3 sigma control charts: Plot the values of some statistic Q for regular samples from the process against the time order of the samples. The center line is at the mean of Q. The control limits lie three standard deviations of Q above and below the center line. A point outside the control limits is an out–of–control signal.
68–95–99.7 rule: In the Normal distribution with mean μ and standard deviation σ: Approximately 68% of the observations fall within σ of the mean μ. Approximately 95% of the observations fall within 2σ of μ. Approximately 99.7% of the observations fall within 3σ of μ.
95% bootstrap percentile confidence interval: The interval between the 2.5% and 97.5% percentiles of the bootstrap distribution of a statistic. This method can be used when the bootstrap estimate of bias is small.
addition rule for disjoint events: If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then P(one or more of A,B,C) = P(A) + P(B) + P(C).
alternative hypothesis: Denoted Ha, the statement we hope or suspect is true in a test of significance.
analysis of variance F test: In the multiple regression model, the hypothesis H0: β1 = β2 =. . . = βp = 0 is tested against the alternative hypothesis Ha: at least one of the βj is not 0, by the analysis of variance F statistic MSM/MSE. The P–value is the probability that a random variable having the F(p,n — p — 1) distribution is greater than or equal to the calculated value of the F statistic.
analysis of variance: Abbreviated ANOVA, the statistical methodology for comparing several means.
anecdotal data: Represent individual cases, which often come to our attention because they are striking in some way. These cases are not necessarily representative of any larger group of cases.
anonymity: Subjects are anonymous that prevents any follow-up to improve nonresponse or inform subjects of a result. Not the same as confidentiality.
ANOVA F statistic: The ratio MSM/MSE and is used to test the null hypothesis H0: β1 = β2 = . . . = βp = 0. If H0 is true, this statistic has an F(p,n — p — 1) distribution.
ANOVA F test: To test the null hypothesis in a one–way ANOVA, find the F statistic MSG/MSE. If H0 is true, the F statistic has the F(I — 1, N — I) distribution. If Ha is true, the F statistic is large. We reject H0 in favor of Ha if the F statistic is sufficiently large.
ANOVA table: For a multiple linear regression, gives the degrees of freedom, sum of squares, and mean squares for the model, error, and total sources of variation.
associated: Two variables measured on the same cases are said to be this if knowing the values of one of the variables tells you something about the values of the other variable that you would not know without this information.
available data: Data that were produced for some other purpose but that may help a question of interest.
bar graph: Graph that shows the distribution of a categorical variable by representing each category as a bar. The bar heights show the category counts or percents.
benchmarking: Evaluate or check (something) by comparison with a standard.
bias: A study is said to have this if it systematically favors certain outcomes.
binomial coefficient: The number of ways of arranging k successes among n observations is given by the binomial coefficient n!/k!(n — k)!.
binomial distribution: The distribution of the count X of successes in the binomial setting with parameters n and p. The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n. As an abbreviation, we say that the distribution of X is B(n, p).
binomial setting: 1. There is a fixed number of observations n. 2. The n observations are all independent. 3. Each observation falls into one of just two categories, which for convenience we call success and failure. 4. The probability of a success, call it p, is the same for each observation.
block design: When the random assignment of units to treatments is carried out separately within each block.
block: A group of experimental units or subjects that are known before the experiment to be similar in some way that is expected to affect the response to the treatments.
bootstrap bias–corrected accelerated interval: Confidence interval denoted BCa. A modification of the percentile method that adjusts the percentiles to correct for bias and skewness present in the bootstrap distribituon.
bootstrap distribution: The approximate sampling distribution of a statistic collected from many resamples with replacement from one random sample. This distribution usually has approximately the same shape and spread as the sampling distribution and is centered at the statistic (from the original sample) when the sampling distribution is centered at the parameter (of the population).
bootstrap estimate of bias: The difference between the mean of the bootstrap distribution and the value of the statistic in the original sample.
bootstrap standard error: Denoted SEboot, the standard deviation of the bootstrap distribution of a statistic. It measures how much the statistic varies under random sampling.
