given value of y. The regression line of y on x is given by S xy y - y = 2 (x - x) Sx and the regression line of x on y is given by S xy x- x = (y - y ) 2 Sy Regression Coefficients S xy The quantity 2 is the regression coefficient of y ox and is denoted by byx , which gives the Sx S xy slope of the line. That is, byx = 2 is the rate of change in y for the unit change in x. Sx S xy The quantity is the regression coefficient of x on y and is denoted by bxy , which gives the 2 Sy slope of the line. That is, bxy =
S xy Sy
2
is the rate of change in x for the unit change in y.
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130
13.12 Let us Sum Up
In this Lesson the concept of correlation and regression are discussed. The correlation is the association between two variables. A scatter plot of the variables may suggest that the two variables are related but the value of the Pearson's correlation coefficient r quantifies this association. The correlation coefficient r may assume values from –1 and + 1. The sign indicates whether the association is direct (+ve) or inverse (- ve). A numerical value of 1 indicates perfect association while a value of zero indicates n o association. Regression is a device for establishing relationships between variables from the given data. The discovered relationship can be used for predictive purposes. Some simple examples are shown to understand the concepts.
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131
UNIT - V Lesson 14 - Time Series
Contents
14.1 Aims and Objectives 14.2 Definition of a time series 14.3 Time series cycle 14.4 Time series models 14.5 Time series analysis 14.6 Standard time series models 14.7 Description of time series components 14.8 Graphing a time series 14.9 Let us Sum Up 14.10 Lesson – End Activities 1411 References
14.1 Aims and Objectives
This Lesson defines a time series and describes the structure (called the time series model) within which time series' movements can be explained and understood. The various components that go to make up each time series value are then discussed and, finally, brief mention is made of graphical techniques.
14.2 Definition of a time series
A time series is the name given to the value of some statistical variables measured over a uniform set of time points. Any business, large or small, will need to keep records of such things as sales, purchases, value of stock held and VAT and these could be recorded daily, weekly, monthly, quarterly or yearly. These are examples of time series. A time series is a name given to numerical data that is described over a uniform set of time points. Time series occur naturally in all spheres of business activity as demonstrated in the following example. Example 1 (Situations in which time series occur naturally) a) Annual turnover of a firm for ten successive years. b) Numbers unemployed (in thousands) for each quarter of four successive years. c) Total monthly sales for a small business for three successive years.
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14.3 Time series cycle
Normally, time series data exhibits a general pattern which broadly repeats, called a cycle. Sales of domestic electricity always have a distinct four-quarterly cycle; monthly sales for a business will exhibit some natural 12-monthly cycle; daily takings for a supermarket will display a definite 6-daily cycle. The cycle for the Home Removals data in above can be seen to be 4-quarterly.
14.4 Time series models
Business records, and in particular certain time series of sales and purchases, need to be kept by law. Of course they are also used to help control current (and plan future) business activities. To use time series effectively for such purposes, the data have to be organized and analysed. In order to explain the movements of time series data, models can be constructed which describe how various components combine to form individual data values. As an example, a Sales Manger could set up the following model to explain the expense claims of his sales force each week: y =f+t Where, y = total expenses for week, f = fixed expenses (meals, insurance etc), and t = travelling expenses (petrol, car maintenance, incidentals, etc.)
14.5 Time series analysis
It is the evaluation and extraction of components of a model that 'break down' a particular series into understandable and explainable portions and enables : a) Trends to be identified. b) Extraneous factors to be eliminated and c) Forecasts to be made The understanding, description and use of these processes is known as time series analysis.
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133
14.6 Standard time series models
Depending on the nature, complexity and extent of the analysis required, there are various types of model that can be used to describe time series data. However, for the purposes of this manual, two main models will be referred to. They are known as the simple additive and multiplicative models. The components that go to make up each value of a time series are described in the following definitions. The time series additive model y=t+s+r where, y is a given time series value t is the trend component s is the seasonal component r is the residual component. The time series multiplicative model y=tXSXR where, y is a given time series value t is the trend component S is the seasonal component R is the residual component. Put another way, given a set of time series data, every single given (y) value can be expressed as the sum or product of three components. It is the evaluation and interpretation of these components that is the main aim of the overall analysis. Note that although the trend component will be constant no matter which of the two models are used, the values of the seasonal and residual components will depend on which model is being used. In other words, given a set of data to which both models are being applied, both trend values would be identical whereas the respective seasonal and residual components would be quite different.
14.7 Description of time series components
a) Trend. The underlying, long-term tendency of the data. b) Seasonal variation. These are short-term cyclic fluctuations in the data about the trend which take their name from the standard business quarters of the year. Note however that the word 'season' in this context can have many different meanings. For example: i. daily ;seasons' over a weekly cycle for sales in a supermarket, ii. monthly ' seasons' over a yearly cycle for purchases of a company, iii. quarterly 'seasons' over a yearly cycle for sales of electricity in the domestic sector. c) Residual variation. These include other factors not explained by a) and b) above. This variation normally consists of two components:
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134 i. Random factors. These are disturbances due to 'everyday' unpredictable influences, such as weather conditions, illness, transport breakdowns, and so on. ii. Long-term cyclic factor. This can be thought of (if it exists) as due to underlying economic causes outside the scope of the immediate environment. Examples are standard trade cycles or minor recessions. Example 2 (general comments on a given time series) Comment on the following data, which relates to visitors (in hundreds) to a hotel over a period of three years. Do not use any quantitative techniques or analyses. Qtr 1 Qtr 2 Qtr 3 Qtr 4 Year 1 57 85 97 73 Year 2 64 96 107 89 Year 3 76 102 115 95 Answer The data displays a distinct 4-quarterly cycle over the three year period, with the underlying trend showing a steady increase overall, as well as in each particular quarter. It shows a significant seasonal effect with (not unexpectedly) the cycle peak in the summer quarter and a trough in the winter quarter. Increases are significantly less in the second and third quarters from year 2 to year 3, which may be due to an upper capacity limit in accommodation for those periods or some other random factor. There is not enough data to identify and possible long-term cyclic factors.
14.8 Graphing a time series
a) The standard graph for a time series is a line diagram, known technically as a historigram. It is obtained by plotting the time series values (on the vertical axis) against time (on the horizontal axis) as single points which are joined by straight line segments. b) Historigrams can be shown on their own but it is quite common to see both a historigram and the graph of associated derived data, such as a trend, plotted together on the same chart.
Exercise
1. What is a time series ? 2. What are the aims of time series analysis ? 3. Describe the simple additive time series model and name its components. 4. Describe what a 'season' is in the context of a time series and give some examples. 5. For an additive time series model, what does the term 'residual variation' mean? Describe briefly its two main constituents. 6. What might contribute towards random variation for data pertaining to daily sales in a supermarket over a period of four weeks. Try to list at least six factors. 7. Graph the following data and comment on significant features. Sales of a company (Rs.000) Qtr 1 Qtr 2 Qtr 3 Qtr 4 1982 19 31 62 9 1983 20 32 65 17
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135 1984 1985 1986 24 24 25 36 39 42 78 83 85 14 20 24
14.9 Let us Sum Up
In this Lesson, we have discussed about a time series which is a set of data that is described over a uniform set of time points. Cycles are general patterns that repeat and occur in most types of time series. Time series models are used to gain an understanding of the factors that effect time series. The time series additive model describes the way that the trend, seasonal and residual components independently make up each time series value. A historigram is the standard way of displaying a time series diagrammatically. The applications of time series is obviously occurring while analyzing sales data, marketing related data, advertisement pattern and costs, inventory analyis, etc.
14.10 Lesson – End Activities
1. Define time series. 2. How to graph a time series?
14.11 References
1. Gupta S.P. – Statistical Methods.
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136
Lesson 15 - Time Series Trend
Contents 15.1 Aims and Objectives 15.2 The significance of trend values 15.3 Techniques for extracting the trend 15.4 The method of semi-averages 15.5 Working data (for rest of the Lesson) 15.6 The method of least squares regression 15.7 The method of moving averages 15.8 Moving average centering 15.9 Comparison of techniques for trend 15.10 Let us Sum Up 15.11 Lesson – End Activities 15.12 References
15.1 Aims and Objectives
This Lesson describes the significance of trend values and the three most common methods of extracting a trend from a given time series. Each method is demonstrated using a common time series and the results compared graphically. Significant features of the three techniques are listed, including their advantages and disadvantages.
