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x 2 + y 2 = 49 x 2 + y 2 = 49 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
4 x 2 + 25 y 2 = 100 4 x 2 + 25 y 2 = 100 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
25 y 2 â 4 x 2 = 100 25 y 2 â 4 x 2 = 100 | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-concepts |
circle : A circle is all points in a plane that are a fixed distance from a fixed point in the plane. | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-terms |
ellipse : An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-terms |
hyperbola : A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant. | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-terms |
parabola : A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-terms |
system of nonlinear equations : A system of nonlinear equations is a system where at least one of the equations is not linear. | https://openstax.org/books/intermediate-algebra-2e/pages/11-key-terms |
Factorial NotationIfnis a positive integer, thenn!n!isn!=n(nâ1)(nâ2)â¦(3)(2)(1).n!=n(nâ1)(nâ2)â¦(3)(2)(1).We define0!0!as 1, so0!=10!=1 | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Summation NotationThe sum of the firstnterms of a sequence whosenth termananis written in summation notation as:âi=1nai=a1+a2+a3+a4+a5+â¦+anâi=1nai=a1+a2+a3+a4+a5+â¦+anTheiis the index of summation and the 1 tells us where to start and thentells us where to end. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
General Term (nth term) of an Arithmetic SequenceThe general term of an arithmetic sequence with first terma1a1and the common differencedisan=a1+(nâ1)dan=a1+(nâ1)d | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Sum of the FirstnTerms of an Arithmetic SequenceThe sum,Sn,Sn,of the firstnterms of an arithmetic sequence, wherea1a1is the first term andananis thenth term isSn=n2(a1+an)Sn=n2(a1+an) | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
General Term (nth term) of a Geometric Sequence:The general term of a geometric sequence with first terma1a1and the common ratiorisan=a1rnâ1an=a1rnâ1 | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Sum of the FirstnTerms of a Geometric Series:The sum,Sn,Sn,of thenterms of a geometric sequence isSn=a1(1ârn)1ârSn=a1(1ârn)1ârwherea1a1is the first term andris the common ratio. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Infinite Geometric Series:An infinite geometric series is an infinite sum whose first term isa1a1and common ratio isrand is writtena1+a1r+a1r2+â¦+a1rnâ1+â¦a1+a1r+a1r2+â¦+a1rnâ1+⦠| https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Sum of an Infinite Geometric Series:For an infinite geometric series whose first term isa1a1and common ratior,If|r|<1,the sum isS=a11ârWe say the series converges.If|r|â¥1,the infinite geometric series does not have a sum. We say the series diverges.If|r|<1,the sum isS=a11ârWe say the series converges.If|r|â¥1,th... | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Value of an Annuity with Interest CompoundednnTimes a Year:For a principal,P, invested at the end of a compounding period, with an interest rate,r, which is compoundedntimes a year, the new balance,A, aftertyears, isAt=P((1+rn)ntâ1)rnAt=P((1+rn)ntâ1)rn | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Patterns in the expansion of(a+b)n(a+b)nThe number of terms isn+1.n+1.The first term isananand the last term isbn.bn.The exponents onadecrease by one on each term going left to right.The exponents onbincrease by one on each term going left to right.The sum of the exponents on any term isn. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
The number of terms isn+1.n+1. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
The first term isananand the last term isbn.bn. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
The exponents onadecrease by one on each term going left to right. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
The exponents onbincrease by one on each term going left to right. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
The sum of the exponents on any term isn. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Pascalâs Triangle | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Binomial Coefficient(nr)(nr):A binomial coefficient(nr),(nr),whererandnare integers with0â¤râ¤n,0â¤râ¤n,is defined as(nr)=n!r!(nâr)!(nr)=n!r!(nâr)!We read(nr)(nr)as ânchooserâ or ântakenrat a timeâ. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Properties of Binomial Coefficients(n1)=n(nn)=1(n0)=1(n1)=n(nn)=1(n0)=1 | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
Binomial Theorem:For any real numbersa,b, and positive integern,(a+b)n=(n0)an+(n1)anâ1b1+(n2)anâ2b2+...+(nr)anârbr+...+(nn)bn(a+b)n=(n0)an+(n1)anâ1b1+(n2)anâ2b2+...+(nr)anârbr+...+(nn)bn | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-concepts |
annuity : An annuity is an investment that is a sequence of equal periodic deposits. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
arithmetic sequence : An arithmetic sequence is a sequence where the difference between consecutive terms is constant. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
common difference : The difference between consecutive terms in an arithmetic sequence,anâanâ1,anâanâ1,isd, the common difference, forngreater than or equal to two. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
common ratio : The ratio between consecutive terms in a geometric sequence,ananâ1,ananâ1,isr, the common ratio, wherenis greater than or equal to two. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
finite sequence : A sequence with a domain that is limited to a finite number of counting numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
general term of a sequence : The general term of the sequence is the formula for writing thenth term of the sequence. Thenth term of the sequence,an,an,is the term in thenth position wherenis a value in the domain. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
geometric sequence : A geometric sequence is a sequence where the ratio between consecutive terms is always the same | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
infinite geometric series : An infinite geometric series is an infinite sum infinite geometric sequence. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
infinite sequence : A sequence whose domain is all counting numbers and there is an infinite number of counting numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
partial sum : When we add a finite number of terms of a sequence, we call the sum a partial sum. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