bootstrap t confidence interval: Suppose that the bootstrap distribution of a statistic from an SRS of size n is approximately Normal and that the bootstrap estimate of bias is small. An approximate level C confidence interval for the parameter that corresponds to this statistic by the plug–in principle is statistic±t*SEboot.
bootstrap tilting interval: Confidence interval that can be used when bias or skewness is present in the bootstrap distribution. Adjusts the process of randomly forming resamples. A clever implementation allows use of the same resamples as other bootstrap methods.
bootstrap: A way of finding the sampling distribution, at least approximately, from just one sample by using resampling with replacement from this one random sample, calculating the statistic for each resample, and inspecting the distribution of the resampled statistics.
boxplot: A graph of the five–number summary with the following properties: 1. A central box spans the quartiles Q1 and Q3. 2. A line in the box marks the median M. 3. Lines extend from the box out to the smallest and largest observations or to a cutoff for suspected outliers.
capability indexes: Measure process variability (Cp) or process center and variability (Cpk) against the standard provided by external specifications for the output of the process. Larger values indicate higher capability.
capability: The ability of a process to meet or exceed the requirements placed on it.
cases: The objects described by a set of data. They may be customers, companies, subjects in a study, or other objects.
categorical variable: Variable that places a case into one of several groups or categories.
cause–and–effect diagram: Organizes the logical relationships between the inputs and stages of a process and an output.
census: An attempt to gather information from every individual in the entire population.
central limit theorem: For large sample sizes, the distribution of the sample mean is close to a Normal distribution. This is true no matter what shape the population distribution has, as long as the population has a finite standard deviation σ.
chart setup: Stage when you first collect data from a process, establish control by uncovering and removing special causes, and then set up control charts to maintain control.
chi–square distributions: A family of distributions described by a single parameter, the degrees of freedom. These distributions take only positive values and are skewed to the right.
chi–square goodness–of–fit test: Used to compare the sample distribution of a categorical variable from a population with a hypothesized distribution. The data for n observations with k possible outcomes are summarized as observed counts, n1, n2,...,nk in k cells. The null hypothesis specifies probabilities p1, p2,...,pk for the possible outcomes.
chi–square statistic: A measure of how much the observed cell counts in a two–way table diverge from the expected cell counts. The formula is: X2 = Σ (observed count — expected count)2/expected count.
chi–square test: This tests the null hypothesis H0 that there is no association between the row and column variables in a two–way table. The alternative hypothesis Ha says that there is some relationship but does not say what kind.
clinical trials: Experiments that study the effectiveness of medical treatments on actual patients.
collinearity: A phenomenon that occurs when a regression model contains overlap of predictive information contained in explanatory variables, which can cause numerical instabilities that result in very imprecise parameter estimates. Also known as multicollinearity.
column variable: value that labels the columns that run down a two–way table.
common cause variation: In the language of statistical quality control, a process that is in control has only this type of variation. This variation is the inherent variability of the process, due to many small causes that are always present.
comparison: The simplest form of control. One of the basic principles of statistical design of experiments.
complement rule: P(Ac) = 1 — P(A)
complement: Of any event A, the event that A does not occur.
completely randomized experimental design: When all experimental units are allocated at random among all treatments.
conditional distribution: In a two–way table, when we condition on the value of one variable and calculate the distribution of the other variable.
conditional probability: Gives the probability of one event under the condition that we know another event.
confidence interval for a mean response: A level C confidence interval for the mean response µy when x takes the value x*, where t* is the value for the t(n — 2) density curve with area C between –t* and t* is: µ^y ± t*SEµ^.
confidence interval: Estimates an unknown parameter with an indication of how accurate the estimate is and of how confident we are that the result is correct. Contains two important aspects: an interval of the form (a,b), where a and b are numbers computed from the data, and a confidence level. The margin of error of this decreases as the confidence level C decreases and the sample size n increases.
confidence level: Gives the probability of producing an interval that contains the unknown parameter.
