Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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Glossary

A

Addition Rule
If C and D are mutually exclusive events, then P(C or D) = P(C) + P(D). Unit 19
Adequacy of a Linear Model
A line is adequate to describe the pattern in a set of data points provided the data have linear form. A residual plot is a good way of checking adequacy. Unit 11
Alternative Hypothesis or Ha
The claim in a significance test that we are trying to gather evidence for - the researcher's point of view. The alternative hypothesis is contradictory to H0 and is judged the more plausible claim when H0 is rejected. Unit 25
ANOVA
Analysis of variance (ANOVA) is a technique used to analyze variation in data in order to test whether three or more population means are equal. Unit 31
Assumptions of the Linear Regression Model
  • The observed response y for any value of x varies according to a normal distribution. Repeated responses, y-values, are independent of each other.
  • The mean response, μy, has a straight-line relationship with x: μy = α + βx.
  • The standard deviation of y, σ, is the same for all values of x.
Unit 30

B

Bar Chart
Graph of a frequency distribution for categorical data. Each category is represented by a bar whose area is proportional to the frequency, relative frequency, or percent of that category. If the categorical variable is ordinal, the logical order of the categories should be preserved in the bar chart. Unit 2
Between-Groups Variation
A measure of the spread of the group means about the grand mean, the mean of all the observations. It is measured by the mean square for groups, MSG. Unit 31
Biased Sample
A sample in which some individuals or groups from the population are less likely to be selected than others due to some attribute. Unit 17
Binomial Distribution
In a binomial setting with n trials and probability of success p, the distribution of x = the number of successes. Shorthand notation for this distribution is b(n, p). The probabilities p(x) for the binomial distribution with parameters n and p can be calculated using the following formula:

\begin{align} p(x) = & {n \choose x} p^x (1-p)^{n-x} \text{ for $x$ = 0, 1, ... $n$,} \\ \text{where } & {n \choose x} = \frac{n!}{x!(n-x)!} \end{align}

Unit 21
Binomial Random Variable
The number of successes, x, in a binomial setting with n trials with probability of success p. The mean and standard deviation of a binomial random variable x can be calculated as follows:

\begin{align} \mu & = np \\ \sigma & = \sqrt{np(1-p)} \end{align}

Unit 21
Binomial Setting
A setting in which there are a fixed number of n independent trials. Each trial can result in only one of two outcomes, success or failure, and the probability of success, p, is the same for each trial. Unit 21
Bivariate Data
Measurements or observations are recorded on two attributes for each individual or subject under study. Unit 10
Boxplot (or Box-and-Whisker Plot)
BoxplotGraphical representation of the five-number summary. The basic boxplot consists of a box that extends from the first quartile to the third quartile with whiskers that extend from each box end to the minimum and maximum data values. The basic boxplot can be modified to include identification of mild and extreme outliers. Unit 5

C

Categorical Variable
Variable whose values are classifications or categories. Gender, occupation, and eye color are examples of categorical variables. Unit 13
Census
An attempt to gather information about every individual in a population. Unit 16
Center Line
The center line on a control chart is generally the target value or mean of the quality characteristic being sampled. Unit 23
Central Limit Theorem
If the sample size n is large (say n > 30), then the sampling distribution of the sample mean of n independent observations from the same population has an approximate normal distribution. If the population mean and variance are μ and σ, respectively, then has an approximate normal distribution with mean μ and standard deviation σ/√n. Unit 22
Chi-Square Test Statistic for Independence
The chi-square test for independence is used for categorical variables. For testing the null hypothesis H0: no association between the variables or H0: variables are independent, the chi-square-test statistic is computed as follows:

\[ \chi^2 = \frac{(\text{observed} - \text{expected})^2}{\text{expected}} \]