15.2 The significance of trend values
It will be recalled from the previous Lesson that the object of finding the time series trend is to enable the underlying tendency of the data to be highlighted. Thus, a business sales trend will normally show whether sales are moving up or down (or remaining static) in the long term. The trend can also be thought of as the core component of the additive time series model about which the two other components, seasonal (s) and residual (r) variation, fluctuate. This component is found by identifying separate trend (f) values, each corresponding to a time point. In other words, at each time point of the series, a value of t can be obtained which forms one of the components that go to make up the observed value of y. The following section summarizes three different ways of obtaining trend values for a given time series.
15.3 Techniques for extracting the trend
There are three techniques that can be used to extract a trend form a set of time series values. a) Semi-averages. This is the simplest technique, involving the calculation of two (x,y) averages which, when plotted on a chart as two separate points and joined up, form a straight line. A similar method was introduced in Lesson 15, to find a regression line. b) Least squares regression. This method, also introduced in Lesson 15 similarly results in a straight line.
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137 c) Moving averages. This is the most commonly used method for identifying a trend and involves the calculation of a set of averages. The trend, when obtained and charted, consists of straight line segments.
15.4 The method of semi-averages
The method of semi-averages for obtaining a trend for a time series is now demonstrated with a simple example. Suppose the following sales (Rs. in 1000) were recorded obtain a semi-average trend. Week 1 Mon Tue Wed Thu Fri Mon Tue Sales(y) 250 320 340 520 410 260 380 for a firm and it is required to Week 2 Wed Thu 410 670
Fri 420
Note that the data is time-ordered, which is normal and natural for a time series. The procedure for obtaining a trend using the method of semi-averages is: STEP 1 Split the data into a lower and an upper group. For the data given: the lower group is 250,320,340,520 and 410; the upper group is 260,380,410,670 and 420. STEP 2 Find the mean value of each group. The mean of the lower group (L) is 1840/5= 368. The mean of the upper group (U) is 2140/5 = 428. STEP 3 Plot, on a graph, each mean against an appropriate time point. 'An appropriate time point' can always be taken as the median time point of the respective group. Thus L would be plotted against Wednesday of week 1 and U against Wednesday of week 1 and U against Wednesday of week 2. STEP 4 The line joining the two plotted points is the required trend. Note that it is important that the two groups in question have an equal number of data values. If the given data, however, contains an odd number of data values, the middle value can be ignored (for the purposes of obtaining the trend line). Once a trend line has been obtained, the trend values corresponding to each time point can be read off from the graph. A fully worked example follows.
15.5 Working data (for rest of the Lesson)
The following set of data will be referred to throughout the Lesson in order to demonstrate the calculations involved in using each of the three methods for obtaining a time series trend. UK outward passenger movements by sea Year 1 Year 2 Year 3 Quarter 1 2 3 4 1 2 3 4 1 2 3 4
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Example 1 (calculating a time series trend using semi-averages)
Question Using the working data, given above: a) Use the method of semi-averages to obtain and plot a trend line. b) Draw up a table showing the original data (y) values against the trend (t) values (obtained from the graph). Answer a) The data has been split up into lower and upper groups, each one being totaled and then averaged. Year 1 Q1 2.2 Year 2 Q3 8.2 Q2 5.0 Q4 3.8 Q3 7.9 Year 3 Q1 3.2 Q4 3.2 Q2 5.8 Year 2 Q1 2.9 Q3 9.1 Q2 2.2 Q4 4.1 Total 26.4 Mean(L) 4.4 Total 34.2
Mean(U) 5.7
In this situation, both L and U must be plotted against a hypothetical point between the middle two time points in their respective sets. That is, L is plotted at a time point between Year 1 Q3 and Year 1 Q4 and L is plotted corresponding to a point between Year 3 Q1 and Year 3 Q2. In Figure 1, the two means have been plotted and joined by a straight line to form the trend line. b) The trend values have been read from the graph and are tabulated below, together with the original data values. Year 1 Year 2 Year 3 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 Data(y) 2.2 5.0 7.9 3.2 2.9 5.2 8.2 3.8 3.2 5.8 9.1 4.1 Trend(t) 3.9 4.1 4.3 4.5 4.7 4.9 5.2 5.4 5.6 5.8 6.0 6.2
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15.6 The method of least squares regression
The technique of least squares regression was explained and demonstrated in earlier Lesson. In order to use this method to obtain a trend line for a time series, it is necessary to consider the time series data as bivariate. The procedure is given as follows. STEP 1 Take the physical time points as values (coded as 1,2,3 etc if necessary) of the independent variable x. STEP 2 Take the data values themselves as values of the dependent variable y. STEP 3 Calculate the least squares regression line of y on x,y=a+bx. STEP 4 Translate the regression line as t=a+bx, where any given value of time point x will yield a corresponding value of the trend, t. An example of the use of this technique follows.
= 292.8 1716 i.e. b = 0.17 (2D) and: a = y - b x = 60.6 - 0.17 X 78 n n 12 12 i.e. a = 3.94 (2D) thus, the regression line for the trend is t = 3.94 + (0.17)(x)(2D) (Remember that once the regression line is determined, it will be used for calculating trend values. So the normal 'y' has been replaced by 't') The time point values (x=1,2,3 etc) can now be substituted into the above regression line to give the trend values required. When x=1 (Year 1 Qtr1), t=3.94+0.17(1) i.e., t=4.11 (2D) When x=2 (Year 2 Qtr2), t=3.94+0.17(2) i.e., t=4.28 (2D) …etc. These and other values of t are tabulated in the previous table.
15.7 The method of moving averages
This method of obtaining a time series trend involves calculating a set of averages, each one corresponding to a trend (t) value for a time point of the series. These are known as moving averages, since each average is calculated by moving from one overlapping set of values to the next. The number of values in each set is always the same and is known as the period of the moving average. To demonstrate the technique, a set of moving averages of period 5 has been calculated below for a set of values. Original values: 12 10 11 11 9 11 10 10 11 10 Moving totals: 53 52 52 51 51 52 Moving averages: 10.6 10.4 10.4 10.2 10.2 10.4 The first total, 53, is formed from adding the first 5 items; i.e. 53=12+10+11+11+9. Similarly, the second total, 52=10+11+11+9+11, and so on. The averages are then obtained by dividing each total by 5. Notice that the totals and averages are written down in line with the middle value of the set being worked on. These averages are the trend (t) values required.
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It should also be noticed that there are no trend values corresponding to the first and last two original values. This is always the case with moving averages and is a disadvantage of this particular method of obtaining a trend.
15.7.1 Let us Sum Up of the moving average technique
Moving averages (of period n) for the values of a time series are arithmetic means of successive and overlapping values, taken n at a time. The (moving ) average values calculated form the required trend components (t) for the given series. The following points should be noted when considering a moving average trend. a) The period of the moving average must coincide with the length of the natural cycle of the series. Some examples follows. i. Moving averages for the trend of numbers unemployed for the quarters of the year must have a period of 4. ii. Total monthly sales of a business for a number of years would be described by a moving average trend of period 12. iii. A moving average trend of period 6 would be appropriate to describe the daily takings for a supermarket (open six days per week) over a number of months. b) Each moving average trend value calculated must correspond with an appropriate time point. This can always be determined as the median of the time points for the values being averaged. For moving averages with an odd- numbered period, 3,5,7, etc, the relevant time point is that corresponding to the 2nd, 3rd, 4th , etc value. See the example in the previous section, where the moving averages had a period of 5 and thus each average obtained was set against the 3rd value of the respective set being averaged. However, when the moving averages have an even- numbered period (2,4,6,8,etc). There is no obvious and natural time point corresponding to each calculated average. The following section describes the technique known as 'centering', which is used in these circumstances.
15.8 Moving average centering
When calculating moving averages with an even period (i.e. 4,6 or 8), the resulting moving average would seem to have to be placed in between two corresponding time points. As an example, the following data has a 4-period moving average calculated and shows its placing Time point 1 2 3 4 5 6 7 8 9 10 Data value 9 14 17 12 10 14 19 15 10 16 Totals(of 4) 52 53 53 55 58 58 60 Averages (of 4) 13.00 13.25 13.25 13.75 14.50 15.00
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142 The placing of these averages as described above would not be satisfactory when the averages are being used to represent a trend, since the trend values need to coincide with particular time points. A method known as centering is used in this type of situation, where the calculated averages are themselves averaged in successive overlapping pairs. This ensures that each calculated (trend) value 'lines up' with a time point. This techniques is now shown for the previous data. Time point 2 3 4 5 6 7 8 9 Averages(of 4) 13.00 13.25 13.25 13.75 14.50 14.50 15.00 Averages (of 2) 13.125 13.250 13.500 14.125 14.500 14.750 A worked example follows which uses this technique.