confidentiality: Important feature of a study where all data about an individual in a study may not be reported as being linked to that individual.
confounding: Occurs when the effects of two or more variables are related in such a way that we need to take care in assigning the effect on the response variable to one or to the other.
continuity correction: Acting as though a whole number occupies the interval from 0.5 below to 0.5 above the number is called the continuity correction to the Normal approximation.
continuous random variable: A random variable X that can take all values in an interval of numbers. Its probability distribution is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.
contrasts: Specific questions formulated before examination of the data can be expressed as these. A combination of population means where the coefficients ai sum to 0, of the form Σaiµi.
control charts: Statistical tools that monitor a process and alert us when the process has been disturbed so that it is now out of control. This is a signal to find and correct the cause of the disturbance.
control group: In an experiment, the group that does not receive the treatment that is being evaluated.
correlation: Denoted r, measures the direction and strength of the linear relationship between two quantitative variables.
cumulative proportion: The proportion of observations in a distribution that lie at or below a given value. When the distribution is given by a density curve, this is the area under the curve to the left of a given value.
data mining: The process of exploring really large databases in the hope of finding useful patterns.
decision theory: An alternative to significance testing that regards H0 and Ha as two statements of equal status that we must decide between. Because it leaves you with two error probabilities and no simple rule on how to balance them, it has been used less often than either tests of significance or tests of hypotheses.
density curve: A curve that describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range. It is always on or above the horizontal axis and has area exactly 1 underneath it.
design of a sample survey: Refers to the method used to choose the sample from the population.
discrete random variable: A random variable X with a finite number of possible values.
disjoint: Two events A and B are this if they have no outcomes in common.
distribution: Tells us what values a variable takes and how often it takes these values.
distribution–free procedures: Do not assume that the population distribution has any specific form, such as Normal; often called nonparametric procedures.
double–blind: When neither the subjects themselves nor the experimenters know which treatment any subject has received.
ethics: In a statistical study, the three basic standards that must be obeyed by any study that gathers data from human subjects are: 1. The proposed study must be reviewed by an institutional review board in advance to protect the subjects from possible harm. 2. All individuals who are subjects in the study must give their informed consent. 3. All individual data must be kept confidential.
event: An outcome or a set of outcomes of a random phenomenon; a subset of the sample space.
expected cell counts: In a two–way table, under the null hypothesis, these are computed using the formula: row total x column total / n.
experiment: A study that deliberately imposes some condition on individuals and observes their responses.
experimental units: The individuals on which the experiment is done.
explanatory variable: Explains or causes changes in the response variables. Also known as an independent variable.
exploratory data analysis: Uses statistical tools and ideas like graphs and numerical summaries to describe the variables in a data set and the relations among them.
extrapolation: The use of a regression line for prediction far outside the range of values of the explanatory variable x used to obtain the line. Such predictions are often not accurate and should be avoided.
F distributions: A family of distributions with two parameters: the degrees of freedom in the numerator and denominator of the F statistic. The degrees of freedom for this distribution are those associated with the model and error in the ANOVA table.
F statistic: Inference procedures for comparing the standard deviations of two Normal populations are based on this statistic, which is the ratio of the sample variances.
F test for equality of standard deviations: Tests H0: σ1 = σ2 versus Ha: σ1 ≠ σ2 using the statistic F = larger s2 / smaller s2 and doubles the upper–tail probability to obtain the P–value.
F-distributions: A family of distributions with two parameters: the degrees of freedom of the mean square in the numerator and denominator of the F statistic.
factors: The explanatory variables in an experiment.
five–number summary: Of a set of observations, this consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. In symbols, it is: Minimum Q1 M Q3 Maximum
flowchart: A picture of the stages of a process.
frequencies: counts on a histogram.
general addition rule for unions of two events: For any two events A and B, P(A or B) = P(A) + P(B) — P(A and B).
histogram: A graphical display of a quantitative variable that breaks the range of values of a variable into classes and displays only the count or percent of the observations that fall into each class. Chosen classes should always be of equal width.