If the null hypothesis is true, \( \chi^2 \) will have a chi-square distribution with degrees of freedom (r - 1)(c - 1), where r and c are the number of rows and columns in the two-way table, respectively. Unit 29
Common Cause Variation
Variation due to day-to-day factors that influence the process. Unit 23
Complement of an Event A
An event that consists of all the outcomes in the sample space that are not in A. If B is the complement of A, then B = not A. Unit 19
Complement Rule
For any event C, P(not C) = 1 – P(C). Unit 19
Complementary Events
Two events are complementary if they are mutually exclusive and combining their outcomes into a single set gives the entire sample space. Unit 19
Conditional Distribution
There are two sets of conditional distributions for a two-way table:
  • distributions of the row variable for each fixed level of the column variable
  • distributions of the column variable for each fixed level of the row variable
Conditional distributions provide one way to explore the relationship between the row and column variables. Unit 13

Conditional distribution
Computing the conditional distribution of physical beauty (the column variable) for unhappy people (a fixed level of the row variable).
Confidence Interval
An interval estimate computed from sample data that gives a range of plausible values for a population parameter. The interval is constructed so that the value of the parameter will be captured between the endpoints of the interval with a chosen level of confidence. Unit 24
Confidence Interval for μ (t-interval)
When σ is unknown, the sample size n is small, and the population distribution is approximately normal, a t-confidence interval for μ is given by the following formula:

\[\bar{x} \pm t^* \biggl( \frac{s}{\sqrt{n}} \biggr) \]

where t* is a t-critical value associated with the confidence level and determined from a t-distribution with df = n - 1 degrees of freedom. Unit 26
Confidence Interval for μ (z-interval)
When σ is known and either the sample size n is large or the population distribution is normal, a confidence interval for μ is given by the following formula:

\[\bar{x} \pm z^* \biggl( \frac{\sigma}{\sqrt{n}} \biggr) \]

where z* is a z-critical value (from a standard normal distribution) associated with the confidence level. Unit 24
Confidence Interval for p
In situations where the sample size n is large, a confidence interval for the population proportion p is given by the following formula:

\[\hat{p}\pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

where is the sample proportion and z* is the a z-critical value (from a standard normal distribution) associated with the confidence level. Unit 28
Confidence Interval for Population Slope β
A confidence interval for the population slope β is given by the following formula:

\[ b \pm t^* s_b \]

where t* is a t-critical value associated with the confidence level and determined from a t-distribution with df = n - 2; b is the least-squares estimate of the population slope calculated from the data, and sb is the standard error of b. Unit 30
Confidence Level
A number that provides information on how much confidence we have in the method used to construct a confidence interval estimate of a population parameter. It is the long-run success rate (success means capturing the parameter in the interval) of the method used to construct the confidence interval. Unit 24
Confounding Factors
Two (or more) factors (explanatory variables) are confounded when their effects on a response variable are intertwined and cannot be distinguished from each other. Unit 15
Continuous Random Variable
A random variable that can take on values that include an interval. The number of possible distinct outcomes is uncountable; there are too many possible values to put them all in a list. Unit 20
Control Charts
Charts used to monitor the output of a process. The charts are designed to signal when the process has been disturbed so that it is now out of control or is about to go out of control. Unit 23

Control chart
Control Group
A group in an experiment that does not receive the treatment under study. The control group could receive a placebo to hide the fact that no treatment is being given. In an active control group, the subjects receive what might be considered the existing standard treatment. Unit 15
Control Limits
The upper control limit (UCL) and lower control limit (LCL) on a control chart are generally set ±3 σ/√n from the center line. Unit 23
Convenience Sampling
A sampling design in which the pollster selects a sample that is easy to obtain, such as friends, family, co-workers, and so forth. Unit 17
Correlation
Denoted by r, correlation measures the direction and strength of a linear relationship between two quantitative variables. The formula for computing Pearson’s correlation coefficient is:

\[ r = \frac{1}{1-n} \sum \biggl( \frac{x-\bar{x}}{s_x} \biggr) \biggl( \frac{y-\bar{y}}{s_y} \biggr) \]