Example 3 (calculating trend values using moving average centering)
Question Calculate trend values for the working data of section 5, using moving averages with an appropriate period. Plot a graph of the original data with the trend superimposed. Answer : The cycle of the data is clearly 4-quarterly and we thus need a (centered) 4-quarterly moving average trend, using the technique described in section 11 above. Table 1 demonstrates the standard columnar layout of the calculations. Qtr Original Moving totals Moving Centered moving Data(y) of 4 average average(t) Year 1 1 2.2 2 5.0 18.3 4.575 4.66 3 7.9 19.0 4.750 4.78 4 3.2 19.2 4.800 4.84 Year 2 1 2.9 19.5 4.875 4.95 2 5.2 20.1 5.025 5.06 3 8.2 20.4 5.100 5.18 4 3.8 21.0 5.250 5.18 5.36 Year 3 1 3.2 21.9 5.475 5.51 2 5.8 22.2 5.550 4 4.1 Table 1 Notice that the two starting and ending time points do not have a trend value. As mentioned previously, this type of omission will always occur with a moving average trend. Figure 2 shows a graph of the original data with the trend values superimposed.
15.9 Comparison of techniques for trend
For the working data given in section 4, all three methods of obtaining a trend have now been demonstrated. The method of semi-averages (Example 1), least squares (Example 2) and moving averages (Example 3). Figure 3 shows the graphs for comparison.
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143 The fact that the three sets of trend values are quite distinct underlines the fact that there is no unique set of trend values for a time series. Each method will yield a different trend, as has been evidenced. UK outward passenger Movements by sea 10 Number of Passengers (millions) 8 6 4 2 -
Trend
Movements
| | | | | | | | | | | | 1 2 3 4 1 2 3 4 1 2 3 4 Year 1 Year2 Year 3
Quarter Figure 2
Significant features of each method are now summarized. a) Semi-averages. Although simple to apply, the fact that only two plotted points are used in its construction leads to the general feeling that it is unrepresentative. It also assumes that a strictly linear trend is appropriate to the data. b) Least squares. Although mathematically representative of the data, it assumes that a linear trend is appropriate. It is generally though unsuitable for highly 'seasonal' data. c) Moving averages. The most widely used technique for obtaining a trend. If the period of the averages is chosen appropriately, it will show the true nature of the trend, whether linear or non-linear. One disadvantage is the fact that no trend values are obtained for the beginning and end time points of a series. Passenger movements
Trend Type Moving average Least squares Semi-averages Time Figure 3
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144
Exercises
1. Calculate a set of trend values (to ID) using the method of semi-averages, for the following data: 16, 12, 15, 14, 18, 12, 14, 13, 18, 13. 2. Calculate a set of moving averages of period: (a) 3 (b) 5 for the following time series data: 8, 11, 10, 21, 4, 9, 12, 10, 23, 5, 10, 13, 11, 26, 6. Which set of moving averages is the correct one to use for obtaining a trend for the series? 3. Draw a historigram for the data described in question 2 above, superimposing the correct trend values. 4. The number of houses (in thousands) built each year between 1953 and 1969 (inclusive) are given as: Year 1 2 3 4 5 6 7 8 9 Number of houses 319 348 317 308 308 329 332 354 378 Year 10 11 12 13 14 15 16 17 Number of houses 364 358 383 391 396 415 426 378 Assuming a seven-year cycle, eliminate the cyclical movement by producing a moving average trend and plot this, together with the original data on the same chart. 5. The following figures relate to Rate receipts (in Łm) for a Local Authority. Year1 Year2 Year3 Qtr1 2.8 3.0 3.0 Qtr2 4.2 4.2 4.7 Qtr3 3.0 3.5 3.6 Qtr4 4.6 5.0 5.3 Plot a historigram for the data, together with a lease squares regression trend
15.10 Let us Sum Up
In this Lesson the time series trend is discussed and three common techniques for identifying trend components are discussed. They are : (i) semi-averages (ii) least squares regression and (iii) moving averages. For time series that have a significant seasonal effect, the moving average technique is generally preferred. When moving averages are used for identifying trend components, the period of the average must coincide with the cycle of the data being analysed. This is done in order to remove possible cyclical fluctuations. Even-period moving averages must be centered in order that their values coincide with actual time points. It is also to be noted that there is no unique set of trend values for a given time series. The particular method chosen needs to take into account the nature of the data and the use to which trend values will be put.
15.11 Lesson – End Activities
1. What is meant by moving average?
15.12 References
R.S.N. Pillai and Mrs. Bhagavathi – Statistics.
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16.1 Aims and Objectives
The Lesson described the nature of seasonal variation in a time series and how it can be calculated. Forecasting, or the ability to estimate future values of a given time series using seasonal variation, is dealt with in this Lesson.
16.2 The nature of seasonal variation
Seasonal (or short-term cyclic) variation is present in many time series. Winter sportswear will sell well in autumn and winter, and badly in spring and summer; supermarket sales are higher at the end of the week than at the beginning; sales of umbrellas are at the peak during the end of the summer and just at the beginning of the rainy season, etc. When values are obtained to describe seasonal variation, they are sometimes known as seasonal values or factors and are expressed as deviations (i.e.'+'or'-') from the underlying trend. They show, on average, by how much a particular season will tend to increase or decrease the underlying trend. Thus we would expect the seasonal variation for winter sportswear to be positive in autumn and winter and negative in spring and summer. Seasonal variation components give an average effect on the trend which is solely attributable to the 'season' itself. They are expressed in terms of deviations from (additive model) or percentages of (multiplicative model) the trend. The use of seasonal variation figures are of great importance to organizations operating in environments where a seasonal factor is significant. For example, a regional Electricity Board needs to know the average increase in demand expected in the winter months in order to be able to meet this demand. The following two sections describe and demonstrate the technique for calculating seasonal variation.
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16.3 Technique for calculating seasonal variation
a) Additive model Given the original time series (y) values, together with the trend (t) values, the procedure for calculating the seasonal variation is given as follows. STEP 1 Calculate, for each time point, the value of y-t (the difference between the original value and the trend). STEP 2 For each season in turn, find the average (arithmetic mean) of the y-t values. STEP 3 If the total of the averages differs from zero, adjust one or more of them so that their total is zero. The values so obtained are the appropriate seasonal variation values; i.e. the 's' figures in the additive model y = t + s + r. b) Multiplicative model Given the original time series (y) values, together with the trend (t) values, the procedure for calculating the seasonal variation is given as follows. STEP 1 Calculate, for each time point, the value of (y-t)/t (the difference between the original value and the trend expressed as a proportion of the trend). STEP 2 For each season in turn, find the arithmetic mean of the above proportional changes. [Note that this should strictly involve calculating the geometric mean of 1+ proportional change values. In practice however this is felt to be too complex!] STEP 3 If the total of the averages differs from zero, adjust one or more of them so that their total is zero. The values so obtained are the appropriate seasonal variation values; i.e. the 'S' figures in the multiplicative model y = t + S + R. Example 1 (Calculating seasonal variation figures using the additive model) The sales of a company (y, in Rs. 000) are given below, together with a previously calculated trend (t). The subsequent calculations to find the seasonal variation are shown, laid out in a standardized way.
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STEP 3 Since the averages sum to -0.5 ( and not zero), it is necessary to adjust one or more of them accordingly. In this case, since the difference is so small, only one will be adjusted. In order to make the smallest percentage error, the largest value (35.5) is changed to 36.0. this adjustment is shown in the following table: Q1 Q2 Q3 Q4 Initial s values -6.5 -19.5 35.5 -10.0 Adjustment 0 0 +0.5 0 Adjusted s values -6.5 -19.5 36.0 -10.0 (Sum = 0) The interpretation of the figures is that the average seasonal effect for quarter 1, for instance, is to deflate the trend by 6.5 (Rs. 000) and that for quarter 3 is to inflate the trend by 36 (Rs. 000). Example 2 (Calculating seasonal variation figures using the multiplicative model) The sales of a company (y, in Rs. 000) are given below, together with a previously calculated trend (t). The subsequent calculations to find the seasonal variation are shown, laid out in a standardized way. Step 1 y-t S=1+ y-t y t t t Step 2 Year 1 Qtr1 20 23 -0.13 0.87 Deviations (1 + y - t ) 2 15 29 -0.48 0.52 t 3 60 34 0.76 1.76 Q1 Q2 Q3 Q4 Sum 4 30 39 -0.23 0.77 Year1 0.87 0.52 1.76 0.77 Year2 Qtr 1 35 45 -0.22 0.78 Year 2 0.78 0.50 1.82 0.82 2 25 50 -0.50 0.50 G. Means 0.82 0.51 1.79 0.79 3.91 3 100 55 0.82 1.82 4 50 61 -0.18 0.82 STEP 3 Since the averages sum to 3.91 (and not 4), it is necessary to add 0.09 to one or more of them accordingly. In this case, as in the previous Example, since the difference is so small, only one will be adjusted. In order to make the smallest percentage error, the largest value (1.79) is changed to 1.88. This adjustment is shown in the following table.