homogeneity hypothesis: In terms of a two–way table, the null hypothesis that says that the c distributions of the row variable are identical.
homogeneity test: For two–way tables, another name for a significance test that is stated in terms of a homogeneity hypothesis.
hypotheses for one–way ANOVA: The null and alternative hypotheses for one–way ANOVA are H0: µ1 = µ2 =...= µI, and Ha: not all of the µi are equal.
independent: Two events A and B are called this if knowing that one occurs does not change the probability that the other occurs.
indicator variable: Used frequently in multiple regression to represent the levels or groups of categorical explanatory variables.
influential: An observation is said to be this for a statistical calculation if removing it would markedly change the result of the calculation.
informed consent: Subjects must be told in advance about the nature of a study and any risk of harm it may bring. The subjects should be told what kinds of questions will be asked and the experimenters must tell subjects the nature and purpose of the study and outline possible risks. Subjects must then give advanced approval of their participation in the study in writing.
institutional review board: The organization that carries out a study must have this, which reviews all planned studies in advance to protect the subjects from possible harm.
interaction: When the effectiveness of one factor depends on another factor. One–way designs that vary a single factor and hold other factors fixed cannot discover this.
intercept: Denoted b0 in the straight line equation of the form y = b0 + b1x, the value of y when x = 0.
interquartile range: Abbreviated IQR, this is the distance between the first and third quartiles. IQR = Q3 — Q1
intersection: The event that all the events of a collection of events occur.
intervention: Process in which change is imposed to understand cause and effect within an experiment.
joint distribution: In a two–way table, the collection of proportions of the two categorical variables found by dividing the count in each cell by the total number of observations.
jointly Normal variables: The condition in which the distribution of x is Normal and also that the conditional distributional of y, given any fixed value of x, is Normal.
Kruskal–Wallis statistic: Denoted H, statistic that is the result of applying one–way ANOVA to the ranks of the observations. It is also a comparison of the sums of the ranks for the several samples. When the sample sizes are not too small and the null hypothesis is true, H for comparing I populations has approximately the chi–square distribution with I — 1 degrees of freedom.
Kruskal–Wallis test: Compares several populations on the basis of independent random samples from each population. This is the one–way analysis of variance setting. The null hypothesis for this test is that the distribution of the response variable is the same in all the populations. The alternative hypothesis is that responses are systematically larger in some populations than in others.
label: A special variable used in some data sets to distinguish the different cases.
lack of realism: The most serious potential weakness of experiments. The subjects or treatments or setting of an experiment may not realistically duplicate the conditions we really want to study.
large–sample confidence interval for a population proportion: Use this interval only when the number of successes and the number of failures in each sample are each at least 10. Draw an SRS of size n from a large population that contains an unknown proportion p of successes. An approximate level C confidence interval for p, where z* is the critical value for the standard Normal density curve with area C between —z* and z*, is p̂ ± z*SEp̂.
law of large numbers: Draw independent observations at random from any population with finite mean µ. Decide how accurately you would like to estimate µ. As the number of observations drawn increases, the mean x–bar of the observed values eventually approaches the mean µ of the population as closely as you specified and then stays that close.
least–squares regression line: The line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
level: A specific value created when each factor is combined to form a treatment within an experiment.
linear transformation: A conversion of a numerical description of a distribution from one unit of measurement to another. Changes the orginal variable x into the new variable xnew given by an equation of the form xnew = a + bx
logistic regression model: Statistical model that relates the log of the odds to the explanatory variable. p is a binomial proportion, x is the explanatory variable, and the parameters are β0 and β1 in the equation log(p/(1 — p)) = β0 + β1x.
lurking variable: A variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.
main effects: In two–way ANOVA, the average values for the effects of the two factors.
margin of error: Reflects how accurate we believe our guess for the value of the unknown parameter is. For a level C confidence interval for the mean μ of a Normal population with known standard deviation σ, based on an SRS of size n, it is given by m = z*(σ / √n).