Unit 12

D

Decision Rules
A set of rules that identify from a control chart when a process is becoming unstable or going out of control. Unit 23
Degrees of Freedom for Test for Independence
(r - 1)(c - 1), where the numbers r and c are the number of rows and columns in the two-way table, respectively. Unit 29
Dependent Events
Two events are dependent if the fact that one of the events occurs does affect the probability that the other occurs. Events that are not dependent are independent. Unit 19
Dependent Variable
A variable whose outcome we would like to predict based on another variable (independent variable). The dependent variable is always plotted on the vertical axis of a scatterplot. Also called a response variable. Unit 10
Deviations from the Mean
The deviations of each data value from the sample mean: x1 - x̄, x2 - x̄, ... xn - x̄. Unit 6
Discrete Random Variable
A random variable that can take on only a countable number of distinct values – in other words, it is possible to list all possible values. Any random variable that can take on only a finite number of values is a discrete random variable. Unit 20
Distribution
Description of the possible values a variable assumes and how often these values occur. Unit 2
Dotplot
Graphical display of quantitative data in which each observation (or a group of a specified number of observations) is represented by a dot above a horizontal axis. Unit 2

Dotplot
Double-Blind Experiment
An experiment in which neither the subjects nor the individuals measuring the response know which subjects are assigned to which treatment. Unit 15

E

Empirical Rule (68-95-99.7% Rule)
Rule that gives the approximate percentage of data that fall within one standard deviation (68%), two standard deviations (95%), and three standard deviations (99.7%) of the mean. This rule should be applied only when the data are approximately normal. Unit 8

Empirical Rule
Estimated Regression Line
The estimated regression line for the linear regression model is the least-squares line, ŷ = a + bx. Unit 30
Expected Counts
The number of observations that would be expected to fall into each cell (or class) of a two-way table if the null hypothesis is true. The expected counts for the chi-square test for independence are computed as follows:

\[ \text{expected count} = \frac{(\text{row total})(\text{column total})}{\text{grand total}} \]

Unit 29
Experimental Study
A study in which researchers deliberatively apply some treatment to the subjects in order to observe their responses. The purpose is to study whether the treatment causes a change in the response. Unit 15
Explanatory Variable
Variable that is used to predict the response variable. The explanatory variable is always plotted on the horizontal axis of a scatterplot. Also called Independent Variable. Unit 10

F

F-Test Statistic
The test statistic of the ratio of the MSG and MSE, \( F = \frac{MSG}{MSE} \) , which is used for testing H0: μ1 = μ2 = ... = μk. When H0 is true, F has an F distribution with numerator df = k - 1 and denominator df = N - k, where k is the number of groups and N is the total number of observations. Unit 31
Factors
The explanatory variables in an observational study or an experiment. Also called the independent variables. Unit 15, Unit 31
First Quartile or Q1
The one-quarter point in an ordered set of quantitative data. To compute Q1, calculate the median of the lower half of the ordered data. Unit 5
Five-Number Summary
A five number summary of a quantitative data set consists of the following: minimum, first quartile (Q1), median, third quartile (Q3), maximum. Unit 5
Frequency Distribution
A table that displays frequencies of data falling into categories or class intervals. Unit 3

H

Histogram
Graphical representation of a frequency distribution. Bars are drawn over each class interval on a number line. The areas of the bars are proportional to the frequencies with which data fall into the class intervals. Unit 3

Histogram

I

In Control
The state of a process that is running smoothly, with its variables staying within an acceptable range. Unit 23
Independent Events
Two events are independent if the fact that one of the events occurs does not affect the probability that the other occurs. Unit 19
Independent Variable
Variable that is used to predict the dependent variable. The independent variable is always plotted on the horizontal axis of a scatterplot. Also called Explanatory Variable. Unit 10
Interquartile range or IQR
A measure of the spread of the middle half of the data: IQR = Q3 – Q1. The IQR is a resistant measure of the variability of a data set. Unit 5

J

Joint Distribution of Two Categorical Variables
A two-way table of counts gives the joint distribution of two categorical variables. The joint distribution can be converted to percentages by dividing each cell count by the grand total and then multiplying by 100%. Unit 13

L

Least-Squares Regression
A method for finding the best-fitting curve to a given set of data points by minimizing the sum of the squares of the residual errors (SSE). Unit 11
Least-Squares Regression Line
The least-squares line is the line that makes the sum of the squares of the residual errors (SSE) as small as possible. The equation of the least-squares line has the form y = a + bx, where a and slope b can be calculated from n data pairs (x, y) using the following formulas:

\begin{align} b & = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sum (x-\bar{x})^2} \\ a & = \bar{y}-b\bar{x} \end{align}