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148 Q1 Q2 Q3 Q4 Initial S values 0.82 0.51 1.79 0.79 Adjustment 0 0 +0.9 0 Adjusted S values 0.82 0.51 1.88 0.79 (Sum = 4.00) The interpretation of the figures is that the average seasonal effect for quarter 1, for instance, is to deflate the trend by 18% (since 0.82 is 0.18 less than 1) and that for quarter 3 is to inflate the trend by 88%.
16.4 Seasonally adjusted time series
One particular and important use of seasonal values is to seasonally adjust the original data. The effect of seasonal adjustment is to smooth away seasonal fluctuations, leaving a clear view of what might be expected 'had seasons not existed'. The techniques is similar for both models but is shown separately for clarity. Additive model: The adjustment is performed by subtracting the appropriate seasonal figure from each of the original time series values and represented algebraically by y-s. As an example, the data of Examples 1 and 2 are seasonally adjusted below. Y s y-s Year1 Qtr 1 20 -6.5 20-(-.5)=26.5 2 15 -19.5 15-(-19.5)=34.5 3 60 36.0 60-36.0=24.0 4 30 -10.0 30-(-10.0)=40.0 Seasonal Year2 Qtr 1 35 -6.5 35-(-6.5)=41.5 adjusted values 2 25 -19.5 25-(-19.5)=44.5 3 100 36.0 100-36.0=64.0 4 50 -10.0 50-(-10.0)=60.0 Multiplicative model: The adjustment is performed by dividing each of the original time series values by S and is represented algebraically by y/S. As an example, the data of Example 1 are again seasonally adjusted below. Y S y/S Year1 Qtr 1 20 0.82 20/0.82=24.3 2 15 0.51 15/0.51=29.5 3 60 1.88 60/1.88=31.9 4 30 0.79 30/0.79=37.8 Seasonally Year2 Qtr 1 35 0.82 35/0.82=42.6 adjusted values 2 25 0.51 25/0.51=49.2 3 100 1.88 100/1.88=53.2 4 50 0.79 50/0.79=63.0
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149 To summarise: Seasonally adjusted time series data are obtained by subtraction (additive model) or division (multiplicative model) as follows: Additive model: seasonally adjusted value = y-s Multiplicative model: seasonally adjusted value=y/s. The importance of seasonal adjustments is reflected in the fact that the majority of economic time series data published by the Central Statistical Office is presented both in terms of 'actual' and 'seasonally adjusted' figures. Example 3 (Seasonal adjustment of a time series) Question The following data gives UK outward passenger movements (in millions) by sea, together with a 4-quarterly moving average trend (calculated previously in the earlier Lesson). Find the values of the seasonal variation for each of the four quarters (using an additive model) and hence obtain seasonally adjusted outward passenger movements. Plot the result on a graph. Year1 1 2 Year2 2 3 Year3 4 1 2
Quarter 3 4 1 3 4 Number of Passengers(y) 2.2 5.0 7.9 3.2 2.9 5.2 8.2 3.8 3.2 5.8 9.1 4.1 4.66 4.78 4.84 4.95 5.06 5.18 5.36 5.51 Answer : The deviations are calculated and displayed in column 5, and the calculations for the seasonal variation are shown in the lower table and the results, together with the seasonally adjusted data, have been added at column 6 and 7.
16.5 Notes on Example 3
1. It is usual to show the calculation of the seasonal values in rectangular form as demonstrated above. 2. Notice that the adjustment needed above was +0.07. However, rather than adding all of this to just one of the averages, it was divided up into the four parts +0.02, +0.02,
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151 +0.02 and +0.01, each being added to a separate average. This is generally regarded as a fairer way to adjust. 3. Even though the moving average trend values are missing at the beginning and end time points, the seasonal values calculated can still be used at these points and thus seasonal adjustment can be performed for all original data items.
16.6 Forecasting
a) A particular use of time series analysis is in forecasting, sometimes called projecting the time series. Clearly, business life would be much easier if monthly sales for the next year were known or the number of transport breakdowns next month could be determined. However, no-one can predict the future; the best that can be done is to estimate the most likely future values, given the analysis of previous years' sales or last month's breakdowns. b) Forecasting can be performed at different levels depending on the use to which it will be put. Simple guessing, based on previous figures, is occasionally adequate. However where there is a large investment at stake (in plant, stock and manpower for example), structured forecasting is essential. c) any forecasts made, however technical or structured, should be treated with caution, since the analysis is based on past data and there could be unknown factors present in the future. However, it is often reasonable to assume that patterns that have been identified in the analysis of past data will be broadly continued, at least into the short-term future.
16.7 Technique for forecasting
Forecasting a value for a future time point involves the following steps. STEP 1 Estimate a trend value for the time point. There are a number of ways of estimating future trend values and some of these are described in section 12. STEP 2 Identify the seasonal variation value appropriate to the time point. Seasonal variation values are calculated in the manner already described in section 5. STEP 3 Add (or multiply, depending on the model) these two values together, giving the required forecast.
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152 Example 4 (Time series forecasting) Forecast the values for the four quarters of year 4, given the following information which has been calculated from a time series. Assume that the trend in year 4 will follow the same pattern as in year 1 to 3 and an additive model is appropriate. Year1 Year2 Year3 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 Trend (t) 42 44 46 48 50 52 54 56 58 60 62 64 S1 =seasonal factor for quarter 1=-15; s2 =-8; s3 =+6;s4 =+17 STEP 1 Estimate trend values for the relevant time points. Note that, in this case, the trend values increase by exactly 2 per quarter. Trend for year 4, quarter 1=t4,1=66. Similarly, t4,2=68,t4,3=70and t4,4=72 STEP 2 Identify the appropriate seasonal factors. The seasonal factors for year 4 are taken as the given seasonal factors. That is, seasonal factor for year 4, quarter 1=s1 =-15 etc. STEP 3 Add the trend estimates to the seasonal factors, giving the required forecasts. Forecast for year 4, quarter1=t4,1+s1 =66-15=51; Forecast for year 4, quarter2= t4,2+s2 =68-8=60; Forecast for year 4, quarter3=t4,3+s3 =70+6=76; Forecast for year 4, quarter4=t4,4+s4=72+17=89.
16.8 Projecting the trend
Projecting the trend for the data of Example 3 was straightforward since the given trend values increased uniformly, thereby displaying a distinct linear pattern. In general, trend values will not conform conveniently in this way. There are a number of techniques available for projecting the trend, depending on the method used in obtaining the trend values themselves. The most common are now listed. a) Linear trend. Whether the method of least squares or semi-averages has been used, the projection involves simply extending the trend line already calculated. b) Moving average trend. There is no one universal method. Three common means of projecting are listed below. i. 'By eye' (or inspection) from the graph. This involves adding a projection freehand in a manner that seems most appropriate. This might seem fairly arbitrary, but remember that any form of projection (no matter how technical) is still only an estimate. This particular method can be employed when the calculated trend values are distinctly 'non- linear'. ii. Using the method of semi-average on the calculated trend values to obtain a linear projection of the trend. This method can be employed with 'fluctuating linear' trend values. iii. Using the first and last of the calculated trend values to obtain a linear projection of the trend. This method can be employed with fairly 'steady linear' trend values.