marginal distribution: The distribution of a single variable in a two–way table.
marginal means: Calculated by taking averages of the cell means across rows or down columns.
matched pairs: Experimental design that uses matching to compare two treatments. The subjects are matched in pairs with certain similar attributes (such as the same age, sex, and income), depending on what is being studied. In some cases, each subject in a pair receives both treatments in a random order and in others one subject in each pair receives each treatment.
mean of a probability distribution: Describes the long-run average outcome, denoted as the symbol μ.
mean square: The ratio of the sum of squares to the degrees of freedom for each source. Abbreviated MS.
mean standard deviation sigma: Estimate given by s=√s2.
mean: Denoted "x–bar", this is the average value of a distribution. It can be found by adding the values of a set of observations and then dividing by the number of observations.
median: Denoted M, this is the midpoint of a distribution. Half of the observations are smaller than M, and the other half are larger than M.
Meta–analysis: A collection of statistical techniques design to combine information from different but similar studies. The basic idea is to compute a measure of the effect of interest for each study.
model selection: When a model has a large number of insignificant variables, it is common to refine the model.
modes: Major peaks of a distribution. A distribution with one major peak is called unimodal.
multiple linear regression model: Statistical model where yi = β0 + β1xi1 + β2xi2 +...+ βpxip + εi for i = 1,2,...,n. The mean response µy is a linear function of the explanatory variables. The deviations εi are independent and Normally distributed with mean 0 and standard deviation σ. In other words, the deviations are an SRS from the N(0,σ) distribution. The parameters of the model are β0,β1,β2,...,βp, and σ.
multiple logistic regression: The response variable has two possible values, as in logistic regression, but there can be several explanatory variables.
multiple–comparisons: Methods used to assess the statistical significance of the differences between pairs of means if no specific questions are formulated before examination of the data and the null hypothesis of equality of population means is rejected.
multiplication rule for independent events: If A and B are independent, P(A and B) = P(A)P(B).
multiplication rule: The probability that both of two events A and B happen together can be found by P(A and B) = P(A)P(B|A). Here P(B|A) is the conditional probability that B occurs, given the information that A occurs.
multistage sample: A sample that is chosen in stages, commonly used for national samples of households or people. This sample selects successively smaller groups within the population in stages, resulting in a sample consisting of clusters of individuals. Each stage may employ an SRS, a stratified sample, or another type of sample.
negatively associated: Two variables are said to be this when above–average values of one tend to accompany below–average values of the other, and vice versa.
nonlinear model: Model that allows for various types of curved relationships when the relationship is not linear and a transformation does not work.
nonparametric tests: Statistical tests that do not require any specific form for the distribution of the population from which our samples come.
nonresponse: Occurs when an individual chosen for the sample cannot be contacted or does not cooperate.
Normal approximation to the binomial distribution: Says that if X is a count having the B(n,p) distribution, then when n is large, X is approximately N(np, √np(1 — p)) and p̂ is approximately N(p, √(p(1 — p) / n)).
Normal curves: A class of density curves that are symmetric, unimodal, and bell–shaped. They describe Normal distributions. The mean is at the center of the symmetric curve and is the same as the median.
Normal distribution: Described by bell–shaped, symmetric, unimodal density curves. Completely specified by the mean μ and standard deviation σ. The mean is the center of symmetry, and σ is the distance from μ to the change–of–curvature points on either side.
Normal quantile plot: Plot that best assesses the adequacy of a Normal model for describing a distribution of data. A pattern on such a plot that deviates substantially from a straight line indicates that the data are not Normal.
null hypothesis: Denoted H0, the statement being tested in a test of significance. Usually is a statement of "no effect" or "no difference".
observational study: Study where we observe individuals and measure variables of interest but do not attempt to influence the responses.
odds ratio: The ratio of the odds for a value of the explantory variable equal to x + 1 to the odds for a value of the explanatory variable equal to x. It is denoted eβ1, where β1 is the slope in the logistic regression model.