Unit 11
Level
One of the possible values or settings that a factor can assume. Unit 31
Linear Form
A scatterplot has linear form when dots in a scatter plot appear to be randomly scattered on either side of a straight line. Unit 10

Linear form scatterplot
Linear Regression Model
The simple linear regression model assumes that for each value of x the observed values of the response variable y are normally distributed about a mean μy that has the following linear relationship with x:

\[ \mu_y = \alpha + \beta x \]

Linear regression model Unit 30
Lurking Variable
An extraneous variable that is related to the other variables in a study. A lurking variable that is linked to both an explanatory variable and a response variable can be the underlying cause for an observed relationship between the explanatory and response variable. Unit 14

M

Margin of Error
For confidence intervals of the form point estimate ± margin of error, the margin of error gives the range of values above and below the point estimate. The margin of error is the half-width of the confidence interval. Unit 24
Marginal Distribution
A distribution computed from a two-way table of counts by dividing the row or column totals by the overall total. Often the marginal distributions are expressed as percentages. Unit 13

Marginal distribution
The marginal distributions of the row and column variables expressed in percentages.
Marginal Totals
The sum of the row entries or the sum of the column entries in a two-way table of counts. Unit 13

Marginal totals
Marginal totals in a two-way table.
Matched-Pairs t-Test Statistic
In testing H0: μD = μD0 where μD is the population mean difference, given by

\[t = \frac{\bar{x}_D - \mu_{D_{0}}}{s_D/\sqrt{n}}\]

where D and sD are the mean and standard deviation of the sample differences. If the differences are approximately normally distributed and the null hypothesis is true, then t has a t-distribution with df = n - 1 degrees of freedom. Unit 26
Mean
The arithmetic average or balance point of sample data. To calculate the mean, sum the data values and divide the sum by the number of data values.

Mean
The mean as the balancing point.

If the sample consists of observations x1,x2,...,xn, then the sample mean is

\[ \bar{x} = \frac{\sum{x}}{n} \]

Unit 4
Mean of a Discrete Random Variable x
Given a probability distribution, p(x), the mean is calculated as follows:

\[ \mu = \sum x \cdot p(x) \]

Unit 20
Median
A resistant measure of center of a data set. The median separates the upper half of the data from the lower half. To calculate the median, order the data from smallest to largest and count up (n + 1)/2 places in the ordered list. Unit 4
Mode
The data value in a quantitative data set that occurs most frequently. Unit 4

Mode
The mode of the test scores is 75.
Multiplication Rule
If C and D are independent, then P(C and D) = P(C)P(D). Unit 19
Multistage Sampling
A sampling design that begins by dividing the population into clusters. In stage one, the pollster choses a (random) sample of clusters. In subsequent stages, samples are chosen from each of the selected clusters. Unit 17
Multivariate Data
Data that consists of measurements or observations recorded on two or more attributes for each individual or subject under study. Unit 10
Mutually Exclusive Events
Events that have no outcomes in common. Events that are disjoint. Unit 19

N

Negative Association
Two variables have negative association if above-average values of one accompany below-average values of the other, and vice versa. In a scatterplot, a negative association would appear as a pattern of dots in the upper left to the lower right. Unit 10

Negative association scatterplot
Nonlinear Form
Often scatterplots do not have linear form. Instead the data might form a curved pattern. In that case, we say the scatterplot has nonlinear form. Unit 10

Nonlinear form scatterplot
Normal Curve
Bell-shaped curve. The center line of the normal curve is at the mean μ. The change-of-curvature in the bell-shaped curve occurs at μσ and μ + σ where σ is the standard deviation. Unit 7