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153 Example 5 (Time series forecasting) Question Forecast the four quarterly values for year 4 for the following data, which relates to UK outward passenger movements by sea (in millions). The trend (calculated previously) and the seasonal variation components (using the multiplicative model) are given below. Year1 Year2 Year3 Quarter 1 2 3 4 1 2 3 4 1 2 3 4 Number of Passengers(y) 2.2 5.0 7.9 3.2 2.9 5.2 8.2 3.8 3.2 5.8 9.1 4.1 Trend(t) 4.66 4.78 4.84 4.95 5.06 5.18 5.36 5.51 Seasonal variation(S): Qtr1=0.60; Qtr2=1.05; Qtr3=1.65; Qtr4=0.70 Plot the original values, trend and forecast on a single chart. Answer STEP1 Estimate trend values for the relevant time points. Since there is a fairly steady increase in the trend values, demonstrating an approximate linear relationship, method iii (from section 11 (b)) is appropriate for projecting the trend. Range of trend values =5.51-4.66=0.85 Therefore, average change per time period=0.85/7=0.12 (approx). [Note that since there are 8 trend values, there are correspondingly only 7 'jumps' from the first to the last. Hence the divisor of 7 in the above calculation.] The last trend value given is 5.51 for Year 3 Quarter 2 and this must be used as the base value to which is added the appropriate number of multiples of 0.12. Thus, the trend estimates are: t(Year4Qtr1)=5.51+3(0.12)=5.87;t(Year4Qtr2)=5.51+4(0.12)=5.99; t(Year4 Qtr3)=5.51+5(0.12)=6.11;t(Year4Qtr4)=5.51+6(0.12)=6.23. STEP 2 Identify the appropriate seasonal factors. These values are given in the question as: S1 =0.60;S2 =1.05;S3 =1.65;S4 =0.70. STEP 3 Multiply the trend estimates by the respective S values, giving the required forecasts. y(Year4Qtr1)=5.87X0.60=3.51; y(Year4Qtr2)=5.99X1.05=6.30; y(Year4Qtr3)=6.11X1.65=10.10; y(Year4Qtr4)=6.23X0.70=4.37. These values are plotted in Figure 2, along with the original data and trend.
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16.9 Forecasting and residual variation
Residual variation is the variation which takes into account everything else other than trend and seasonal factors. In the main it consists of small random fluctuations which, although not controllable, have little effect. If the residual variation is relatively large however, it will make forecasts less dependable, since they effectively ignore residual elements. Thus, being able to identify a residual element in a time series will normally be a pointer to how reliable any projection will be.
17.1 Aims and Objectives
Index numbers provide a standardized way of comparing the values, over time, of commodities such as prices, volume of output and wages. They are used extensively, in various forms, in Business, commerce and Government. This Lesson introduces index numbers and describes the most simple form; the index relative. Relatives are defined, calculated as time series and compared (using a base-changing techniques). Finally, time series deflation is described, which is a method of calculating an index of the real values of time series. This Lesson also describes index relatives, the simplest form of index number, and some of the ways that they can be presented and manipulated and composite index numbers, which describe the change over time of groups or classes of commodities that have something in common. The two forms of composite index covered are the weighted average of relatives and the weighted aggregate.
17.2 Definition of an Index Number
An index number measures the percentage change in the value of some economic commodity over a period of time. It is always expressed in term of a base of 100. 'Economic commodity' is a term of convenience, used to describe anything measurable which has some economic relevance. For example: price, quantity, wage, productivity, expenditure, and so on. Examples of typical index number values are: 125 (an increase of 25%), 90 (a decrease of 10%), 300 (an increase of 200%).
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157
17.3 Simple index number construction
a) Suppose that the price of standard boxes of ball-point pens was Rs. 60 in January and rose to Rs. 63 in April. We can calculate as follows: percentage increase= 63-60 = 100 = 5 60 20 In other words, the price of ball-point pens rose by 5% from January to April. To put this into index number form, the 5% increase is added to the base of 100, giving 105. This is then described as follows: the price index of ball-point pens in April was 105(January = 100). Note that any increase must always be related to some time period, otherwise it is meaningless. Index numbers are no exception, hence the (January=100) in the above statement, which: i. gives the starting point (January) over which the increases in price is being measured; ii. emphasizes the base value (100) of the index number. b) If the productivity of a firm (measured in units of production per man per day) decreased by 3% over the period from 1983 to 1985, this percentage would be subtracted from 100 to give an index number of 97. Thus we would say: 'the productivity index for the company in 1985 was 97 (1983=100)'.
17.4 Some notation
a) It is convenient, particularly when giving formulae for certain types of index numbers, to be able to refer to an economic commodity at some general time point. Prices and quantities (since they are commonly quoted indices) have their own special letters, p and q respectively. In order to bring in the idea of time, the following standard convention is used. Index number notation p0 = price at base time point pn = price at some other time point q0 = quantity at base time point qn = quantity at some other time point.
In the example in 5.4.3(a) above, time point 0 was January and time point n was April, with p0 =60 and pn =63. b) It is also convenient on occasions to label index numbers themselves in a compact way. There is no standard form for this but, for example (from section 5.4.3 b), the following is sometimes used: I1985(1983=100)97 or
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158 I1985/1983=97 Which is translated as: 'the index for 1985, based on 1983 (as 100), is 97'.
17.5 Index relatives
An index relative (sometimes just called a relative) is the name given to an index number which measures the change in a single distinct commodity. A price relative was calculated in section 5.4.3 (a) and a productivity relative was found in section 5.4.3 (b). However, there is a more direct way of calculating relatives than that demonstrated in section3. the following shows the method of calculating a price and quantity relative.
Price and quantity relatives Pn Price relative: Ip = x100 P0 qn Quantity relative: IQ = x100 q0 Expenditure and productivity relatives can be calculated in a similar fashion. Example 1 (Calculation of price and quantity relatives) The following table gives details of prices and quantities sold of two particular items in a department store over two years. 1984 Item Price P0 Rs. 438 Rs. 322 Number sold q0 37 26 Price pn Rs. 462 Rs. 384 1985 Number sold qn 18 45
The above calculations and presentation demonstrates typical index number notation. Thus it can be seen that an index number is a compact way of describing percentage changes over time.
17.6 Time series of relatives
It is often necessary to see how the values of an index relative change over time. Given the values of some commodity over time (i.e a time series), there are two distinct ways in which relatives can be calculated. a) Fixed base relatives. Here, each relative is calculated based on the same fixed time point. This approach can only be used when the basic nature of the commodity is comparing 'like with like'. For example, the price of rice in a supermarket over six monthly periods or weekly family expenditure on entertainment. b) Chain base relatives. In this case, each relative is calculated with respect to the immediately preceding time point. This approach can be used with any set of commodity values, but must be used when the basic nature of the commodity changes over the whole time period. For example, a company might wish to construct a monthly index of total petrol costs of the standard model of car that its salesmen use. However, the model is likely to change yearly with, for instance, different tyres or 'lean-burn' engines being fitted as standard. Both of these features would affect petrol consumption and thus, also, the petrol cost index. Therefore, in this case, a chain base relative should be used. Example 2 demonstrates the use of the two techniques for the values of a commodity over time. Example 2 (Fixed and chain base set of relatives for a given time series) The data in Table 1 relate to the production of beer (thousands of hectoliters) in the United Kingdom for the first six months of a year. Table 2 shown the calculation of both fixed and chain base relatives, together with some descriptive calculations.
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160
Year Production
Jan 4,563
Feb 4,245
87.7 93.0
Mar 4,841
100 114.0
Apr 4,644
95.9 95.9
May 5,290
109.3 113.9
Jun 5,166 Table 1
106.7 97.7 Table 2
Fixed base relative (Mar=100) 94.3 chain base relative -
4563 X 100 4841
4245 X 100 4563
4841 X 100 4245
5290 X 100 4841
In Table 2, the fixed base relative have been calculated by dividing each month's production by the March production (4841) and multiplying by 100. they enable each month's production to be compared with the March production. Thus, for example, May's production (relative=109.3) was 9.3% up on March. The chain base relatives in Table 2 have been calculated by dividing each month's production by the previous month's production and multiplying by 100. they enable changes from month to month to be highlighted. Thus, for example, February's production (chain relative=93.0) was 7% down on January, March's production (chain relative=114.0) was 14% up on February, and so on.
17.7 Changing the base of fixed-base relatives
Given a time series of relatives, it is sometimes necessary to change the base. One of the reasons for doing this might be that the original base time point is too far in the past to be relevant today and amore recent one is needed. For example, the following set has a base of 1965, which would probably now be considered out-of-date. 1987 1988 1989 1990 1991 1992 1993 Index(1965=100) 324 351 377 384 391 404 428 The procedure for changing the base of a time series of relatives is essentially the same as that for calculating a set of relatives for a given time series of values. However, the procedure is given below and demonstrated, using the above set of relatives: STEP 1 Choose the required new base time point and thus identify the corresponding relative. We will choose 1987 as the base year, with a corresponding relative of 324. STEP 2 Divide each relation in the set by the value of the relative identified above and multiply the result by 100. Thus, each index relative given needs to be divided by 324 and multiplied by 100. Table 3 shows the new index numbers.