one–sample t confidence interval: Suppose than an SRS of size n is drawn from a population having unknown mean µ. A level C confidence interval for µ, where t* is the value for the t(n — 1) density curve with area C between –t* and t*, is x̄ ± t*(s / √n).
one–sample t statistic: Suppose that an SRS of size n is drawn from a population having unknown mean µ. To test the hypothesis H0: µ = µ0 based on SRS of size n, compute this statistic, which is found as: t = x̄ — µ0 / (s/√n).
one–sample z statistic: The standardized sample mean which has the N(0,1) distribution, and is denoted by: z = x̄ — µ / (σ/√n). It is the basis of the z procedures for inference about µ when σ is known.
one–sided alternative: An alternative hypothesis that says there is an effect or difference in a specific direction.
one–way ANOVA model: The model xij = µi + εij for i, = 1,..., I and j = 1,...,ni. The εij are assumed to be from an N(0,σ) distribution. The parameters of the model are the population means µ1,µ2,...,µI and the common standard deviation σ.
one–way ANOVA: Type of ANOVA technique when there is only one way to classify the populations. Used to compare several population means based on independent SRSs from each population. The populations are assumed to be Normal with possibly different means and the same standard deviation.
out–of–control signal: x–bar and s or x–bar and R control charts produce this if: 1. A single point lies outside the 3σ control limits of either chart. 2. The x–bar chart shows 9 consecutive points above the center line or 9 consecutive points below the center line.
outcome: The measured variables that are used to compare the treatments.
outlier: An observation that lies outside the overall pattern of the other observations.
p chart: A control chart based on plotting sample proportions p–hat from regular samples from a process against the order in which the samples were taken.
P–value: The probability, assuming H0 is true, that the test statistic would take a value as extreme or more extreme than that actually observed. The smaller this is, the stronger the evidence against H0 provided by the data.
parameter: Number that describes a population.
permutation distribution: Distribution of a statistic formed by the values of the statistic in a large number of permutation resamples. The P–value of the test is found by locating the original value of the statistic on this distribution.
permutation resamples: Repeated random samples that are drawn without replacement. These must be drawn in a way that is consistent with the null hypothesis and with the study design.
permutation tests: Significance tests based on permutation resamples drawn at random from the original data. Permutation resamples are drawn without replacement, in contrast to bootstrap samples, which are drawn with replacement.
pie chart: Graph that shows the distribution of a categorical variable as a "pie" whose slices are sized by the counts or percents for the categories.
placebo effect: When a subject responds favorably to a dummy treatment.
plug–in principle: To estimate a parameter, a quantity that describes the population, use the statistic that is the corresponding quantity for the sample.
plus four confidence interval for a proportion: Draw an SRS from a large population with an unknown proportion p of successes. To get the plus four confidence interval for p with fewer than 10 successes and failures, add 4 imaginary observations, 2 successes and 2 failures. Then use the large–sample confidence interval with the new sample size and count of successes.
plus four estimate: In order to improve the accuracy of the confidence interval for a proportion, add four imaginary observations, two successes and two failures. With the added observations, the plus four estimate of p is number of successes in the sample + 2 / n + 4.
poisson distribution: The distribution of the count X of successes in the Poisson setting is the Poisson distribution with mean µ. The parameter µ is the mean number of successes per unit of measure. The possible values of X are the whole numbers 0, 1, 2, 3,...
poisson setting: 1. The number of successes that occur in two nonoverlapping units of measure are independent. 2. The probability that a success will occur in a unit of measure is the same for all units of equal size and is proportional to the size of the unit. 3. The probability that more than one event occurs in a unit of measure is negligible for very small-sized units. In other words, the events occur one at a time.
pooled estimate of p: Estimate that combines, or pools, information from both samples.
pooled two–sample t procedures: If we can assume that two populations have equal variances, these procedures can be used. These are based on the t(n1 + n2 — 2) distribution and the pooled estimator of the unknown common variance σ2.
population correlation: The correlation between variables x and y when they are measured for every member of a population.