Normal curve
Normal Density Curve
A normal curve scaled so that the area under the curve is 1. Unit 7
Normal Distribution
Distribution that is described by a normal density curve. Any particular normal distribution is completely specified by two numbers, its mean μ and standard deviation σ. Unit 7
Normal Quantile Plot
Also known as normal probability plot. A graphical method for assessing whether data come from a normal distribution. The plot compares the ordered data with what would be expected of perfectly normal data. A normal quantile plot that shows a roughly linear pattern suggests that it is reasonable to assume the data come from a normal distribution. Unit 9

Normal quantile plot
Normal quantile plot for hen weights.
Null Hypothesis or H0
The claim tested by a significance test. Usually the null hypothesis is a statement about "no effect" or "no change." The null hypothesis has the following form: H0: population parameter = hypothesized value. Unit 25

O

Observational Study
A study in which researchers observe subjects and measure variables of interest. However, the researchers do not try to influence the responses. The purpose is to describe groups of subjects under different situations. Unit 15
Observed Counts
The number of observations that fall into each cell (or class) of a two-way table. Unit 29
One-Sided Alternative Hypothesis
The alternative hypothesis in a significance test is one-sided if it states that either a parameter is greater than or a parameter is less than the null hypothesis value. Unit 25
One-Way ANOVA
An analysis of variance in which one factor is thought to be related to the response variable. Unit 31
Out of Control
The state of a process that is no longer in control. The process has become unstable or its variables are no longer within an acceptable range. Unit 23
Outlier
Data value that lies outside the overall pattern of the other data values. Unit 2

P

Paired t-Confidence Interval for μD
When data are matched pairs, and the standard deviation of the population differences σD is unknown, a t-confidence interval estimate of the population mean differences, μD, is given by the formula:

\[\bar{x}_D \pm t^* \left( \frac{s_D}{\sqrt{n}} \right)\]

where t* is a t-critical value associated with the confidence level and determined from a t-distribution with df = n - 1 and D and sD are the mean and standard deviation of the sample differences. Unit 26
Percentile
A value such that a certain percentage of observations from the distribution falls at or below that value. The pth percentile of a data set is a value such that p% of the observations fall at or below that value. Unit 9

Percentile
The value -1.645 is the 5th percentile of a standard normal distribution because 5% of the area under the normal curve lies at or below -1.645.
Pie Chart
Graph of a frequency distribution for categorical data. Each category is represented by a slice of pie in which the area of the slice is proportional to the frequency or relative frequency of that category. Unit 2

Pie chart
Placebo
Something that is identical in appearance to the treatment received by the treatment group. Placebos are meant to be ineffectual and are given as control treatments. Unit 15
Point Estimate
A single number based on sample data (a statistic) that represents a plausible value for a population parameter. Unit 24
Population
The entire group of objects or individuals about which information is wanted. Unit 16
Population Proportion
For a population that is divided into two categories, success and failure, based on some characteristic, the population proportion, p, is:

\[p = \frac{\text{number of successes in the population}}{\text{population size}}\]

Unit 28
Population Regression Line
The population regression line, μy = α + βx describes how the mean response y varies as x changes. Unit 30
Positive Association
Two variables have positive association if above-average values of one tend to accompany above-average values of the other and below-average values of one tend to accompany below-average values of the other. In a scatterplot, a positive association would appear as a pattern of dots in the lower left to the upper right. Unit 10

Positive association scatterplot
Probability
A measure of how likely it is that something will happen or something is true. Probabilities are always between 0 and 1. Events with probabilities closer to 0 are less likely to happen and events with probabilities closer to 1 are more likely to happen. Unit 18
Probability Distribution
A list of the possible values of a discrete random variable together with the probabilities associated with those values. Unit 20
Process
Chain of steps that turns inputs into outputs. Unit 23
Prospective Study
A study that starts with a group and watches for outcomes (for example, the development of cancer or remaining cancer-free) during the study period and relates this to suspected risk or protection factors that might be linked to the outcomes. Unit 14
P-value
The probability, computed under the assumption that the null hypothesis is true, of observing a value from the test statistic's distribution that is at least as extreme as the value of the test statistic that was actually observed. Unit 25

Q

Quantitative Variable
Variable whose values are numbers obtained from measurements or counts. Height, weight, and points scored at a basketball game are examples of quantitative variables. Unit 2