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17.8 Comparing sets of fixed base relatives
Sometimes it is necessary to compare two given sets of time series relatives. For example, the annual index for the number of televisions sold might be compared with the annual index for television licenses taken out, or the monthly consumer prices index compared with the monthly index for wages. In cases such as these, it is usually found that the bases on which the two sets of indices are calculated are different. For example, the consumer index might have a base of 1974, while the wage index has a base of 1983. This can make comparisons difficult because the two sets of index relatives will be of different magnitudes. As an illustration, consider the data of Table 4. Year
Number of TV sets sold (1988 = 100) Number of TV licences taken out (1970 =100)
1986 61 210
1987 88 230
1988 100 250
1989 135 300
1990 165 360
1991 192 410
1992 210 500
Comparing the indices given above is not easy. Many percentage increases will have to be calculated before any worthwhile comparisons can be made. This type of problem can be overcome by changing the base of one set of indices to match the base of the other. The following example shows the calculations necessary.
Example 3 (Time series comparison by changing the base of one of the sets)
Question Compare the figures given in Table 4 by changing the base of one of the sets and comment on the results. Answer The base of the television licence relatives will be changed to coincide with the base of the televisions sold relatives. The following table shows the new figures. Year 1986 1987 1988 1989 1990 1991 1992 Number of TV sets sold (1988 = 100) 61 88 100 135 165 192 210 Number of TV licences taken out (1970 =100) 210 230 250 300 360 410 500
Number of television licences Taken out (1988=100)
84
92
100
120
144
360 250 X 100
164
200
230 X 100 250
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162 The two sets of relatives are now much easier to compare. Before 1988 and up to 1991, sales of television sets increased at a much faster rte. However, over the last year, the number of television licenses taken out increased dramatically, showing the same percentage increase (over 1988) as the sales of television sets (possibly due to detector van publicity).
17.9 Actual and real values of a commodity
In times of significant inflation, the actual value of some commodity is not the best guide of its 'real' value 9or worth). The worth of any commodity can only be measured relative to the value of some associated commodity. In other words, some relevant 'indicator' is necessary against which to judge value. For example, suppose that the annual rent of some business premises last year was Rs. 2200. Clearly the actual cost is higher. However, if we are now given the information (as an indictor) that the cost of business premises in the region as a whole has risen by 15% over the past year, we can rightly argue that the real cost of the given premises has decreased. On the other hand, if business turnover for the premises (as an alternative indicator) has only increased by 5%, we might consider that the real cost of the premises has increased. Thus, depending on the particular indicator chosen, the real value of a commodity can change. Two standard national indicators are the rate of inflation (normally represented by the Retail Price Index) and the Index of Output of the Production Industries The following section describes a method of constructing a series of relatives to measure the real value of some commodity over time. This is known as time series deflation.
17.10 Time series deflation
Time series deflation is a technique used to obtain a set of index relatives that measure the changes in the real value of some commodity with respect to some given indicator. Month 1 2 3 4 5 6 7 8 Average daily wage(Rs.) 17.60 18.10 18.90 19.60 20.25 20.30 20.60 21.40 Retail price index 106.1 107.9 112.0 113.1 116.0 117.4 119.5 119.7 Table 5 The procedure for calculating each index relative is given below, using the data of Table 5 to demonstrate calculating the real wage index for month 7 (month 1 = 100) as an example. STEP1 Choose a base for the index of real values of the series. In this case, month 1 has been chosen. Then, for each time point of the series: STEP2 Find the ratio of the current value to the base value. 20.60 For month 7, this gives: = 1.17 17.60 This step expresses he increase in the actual value as a multiple.
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163 STEP3 Multiply by the ratio of the base indicator to the current indicator (notice that these two values are in reverse order compared with the two in the previous step). 106.1 For month 7, this gives: 1.17 X = 1.039 , 'deflating' the above 119.5 wage multiple. STEP4 Multiply by 100 For month 7, this gives: 1.039X100 = 103.9. This step changes the multiple of the previous step into an index (based on 100). The above steps can be summed up both in symbols and words as follows. Real Value Index (RVI) Given a time series (x- values) and some indicator index series (I – values) for comparison, the real value index for period n is given by: currentvalue baseindicator RVI = X X 100 basevalue currentindicator == Xn I 0 == X X 100 X 0 In
The following example duplicates the data of Table 5 and shows the real wage index relatives, the calculations (using the above steps) being demonstrated fro selected values.
Exercises
1. The average price of a product this year was Rs. 33.3, which represented a decease of 10% over last year's average price. The number bought (at these prices) last year was 2500, but increased by 750 this year. Calculate price, quantity and expenditure relatives for these cassettes for this year (based on last year).
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165
17.11 Let us Sum Up
This Lesson d i scussed about indices and its common applications. An index number measures the percentage change in the value of some economic commodity over a period of time. It is always expressed in terms of a base of 100. An index relative is the name given to an index number which measures the change in a single distinct commodity. A price relative can be calculated as the ration of the current price to the base price multiplied by one hundred. Quantity, expenditure and productivity relatives are calculated in a similar manner. Fixed base relatives are found by calculating relatives for each value of a time series based on the same fixed time point. Chain base relatives are found by calculating relatives for each value of time series based on the immediately preceding time point. In order to compare two time series of relatives, each series should have the same base time point. The real value of some commodity can only be measured in terms of some 'indicator'. Standard indicators are the Retail Price Index or the Index of Output of the Production Industries. Time series deflation is also discussed which is a technique used to obtain a set of index relatives that measure the changes in the real value of some commodity with respect to some given indicator.
17.12. Lesson – End Activities
1. Define Index Number.
17.13. References
1. Gupta S.P. – Statistical Methods
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166
Lesson 18 - Special Published Indices Contents
18.1 Aims and Objectives 18.2 The Retail Prices Index 18.3 Main RPI groups and their weights 18.4 The family Expenditure Survey 18.5 Price collection and calculation of the RPI 18.6 The Purchasing Power (index) 18.7 The Tax and Price index 18.8 Index numbers of producer Prices 18.9 Indices of average earnings 18.10 Index of output of the production industries 18.11 Other index numbers 18.12 Let us Sum Up 18.13 Lesson – End Activities 18.14 References
18.1 Aims and Objectives
This Lesson describes some of the most important official index numbers. The price indices described are the Retail Price Index (which includes the important Family Expenditure Survey), Purchasing Power, the Tax and Price Index and Index numbers of Producer Prices. Indices of Average Earnings are also covered. Volume (or quantity) indices described are the Index of Output of the Production Industries and the Index of Retail Sales. Some indices described cover more than one section.
18.2 The Retail Prices Index
The Retail Prices Index (or RPI), is probably the best known of all the published indices. a) It is published monthly by the Department of Employment and Displayed (to different levels of complexity) in the following publications: Monthly Digest of Statistics, the Annual abstract of Statistics, the Department of Employment Gazette and Economic Trends. b) The RPI measures the percentage changes, month by month, in the average level of prices of the commodities and services purchased by the great majority of households in the Country. It takes account of practically all wage earners and most small and medium salary earners. c) The items covered by the RPI are classified into several groups. For example, Food, Housing, Transport and Vehicles etc). Each group is sub-divided into sections. For
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167 example, Transport is sub-divided into Motoring/cycling and Fares). These sections may be further split up into separate items. For example, Fares are split up into Rail and Road. d) Each month, an overall index is published, together with separate indices for each group, section and individual item (of which there are approximately 350). e) Each group (and further sections and specific items) is weighted according to expenditure by a 'typical family' and the weights are updated annually. f) The weights are obtained from a continuous investigation known as the Family Expenditure Survey.
Notes on Table 1: a) Weights are always calculated to add to 1000. b) 'Meals bought out' was not included in the 1962 weightings c) Certain items of expenditure are not included in the RPI. These include: i. Income tax and National Insurance payments; ii. Insurance and pension payments; iii. Mortgage payments for house purchase (except for interest payments which are included); iv. Gambling, gifts, charity, etc.
Example 1 (Comments on the data in Table 1)
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168 a) The Retail Prices Index for January 1986 (1974 = 100) was 379.7. This represents an overall increase in prices of approximately 280% since 1974. b) Food has been subject to below average price increase (341.1 index = 241% increase) and expenditure has continued to decrease significantly. Since food is a basic necessity of life, this is a good indication of our increasing affluence. c) Tobacco has seen the highest increase in price (index = 545.7) with a definite downward trend in expenditure. The latter trend is obviously due to both high price and health warnings. d) Clothing and Footwear has had the lowest increase in price (index = 225.2), representing only a doubling in price over the previous 10 years, but this group has still seen a downward trend in expenditure. Since there is no reason to suppose that we now buy fewer clothes, it probably means that clothes are much cheaper in real terms. e) Housing and Transport and Vehicles both show a similar upward trend in expenditure. However, where Transport is only showing an average price increase, Housing shows the third highest (index = 463.7). Upward expenditure on transport clearly signifies our increasing mobility (in both work and recreation). Extra expenditure on housing probably reflects social and ecological factors as much as increase in price.