population distribution: The distribution of the values of a variable for all members of the population. It is also the probability distribution of the variable when we choose one individual at random from the population.
population regression equation: In the multiple regression setting, the response variable y depends on p explanatory variables, denoted by x1, x2,...,xp. The mean response depends on these explanatory variables according to a linear function. The observed values of y vary about their means given by this equation: µy = β0 + β1x1 + β2x2 +...+xpxp.
population regression line: In simple linear regression, the line that describes how the mean response µy changes with x.
population: The entire group of individuals we want information about.
positively associated: Two variables are said to be this when above–average values of one tend to accompany above–average values of the other, and below–average values also tend to occur together.
power: Measure of a fixed level test's probability of detecting an effect of the size you hope to find. In other words, it is the probability that the test does reject H0. This probability to detect a particular alternative is calculated as 1 minus the probability of a Type II error for that alternative.
prediction interval: The interval used to predict a future observation.
probability distribution: Of a random variable X, tells us what the possible values of X are and how probabilities are assigned to those values.
probability histograms: Histograms that show probability distributions as well as distributions of data.
probability model: For a random phenomenon, consists of a sample space S and an assignment of probabilities P.
probability sample: A sample chosen by chance. We must know what samples are possible and what chance each possible sample has.
probability: The proportion of times an outcome of a random phenomenon would occur in a very long series of repetitions.
process monitoring: When a process has been operating in control for some time, you keep control charts to monitor the process because a special cause could erupt at any time. The mean and standard deviation are based on past data from the process and are updated regularly.
process: A chain of activities that turns inputs into outputs.
quantitative variable: Variable that takes numerical values for which arithmetic operations such as adding and averaging make sense.
quartiles: Used to describe the spread of a distribution. The first quartile, Q1, has one–fourth of the observations below it, and the third quartile, Q3, has three–fourths of the observations below it.
r × c table: Two-way table of counts with r rows and c columns.
R chart: Based on the range of observations in a sample. Is often used in place of an s chart.
random phenomenon: If individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
random variable: A variable whose value is a numerical outcome of a random phenomenon.
randomization: The use of chance to divide experimental units into groups. One of the basic principles of statistical design of experiments.
rank tests: Nonparametric tests based on the ranks of observations.
rank: The position of each observation in an ordered list when the observations are arranged in order from smallest to largest, with 1 being for the smallest observation.
regression line: A straight line that describes how a response variable y changes as an explanatory variable x changes. This line is often used to predict the value of y for a given value of x.
relative risk: Denoted RR, the ratio of two sample proportions.
repetition: Process in which repeating treatments on many units reduces the role of chance variation and makes the experiment more sensitive to differences among the treatments. One of the basic principles of statistical design of experiments.
resample: Samples taken from a sample that is represented as a population.
resampling: Method used in bootstrapping in which we repeatedly sample with replacement from one random sample in order to find a sampling distribution from this one sample.
residual plot: A scatterplot of the regression residuals against the explanatory variable. Helps us to assess the fit of a regression line.
residual: The difference between an observed value of the response variable and the value predicted by the regression line.
response bias: Type of bias in a sample brought about due to the behavior of the interviewer or the respondent because of various factors, including the type of question being asked.
response rate: The proportion of the original sample who actually provide useable data. Should be reported for all surveys.
response variable: Measures an outcome of a study. Also known as a dependent variable.
restricted–range problem: A phenomenon in which the range of values for the explanatory variable in a regression can have a large impact on the strength of the relationship.
risk: Ratio of compared proportions that help predict that something (usually bad) will happen.
robust measure: Another term for a resistant measure.
robust: A statistical inference procedure is called this if the required probability calculations are insensitive to violations of the assumptions made.
row variable: value that label the rows that run across a two–way table.
s chart: Used to measure some quantitative characteristic of a process and monitors variation within individual samples.
sample means: Averages of observations. They are less variable than individual observations and are more Normal than individual observations.