R

Random Phenomenon
A situation in which the possible outcomes are known but we do not know which one will occur. If the situation is repeated over and over, a regular pattern to the outcomes emerges over the long run. Unit 18
Random Variable
A variable whose possible values are numbers associated with outcomes of a random phenomenon. Unit 20
Range
Measure of the variability of a quantitative data set from its extremes: range = maximum – minimum. Unit 5
Regression Line
A straight line that describes how a response variable y is related to an explanatory variable x. Unit 11
Representative Sample
A sample that accurately reflects the members of the entire population. Unit 17
Residual Error
A residual error is the vertical deviation of a data point from the regression model: residual error = actual y – predicted y. Unit 11

Residual error
Resistant Measure
A statistic that measures some aspect of a distribution (such as its center) that is relatively unaffected by a small subset of extreme data values. For example, the median is a resistant measure of the center of a distribution while the mean is not a resistant measure of center. Unit 4
Response Variable
The variable used to measure the outcome of a study, which we attempt to explain or predict using one or more independent variables (factors). The response variable is always plotted on the vertical axis of a scatterplot. Also called the dependent variable. Unit 10, Unit 31
Retrospective Study
A study that starts with an outcome (for example, two groups of people, a cancer group and a non-cancer group) and then looks back to examine exposures to suspected risk or protection factors that might be linked to that outcome. Unit 14
Run Chart
A plot of data values versus the order in which these values were collected. Unit 23

Run chart

S

Sample
The part of the population that is actually examined in a study. Unit 16
Sample Mean
One measure of center of a data set. The mean is the arithmetic average or balance point of a set of data. To calculate the mean, sum the data and divide by the number of data items:

\[ \bar{x} = \frac{\sum x}{n} \]

Unit 4
Sample Proportion
The sample proportion, , from a sample of size n is:

\[\hat{p} = \frac{\text{number of successes in the sample}}{n}\]

Unit 28
Sample Standard Deviation
One measure of variability of a data set. The standard deviation has the same units as the data values. To calculate the standard deviation, take the square root of the sample variance:

\[ s = \sqrt{\frac{\sum{(x - \bar{x})^2}}{n-1}} \]

Unit 6
Sample Variance
One measure of variability of a data set. To calculate the variance, sum the squared deviations from the mean and divide by the number of data minus one:

\[ s^2 = \frac{\sum{(x - \bar{x})^2}}{n-1} \]

Unit 6
Sampling Bias
Occurs when a sample is collected in such a way that some individuals in the population are less likely to be included in the sample than others. Because of this, information gathered from the sample will be slanted toward those who are more likely to be part of the sample. Unit 16
Sampling Design
Plan of how to select the sample from the population. Unit 17
Sampling Distribution
The distribution of the values of a sample statistic (such as , the median, or s) over many, many random samples chosen from the same population. Unit 22
Sampling Distribution of the Sample Mean
The distribution of over a very large number of samples. If is the mean of a simple random sample (SRS) of size n from a population having mean µ and standard deviation σ, then the mean and standard deviation of are:

\[ \mu_{\bar{x}} = \mu \\ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

Furthermore, if the population distribution is normal, then the distribution of is normal. Unit 22

Sampling distribution of the sample mean
Sampling Distribution of the Sample Proportion
When the sample size n is large, the sampling distribution of the sample proportion is approximately normally distributed with the following mean and standard deviation:

\[ \mu_{\hat{p}} = p \text{, where $p$ is the population proportion.}\\ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \text{, where $n$ is the sample size.} \]

Unit 28
Scatterplot
A graphical display of bivariate quantitative data in which each observation (x, y) is plotted in the plane. Unit 10
Self-Selecting Sampling
A sampling design in which the sample consists of people who respond to a request for participation in the survey. (Also called voluntary sampling.) Unit 17
Significance Level
In a significance test, the highest p-value for which we will reject the null hypothesis. Unit 25
Significance Test
A method that uses sample data to decide between two competing claims, called hypotheses, about a population parameter. Unit 25
Simple Random Sample of Size n
A sample of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be in the sample actually selected. Unit 16
Simple Random Sampling
A sampling design that chooses a sample of size n using a method in which all possible samples of size n are equally likely to be selected. Unit 17
Single-Blind Experiment
An experiment in which the subjects do not know which treatment they are receiving but the individuals measuring the response do know which subjects were assigned to which treatments. Unit 15
Skewed Right or Left
Skewed right
Skewed right