18.4 The family Expenditure Survey
The Family Expenditure Survey (FES) is a continuous major investigation which, among other things, measures average consumption levels. These are used to obtain the (annually revised) weights for items included in the RPI. The FES involves a stratified random sample, spread over the course of a year, of about 10000 households. Each household is visited by an interviewer. Each member of the household over the age of sixteen years is required to keep a detailed diary of all expenditure for a continuous 14day period, which is checked and retained by the interviewer. The interviewer also completes a Household Schedule, which contains information on longer term spending such as rent, rates, carpets, cars, and so on. (An Income Schedule is also filled out for the members of the household.). The published weights are calculated, not from a single year's FES data, but as an average of the previous three year's data. This ensures that large items of expenditure do not unfairly influence average patterns of spending. The pattern of FES varies from country to country.
18.5 Price collection and calculation of the RPI
Prices are collected by Department of Employment staff. Different types of retail outlets, from village shops to large supermarkets, are visited. To ensure uniformity, the same ones are used each month and these will be the type of retail outlet used by households examined by the FES. Price relatives are calculated (for each item covered by the RPI) for each retail outlet and averaged for a local area. Average relatives for all local areas are in turn averaged to obtain a national average of relatives (for each of the 350 items covered by the index). Weights are then used to calculate composite indices using the average of relatives method for items within sections, sections within groups and, finally, groups. Thus the RPI is a weighted average of relatives of each group.
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169
18.6 The Purchasing Power (index)
The Purchasing Power is an index which has been based solely on the annual average of the RPI. The philosophy behind the index is: when prices go up, the amount which can be purchased with a given sum of money goes down. The index is described in terms of two particular years. If the purchasing power of the Rupee is taken to be 100 in the first year, the comparable purchasing power in a later year is calculated as: For example, the PP index for 1984 (1980 = 100) is given as 75. This can be interpreted as: Average Pr iceIndexforFirstYear 100 X Average Pr iceIndexforLaterYear i. the Rupee (in 1984) is worth only 75% of its 1980 value, of ii. 100 rupees buys (in 1984) what would only have cost 75 rupees in 1980.
18.7 The Tax and Price index
The Tax and Price Index (TPI), published monthly, is another index which is linked to the Retail Prices Index. The TPI measures the increase in gross taxable income needed to compensate taxpayers for any increase in retail prices (as measured by the RPI). It is considered as a more comprehensive index than the RPI since, while the RPI measures changes in retail prices, the TPI additionally takes account of the changes in liability to direct taxes (including employees' national insurance contributions) facing a representative cross-section of taxpayers. Some people would argue that the TPI is a better measure of the cost of living than the RPI since it takes direct taxes into account. However, whether or not this is acceptable depends on the meaning of the phrase 'cost of living' – it has different meanings to different people and circumstances. Another complicating factor is that the TPI (a relatively new index) is regarded suspiciously by some political opponents of the Government in office at the time of its introduction. Example 2 (Comparison of the TPI and RPI) The TPI for June 1985 (January 1978 = 100) was 191.7 [INDEX 1] The RPI for June 1985 (January 1974 =100) was 376.4 [INDEX 2] The RPI for 1978 (1974 = 100) was 197.1 [INDEX 3] Note that it is difficult to compare the first two indices, since their base dates are ifferent. However, the information contained in INDEX 3 allows the RPI (INDEX 2) to be basechanged to coincide with the base of the TPI (INDEX 1), for a direct comparison. Thus: RPI85/78 = RPI 85/74 X 100 RPI 78/74 376.4 = X 100 197.1 = 191.0 (ID) Therefore the TPI for June 1985 shows a slightly higher increase (91.7%) than the RPI for June 1985 (91.0%)
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170 Note, however, that INDEX 3 is based on annual averages whereas the other two indices are based on actual months of the year. Hence the above base change will cause the resultant figure to be slightly in error.
18.8 Index numbers of producer Prices
The Producer Price Indices (PPI) measure manufacturers' prices and were formerly known as the wholesale Price Indices. The data for the indices are collected by the Business Statistics Office. Indices are produced for a wide range of prices including output (home sales), materials and fuel purchased, commodities produced and imported. They are quoted for main industrial groupings, such as Motor Vehicles and Parts, Food Manufacturing industries, Textile industry, and son on. In some publications, the groupings are sub-divided into items. The various index numbers produced are calculated from the price movements of about 10,000 closely defined materials and products representative of goods purchased and manufactured. All the indices express the current prices as a percentage of their annual average price in 1980, the base year.
18.9 Indices of average earnings
The Indices of Average Earnings measure the changes in average gross income. They are published for manual workers and all workers and given for industry groups. Actual and seasonally adjusted indices are given for certain tables. The series as at June 1986 are all based on 1980 = 100.
18.10 Index of output of the production industries
The Index of Output of the Production Industries was formerly known as the Index of Industrial Production. It provides a general measure of monthly changes in the volume of output of the production industries. Energy, water supply and manufacturing are included in the index. However, agriculture, construction, distribution, transport, communications, fiancé and all other public and private services are excluded. The index covers the production of intermediate, investment and consumer goods, both for home and export. Many of the series presented are seasonally adjusted. This excludes any changes in production resulting from public and other holidays and from other seasonal factors. The adjustments are designed to eliminate normal month to month fluctuations and thus to show the trend more clearly.
Exercises
1. What is the Retail Prices Index (RPI)? 2. Name at least five of the eleven main groups into which the RPI is divided. 3. Name some of the items of expenditure that are not included in the calculation of the RPI. 4. How are prices collected for the RPI? 5. Explain what the 'Purchasing Power' and how it is calculated. 6. What does the Tax and Price Index (TPI) measure? 7. Compare the RPI and TPI. 8. Describe some aspects of the Index Numbers of Producer Prices. 9. What is the Index of Retail Sales and how are the data in its construction collected?
18.12 Let us Sum Up
This Lesson discussed special published indices which finds applications in economics and in financial management. The Retail Prices Index (RPI) is published monthly and measures the percentage changes in the average level of prices of the commodities and services purchased by most households. Purchasing Power (index) gives the percentage worth of a current pound compared with a pound in a previous period. The Tax and Price Index measures the increase in gross taxable income needed to compensate taxpayers for any increases in retail prices (as measured by the RPI). It takes account of direct taxation. The Indices of Average Earning measure the changes in average gross income for manual and other workers. The Index of Output of the Production Industries provides a general measure of monthly changes in the volume of output of the production industries. Index numbers of Retail Sales give both volume and value indices and are compiled on the type of business rather than on a commodity basis.
18.13 Lesson – End Activities
Give the importance of the Retail Price Index. | 677.169 | 1 |
Transformations
Polygons and Angles in Parallel Lines
3D Shapes
2D Shapes
Probability
Experimental and Theoretical Probability
Tree Diagrams and Frequency Trees
Venn Diagrams
Combinations and Systematic Listing
Useful Websites- 1) Very useful for step by step lessons and interactive quizzes. 2) Videos that talk you through key areas on a wide range of the GCSE Specification. Videos from this website are embedded throughout the mindmap. 3) Differentiated questions (Bronze/Silver/Gold) that allows you to practise fluency of key skills. 4) 1-9 levels broken down into topics related to each level. Comes with revision videos and practise questions (mostly past paper questions) with solutions. 5) Website that has hundreds of 5-a-days to practise a range of skills each day that target different levels- answers are with the '5 a days A*-G' only. Very useful section of topics broken down alphabetically with a huge number of fluency practise of skills. 6) A great website that breaks down topics and allows users to revise, practise and understand topics. It has the GCSE takeaway section which is past paper questions and multiple choice quizzes for each topic. 7) This website and this specific section generates 'higher' questions with answers instantaneously, similar to the 5-a-days but more working required to get an answer. The websites aren't in any order, they are free and checked thoroughly for usefulness- allowing students to enhance their skills, knowledge and understanding of the GCSE Specification. If you are not sure on how to use any of the websites then please ask Mr Barker.
GCSE Revision Section (Under Construction)
Main Revision Booklet
Key Formulas you MUST learn
Higher Topic 9-1 Checklist
Aiming for Grade Booklets
'Grade 9' Papers- note: these do not claim to be a grade 9 paper, it contains questions which need a variety of skills to solve. | 677.169 | 1 |
College Algebra Questions & Answers
College Algebra Flashcards
College Algebra Advice
College Algebra Advice
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it is absolutely fabulous. i loved to crunch the numbers and make myself think.