sequence : A sequence is a function whose domain is the counting numbers. | https://openstax.org/books/intermediate-algebra-2e/pages/12-key-terms |
The mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Data are individual items of information that come from a population or sample. Data may be classified as qualitative (categorical), quantitative continuous, or quantitative discrete. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Because it is not practical to measure the entire population in a study, researchers use samples to represent the population. A random sample is a representative group from the population chosen by using a method that gives each individual in the population an equal chance of being included in the sample. Random sampli... | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Samples that contain different individuals result in different data. This is true even when the samples are well-chosen and representative of the population. When properly selected, larger samples model the population more closely than smaller samples. There are many different potential problems that can affect the rel... | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Some calculations generate numbers that are artificially precise. It is not necessary to report a value to eight decimal places when the measures that generated that value were only accurate to the nearest tenth. Round off your final answer to one more decimal place than was present in the original data. This means tha... | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
In addition to rounding your answers, you can measure your data using the following four levels of measurement. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Nominal scale level:data that cannot be ordered nor can it be used in calculations | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Ordinal scale level:data that can be ordered; the differences cannot be measured | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Interval scale level:data with a definite ordering but no starting point; the differences can be measured, but there is no such thing as a ratio. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Ratio scale level:data with a starting point that can be ordered; the differences have meaning and ratios can be calculated. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
When organizing data, it is important to know how many times a value appears. How many statistics students study five hours or more for an exam? What percent of families on our block own two pets? Frequency, relative frequency, and cumulative relative frequency are measures that answer questions like these. | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
A poorly designed study will not produce reliable data. There are certain key components that must be
included in every experiment. To eliminate lurking variables, subjects must be assigned randomly to different treatment groups. One of the groups must act as a control group, demonstrating what happens when the active ... | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
âAn ethics problem arises when you are considering an action that benefits you or some cause you support, hurts or reduces benefits to others, and violates some rule.â (Andrew Gelman, âOpen Data and Open Methods,â Ethics and Statistics, http://www.stat.columbia.edu/~gelman/research/published/ChanceEthics1.pdf (... | https://openstax.org/books/introductory-statistics-2e/pages/1-chapter-review |
Average : also called mean; a number that describes the central tendency of the data | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Blinding : not telling participants which treatment a subject is receiving | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Categorical Variable : variables that take on values that are names or labels | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Cluster Sampling : a method for selecting a random sample and dividing the population into groups (clusters); use simple random sampling to select a set of clusters. Every individual in the chosen clusters is included in the sample. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Continuous Random Variable : a random variable (RV) whose outcomes are measured; the height of trees in the forest is a continuous RV. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Control Group : a group in a randomized experiment that receives an inactive treatment but is otherwise managed exactly as the other groups | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Convenience Sampling : a nonrandom method of selecting a sample; this method selects individuals that are easily accessible and may result in biased data. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Cumulative Relative Frequency : The term applies to an ordered set of observations from smallest to largest. The cumulative relative frequency is the sum of the relative frequencies for all values that are less than or equal to the given value. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Data : a set of observations (a set of possible outcomes); most data can be put into two groups:qualitative(an attribute whose value is indicated by a label) orquantitative(an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups:discreteandcontinuous. Data is discrete i... | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Double-blind experiment : an experiment in which both the subjects of an experiment and the researchers who work with the subjects are blinded | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Experimental Unit : any individual or object to be measured | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Explanatory Variable : the independent variable in an experiment; the value controlled by researchers | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Frequency : the number of times a value of the data occurs | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Informed Consent : Any human subject in a research study must be cognizant of any risks or costs associated with the study. The subject has the right to know the nature of the treatments included in the study, their potential risks, and their potential benefits. Consent must be given freely by an informed, fit particip... | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Institutional Review Board : a committee tasked with oversight of research programs that involve human subjects | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Lurking Variable : a variable that has an effect on a study even though it is neither an explanatory variable nor a response variable | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Nonsampling Error : an issue that affects the reliability of sampling data other than natural variation; it includes a variety of human errors including poor study design, biased sampling methods, inaccurate information provided by study participants, data entry errors, and poor analysis. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Numerical Variable : variables that take on values that are indicated by numbers | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Parameter : a number that is used to represent a population characteristic and that generally cannot be determined easily | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Placebo : an inactive treatment that cannot directly affect the response variable, used to counter the power of suggestion | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Population : all individuals, objects, or measurements whose properties are being studied | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Probability : a number between zero and one, inclusive, that gives the likelihood that a specific event will occur | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Proportion : the number of successes divided by the total number in the sample | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Qualitative Data : a set of observations (a set of possible outcomes);qualitativedata has an attribute whose value is indicated by a label | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Quantitative Data : a set of observations (a set of possible outcomes);quantitative(an attribute whose value is indicated by a number) data can be separated into two subgroups:discreteandcontinuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or th... | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Random Assignment : the act of organizing experimental units into treatment groups using random methods | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Random Sampling : a method of selecting a sample that gives every member of the population an equal chance of being selected. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Relative Frequency : the ratio of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Representative Sample : a subset of the population that has the same characteristics as the population | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Response Variable : the dependent variable in an experiment; the value that is measured for change at the end of an experiment | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Sample : a subset of the population studied | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Sampling Bias : not all members of the population are equally likely to be selected | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Sampling Error : the natural variation that results from selecting a sample to represent a larger population; this variation decreases as the sample size increases, so selecting larger samples reduces sampling error. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Sampling with Replacement : Once a member of the population is selected for inclusion in a sample, that member is returned to the population for the selection of the next individual. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Sampling without Replacement : A member of the population may be chosen for inclusion in a sample only once. If chosen, the member is not returned to the population before the next selection. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Simple Random Sampling : a straightforward method for selecting a random sample; give each member of the population a number. Use a random number generator to select a set of labels. These randomly selected labels identify the members of your sample. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Statistic : a numerical characteristic of the sample; a statistic estimates the corresponding population parameter. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Stratified Sampling : a method for selecting a random sample used to ensure that subgroups of the population are represented adequately; divide the population into groups (strata). Use simple random sampling to identify a proportionate number of individuals from each stratum. | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Systematic Sampling : a method for selecting a random sample; list the members of the population. Use simple random sampling to select a starting point in the population. Let k = (number of individuals in the population)/(number of individuals needed in the sample). Choose every kth individual in the list starting with... | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Treatments : different values or components of the explanatory variable applied in an experiment | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Variable : a characteristic of interest for each person or object in a population | https://openstax.org/books/introductory-statistics-2e/pages/1-key-terms |
Astem-and-leaf plotis a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. Aline graphis often used to represent a set of data val... | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
Ahistogramis a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are ty... | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50thpercentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quart... | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
Q3+IQR(1.5) | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
Q1âIQR(1.5) | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
Box plots are a type of graph that can help visually organize data. To graph a box plot the following data points must be calculated: the minimum value, the first quartile, the median, the third quartile, and the maximum value. Once the box plot is graphed, you can display and compare distributions of data. | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
The mean and the median can be calculated to help you find the "center" of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum (or data) in your d... | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
Looking at the distribution of data can reveal a lot about the relationship between the mean, the median, and the mode. There arethree types of distributions. Aleft (or negative) skeweddistribution has a shape likeFigure 2.17. Aright (or positive) skeweddistribution has a shape likeFigure 2.18. Asymmetricaldistribution... | https://openstax.org/books/introductory-statistics-2e/pages/2-chapter-review |
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