Skewed left
Skewed left
A unimodal distribution is skewed to the right if the right tail of the the distribution is longer than the left and is skewed to the left if the left tail of the distribution is longer than the right. Unit 3
Special Cause Variation
Variation due to sudden, unexpected events that affect the process. Unit 23
Standard Deviation of a Discrete Random Variable x
Given a probability distribution, p(x), the standard deviation, σ, is calculated as follows:

\[ \sigma^2 = \sum{(x-\mu)^2} \cdot p(x); \sigma = \sqrt{\sigma^2} \]

Unit 20
Standard Error of the Estimate
A point estimate of σ, which is a measure of how much the observations vary about the regression line. The standard error of the estimate, se, is computed as follows:

\[s_e = \sqrt{MSE} = \sqrt{ \frac{\sum(y-\hat{y})^2}{n-2}}\]

Unit 30
Standard Error of the Slope b
The estimated standard deviation of b, the least-squares estimate for the population slope β, is:

\[s_b = \frac{s_e}{\sqrt{\sum(x-\bar{x})^2}}\]

Unit 30
Standard Normal Distribution
Normal distribution with μ = 0 and σ = 1. Unit 8
Standard Normal Quantiles
The z-values that divide the horizontal axis of a standard normal density curve into intervals such that the areas under the density curve over each of the intervals are equal. Unit 9
Stemplot (or Stem-and-Leaf Plot)
Graphical tool for organizing quantitative data in order from smallest to largest. The plot consists of two columns, one for the stems (leading digit(s) of the observations) and the other for the leaves (trailing digit(s) for each observation listed beside corresponding stem). Stemplots are a useful tool for conveying the shape of relatively small data sets and identifying outliers. Unit 2

Stemplot
Strata
The non-overlapping groups used in a stratified sampling plan. Unit 17
Stratified Random Sample
A stratified sampling plan in which the sample is obtained by taking random samples from each of the strata. Unit 17
Stratified Sampling
A sampling plan that is used to ensure that specific non-overlapping groups of the population are represented in the sample. The non-overlapping groups are called strata. Samples are taken from each stratum. Unit 17
Symmetric Distribution
Shape of a distribution of a quantitative variable in which the lower half of the distribution is roughly a mirror image of the upper half. Unit 2

Symmetric distribution
Distribution of Time of First Lightning flash is roughly symmetric.

T

t-Confidence Interval for μ
When σ is unknown, the sample size n is small, and the population distribution is approximately normal, a t-confidence interval for μ is given by the following formula:

\[\bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right), \]

where t* is a t-critical value associated with the confidence level and determined from a t-distribution with df = n - 1 degrees of freedom. Unit 26
t-Distribution
Density curves for t-distributions are bell-shaped and centered at zero, similar to the standard normal density curve. Compared to the standard normal distribution, a t-distribution has more area under its tails. The shape of a t-distribution, and how closely it resembles the standard normal distribution, is controlled by a number called its degrees of freedom (df). A t-distribution with df > 30 is very close to a standard normal distribution. Unit 26

t-distribution
Comparison of a standard normal distribution and a t-distribution with df = 2. Notice that the t-distribution has thicker tails (see shaded region) than the standard normal distribution.
t-Test Statistic
In testing H0: μ = μ0, where μ is the population mean, the formula for the t-test statistic is:

\[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \]

The t-test is used in situations where the population standard deviation σ is unknown, the sample size n is small, and the population has a normal distribution. If the null hypothesis is true, t has a t-distribution with df = n - 1 degrees of freedom. Unit 26
t-Test Statistic for the Slope
In testing H0: β = β0, where β is the population slope, the formula for the t-test statistic is:

\[ t = \frac{b - \beta_0}{s_b}\text{, where }s_b = \frac{s_e}{\sqrt{\sum(x-\bar{x})^2}} \]