Course highlights:
i learned how to solve trinomials, quadratics, SIN, COS, TAN, and much much more.
Hours per week:
3-5 hours
Advice for students:
The advice i would give is to not procrastinate. If you do, it will be a living hell to catch up.
Course Term:Fall 2016
Professor:Lyndsey Reed
Course Required?Yes
Course Tags:Math-heavyMany Small AssignmentsA Few Big Assignments
Oct 31, 2016
| Would highly recommend.
This class was tough.
Course Overview:
It was great! Opie was a great teacher and really knew how to explain and teach all of the information that we needed to know. I remember enjoying the class very much.
Course highlights:
It was great because I learned the deeper formulas of algebra and was able to do so at an earlier age in high school! I'm very thankful that these types of classes are provided for high school students to take.
Hours per week:
3-5 hours
Advice for students:
You need to have a great work ethic and a desire to learn. If you are willing to do what is asked of you and put your mind to the test, you will pass with flying colors. Be diligent and motivated, and when you don't understand something don't be afraid to ask questions. | 677.169 | 1 |
Parent Functions Algebra Greeting Card - Mathematically Yours
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While the concept of "parent function" is important in Algebra, I find it's a vocabulary term that often escapes my students. I needed to grab their attention, so I created this "greeting card" to help them see and understand the concept.
May your students never have brain freeze when asked:
What is the parent function for y = 2x + 4?
The closing, "Mathematically Yours," is left for your signature before photocopying the card for your students. Simply sign, copy two-sided, and fold in half... the hamburger way. | 677.169 | 1 |
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This is an Algebra 1 Common Core Lesson on Factoring by Greatest Common Factor. Students will learn how to find the greatest common factor of various expressions. After a few teacher led examples and detailed notes, students will practice on their own or in groups. | 677.169 | 1 |
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Pre-Algebra (9th Grade), Room 432 - 09:00 AM
Jun 26 - Jul 21
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Basic arithmetic operations are reviewed to help the student acquire the necessary foundation for algebra. The fundamental principles of whole numbers, fractions, decimals, and percentages will lead to the exploration of basic algebra. This course will also include a brief introduction of integers, exponential notation, scientific notation and word problems. A minimum of 7 students is required for all classes. Enroll early to prevent cancellation of of classes. Class closures will be determined by May 1, | 677.169 | 1 |
Pre-Algebra, Grades 6 - 8 (The 100+ Series?)
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An essential supplement to classroom instruction, 100+ practice activities will keep students' skills sharp while preparing them for the next level of math study. This book provides all the instruction, practice, and review students need to understand pre-algebra from the beginning of the year to the end. Step-by-step explanations, complete practice problems, and fun extension activities help students in grades 6 and up master even the most challenging topics.. Paperback. Book Condition: New. Paperback. 128 pages. Dimensions: 11.2in. x 8.4in. x 0.4in.The 100 68 are studying more accelerated math at younger ages. The 100 Series provides the solution with titles that include over 100 targeted practice activities for learning algebra, geometry, and other advanced math topics. It also features over 100 reproducible, subject specific, practice pages to support standards-based instruction. This item ships from multiple locations. Your book may arrive from Roseburg,OR, La Vergne,TN. Paperback. Bookseller Inventory # 9781483800769
Book Description Paperback. Book Condition: New. 211mm x 13mm x 269mm. Paperback. Shipping may be from multiple locations in the US or from the UK, depending on stock availability. 128 pages. 0.295. Bookseller Inventory # 9781483800769 | 677.169 | 1 |
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O. Schreier
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Algebra equations fractions, 6th grade math adding and subtracting negative integers, algebraic simplifier, sample of mixture problems in math, ALGEBRAIC EQUATION PERCENTAGE. It is often best to keep the registration point aligned with the movie clip's origin so you have a direct relation with the values you see in the Flash IDE and those accessible through ActionScript. OpenSG, VRJuggler, and other projects have all switched to using this instead of their own hand-rolled vertor/matrix math.
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To be more precise, a matrix (plural matrices) is a rectangular array of numbers. Dimitrios Chatzakos (supervisor Yiannis Petridis): Lattice point problems in the hyperbolic plane. 3. This memo focuses on math course enrollment patterns throughout high school by following the 2013-14 twelfth grade cohortHere is a magic square with the numbers one through nine where each row and each column adds up to 15: Topics include graph visualization, labelling, and embeddings, random graphs and randomized algorithms. A plot of its energy distribution is Figure 60. Now that you have a set of matrix functions, you might want to see exactly how you would use those functions in a program to translate, scale, and rotate a shape. In 1972, the number theorist Hugh Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely related to the distribution of prime numbers.
It's time of an endless-time classic to reach the Steam Platform! Well, again, it wound up in a letter written by Pete about it, this time sent to the Las Vegas PD, that I�ll let speak for itself. Topics from partially ordered sets, Mobius functions, simplicial complexes and shell ability. Finally, m2 gets copied into the transformation matrix. ( It may be possible to find many of these video clips by searching the Internet. Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element.
Use this value to back-substitute into Row 2 and get y = 0. Matrix multiplication is a little more complicated. This fine balance between chaos and order, which is defined by a precise formula, also appears in a purely mathematical setting: It defines the spacing between the eigenvalues, or solutions, of a vast matrix filled with random numbers. "Why so many physical systems behave like random matrices is still a mystery," said Horng-Tzer Yau, a mathematician at Harvard University. "But in the past three years, we have made a very important step in our understanding." By investigating the "universality" phenomenon in random matrices, researchers have developed a better sense of why it arises elsewhere — and how it can be used.
She balances her life between traveling, teaching, and writing. The Mad Hatter (Martin Short) is sitting at the head of the table and asks "Why is a raven like a writing desk?". Thompson, A MATLAB implementation of upwind finite differences and adaptive grids in the method of lines, Journal of Computational and Applied Mathematics, v.183 n.2, p.245-258, 15 November 2005 A. Matrix R has the following important properties: The absolute value of a diagonal element of R is the largest value in this row, i.e., abs(R[i,i]) ≥ abs(R[i,j]).
Kapoo Stem
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Jack Benny and Eddie "Rochester" Anderson -- 28 years. Every finite group is isomorphic to a matrix group, as one can see by considering the regular representation of the symmetric group. [66] General groups can be studied using matrix groups, which are comparatively well-understood, by means of representation theory. [67] It is also possible to consider matrices with infinitely many rows and/or columns [68] even if, being infinite objects, one cannot write down such matrices explicitly.
Dmitri Burago
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Thue, Axel, "�ber unendliche Zeichenreihen," Norske vid. Applications of matrices are found in most scientific fields. "My children have been using Excel Math Standard Edition at home for the last two years to supplement the math curriculum they have at school (which isn't very effective). Chuck is a great guest, he's passionate about physics and math as well as fantasy and science fiction. Dauben, Joseph Warren, Georg Cantor: His Mathematics and Philosophy of the Infinite (Cambridge, Massachusetts: Harvard University Press, 1979). | 677.169 | 1 |
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MATH 113, FALL 2012
HOMEWORK 1
1. Let : I R3 (where I R is an interval) be a parametrized curve and let v R3 be a xed
vector. Assume that (t) is orthogonal to v for all t I and that (0) is also orthogonal to v .
Prove that (t) is orthogonal to v for all t
MATH 113, FALL 2012
HOMEWORK 2, DUE SEPTEMBER 13
1. Let f : R2 R be a C 1 function (i.e., f has continuous partial derivatives) such that the
gradient f (p) of f is nonzero for every p R2 .
(a) Show that every level curve of f is orthogonal to the gradien
MATH 113, FALL 2012
HOMEWORK 6, DUE NOVEMBER 20
1. Exercise 8.1.9 from Pressley
2. Exercise 8.2.2 from Pressley
3. Show that at the origin of the hyperboloid z = axy we have K = a2 and H = 0.
4. Show that at a point p of a surface S the sum of normal curv
MATH 113, FALL 2012
HOMEWORK 5, DUE NOVEMBER 1
1. Exercise 6.1.1 from Pressley
2. Exercise 6.1.3 from Pressley
3. Exercise 6.1.4 from Pressley
4. Let S be a surface of revolution with axis of revolution . Show that rotations about
isometries of S.
are
5.
INTRODUCTION
FIELD colors TYPIEALLY focus on ora and fauna of the natural
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how and where to nd them, and elaborating on what exactly to look
for once they are located. As helpful handboo | 677.169 | 1 |
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