When H0 is true, t has a t-distribution with df = n - 2, where n is the number of (x,y)-pairs in the sample. The usual null hypothesis is H0: β = 0, which says that the straight-line dependence on x has no value in predicting y. Unit 30
Test of Hypotheses
A method that uses sample data to decide between two competing claims, called hypotheses, about a population parameter. Unit 25
Test Statistic
A quantity computed from the sample data that is used to make a decision between the null and alternative hypotheses in a significance test. Unit 25
Third Quartile or Q3
The three-quarter point in an ordered set of data. To compute Q3, calculate the median of the upper half of the ordered data. Unit 5
Treatment
Any specific condition applied to the subjects in an experiment. If an experiment has more than one factor, then a treatment is a combination of specific values for each factor. Unit 15
Two-Sample t-Confidence Interval for μ1 - μ2
When data are from two independent random samples from different populations, and the population standard deviations are unknown, a two-sample t-confidence interval estimate of the difference in population means is given by the formula:

\[ (\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} } \]

There are two options for finding the degrees of freedom (df) associated with t*, the t-critical value associated with the confidence level: (1) use technology or (2) use a conservative approach and let df = smaller of n1 - 1 or n2 - 1 . Unit 27
Two-Sample t-Procedures
Two sample t-procedures are used to test or estimate μ1 - μ2, the difference of two population means. The required data consists of two independent simple random samples of sizes n1 and n2 from each of the populations (or treatments). Unit 27
Two-Sample t-Test Statistic
In testing H0: μ1 - μ2 = d, where μ1 and μ2 are the means of two populations, the formula for the two-sample t-test statistic is:

\[ t = \frac{(\bar{x}_1 - \bar{x}_2) - d}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}, \]

There are two options for finding the degrees of freedom (df) associated with t: (1) use technology or (2) use a conservative approach and let df = smaller of n1 - 1 or n2 - 1. Unit 27
Two-Sided Alternative Hypothesis
The alternative hypothesis in a significance test is two-sided if it states that a parameter is different from the null hypothesis value. Unit 25
Two-Way Table of Counts (Frequencies)
A table with r rows and c columns that organizes data on two categorical variables taken from the same individuals or subjects. Values of the row variable label the rows of the table; values of the column variable label the columns of the table. Unit 13

Two way table
A 3×3 table two-way table: the row variable is Happiness and the column variable is Physical Beauty.

U

Unimodal
Describes a single-peaked shape of a histogram or density curve. Unit 2

Unimodal histogram
Distribution of Time of First Lightning flash is unimodal.
Univariate Data
Data in which measurements or observations are recorded on one attribute for each individual or subject under study. Unit 2

V

Variable
Describes some characteristic or attribute of interest that can vary in value. Unit 2
Variance of a Discrete Random Variable x
Given a probability distribution, p(x), the variance is calculated as follows:

\[ \sigma^2 = \sum{(x-\mu)^2} \cdot p(x) \]

Unit 20
Voluntary Sampling
A sampling design in which the sample consists of people who respond to a request for participation in the survey. Also called self-selecting sampling. Unit 17

W

Within-Groups Variation
A measure of the spread of individual data values within each group about the group mean. It is measured by the mean square error, MSE. Unit 31

X

Charts
A plot of means of successive samples versus the order in which the samples were taken. Unit 23

Z

z-Score
Transformation of a data value x into its deviation from the mean measured in standard deviations. To calculate a z-score for a data value x, subtract the mean and divide by the standard deviation:

\[ z = \frac{x - \mu}{\sigma} \]

Unit 8
z-Test Statistic
In testing H0: μ = μ0, where μ is the population mean, the formula for the z-test statistic is:

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

The z-test statistic is used in situations where the population standard deviation σ is known and either the population has a normal distribution or the sample size n is large. Unit 8
z-Test Statistic for Proportions
In testing H0: p = p0, where p is the population proportion, the formula for the z-test statistic is:

\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}} \]

The z-test is used in situations where the sample size n is large. Unit 28