sample proportion: Inference about a population proportion p from an SRS of size n is based on this, which is denoted p̂. When n is large, p̂ has approximately the Normal distribution with mean p and standard deviation √p(1 — p)/n.
sample space: Denoted S, the set of all possible outcomes of a random phenomenon.
sample: A part of the population that we actually examine to gather information.
sampling distribution of a sample mean: If a population has the N(µ,σ) distribution, then the sample mean x–bar of n independent observations has the N(µ,σ / √n) distribution.
sampling distribution: The probability distribution of a random variable.
sampling distributions of a count: A population contains proportion p of successes. If the population is much larger than the sample, the count X of successes in an SRS of size n has approximately the binomial distribution B(n,p). The accuracy of this approximation improves as the size of the population increases relative to the size of the sample.
sampling variability: The basic fact in which the value of a statistic varies in repeated random sampling.
sampling with replacement: The process in which an observation is put back before drawing the next observation after one randomly draws an observation from the original sample.
scatterplot smoother: A tool to examine the relationship between two quantitative variables by fitting a smooth curve to the data.
scatterplot: Shows the relationship between two quantitative variables measured on the same individuals. The values of one variable appear on the horizontal axis, and the values of the other variable appear on the vertical axis. Each individual in the data appears as the point in the plot fixed by the values of both variables for that individual.
sign test for matched pairs: Ignore pairs with difference 0; the number of trials n is the count of the remaining pairs. The test statistic is the count X of pairs with a positive difference. P–values for X are based on the binomial B(n, 1/2) distribution.
sign test: The simplest distribution–free, or nonparametric, procedure. It uses probability calculations that are correct for a wide range of population distributions.
significance level: Commonly denoted α, a fixed value that we regard as decisive in a test of significance, and that we can compare our calculated P–value to. This amounts to announcing in advance how much evidence against H0 we will require to reject H0.
significance tests for a proportion: Use this test when the sample size n is so large that both np0 and n(1 — p0) are 10 or more. Draw an SRS of size n from a large population that contains an unknown proportion p of successes. To test the hypothesis H0: p = p0, compute the z statistic p̂ — p0 / √ p0(1 — p0) / n.
simple linear regression: Type of regression where we assume that the means all lie on a line when plotted against x. Assumes that for each value of x the observed values of the response variable y are Normally distributed with a mean that depends on x.
simple random sample: Abbreviated SRS, a sample of size n that consists of n individuals from the population chosen in such a way that every set of n individuals has an equal chance to be the sample actually selected.
Simpson's paradox: An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group.
simulation: The process in which one can imitate taking many samples by using random digits to emulate chance behavior.
skewed: Description of a distribution if the right tail (larger values) is much longer than the left tail (smaller values), or vice versa.
slope: Denoted b1 in the straight line equation of the form y = b0 + b1x, the amount by which y changes when x increases by one unit.
smoothing parameter: measurable factor by which the amount of smoothing can be varied.
special cause variation: In the language of statistical quality control, when the normal functioning of a process that is in control is disturbed by some unpredictable event, the process has this type of variation.
square of the correlation: Denoted r2, the fraction of the variation in the values of y that is explained by the least–squares regression of y on x.
squared multiple correlation: Interpreted as the proportion of the variability in the response variable y that is explained by the explanatory variables x1, x2,...,xp in a multiple linear regression. Given by the equation: R2 = SSM/SST.
standard deviation: Denoted s, measures the spread of a distribution by looking at how far the observations are from their mean. It is also the square root of the variance.
standard error: The result when the standard deviation of a statistic is estimated from the data.
standard Normal distribution: The Normal distribution N(0,1) with mean 0 and standard deviation 1.
statistic: A number that describes a sample and computed from the sample data.
statistical control: A variable that continues to be described by the same distribution when observed over time is said to be in this, or simply in control.
statistical inference: The process in which one infers conclusions about larger populations from data on selected individuals.
statistical significance: If the P–value is as small or smaller than α, we say that the data are statistically significant at level α.
statistically significant: An observed effect so large that it would rarely occur by chance is called this.
stemplot: A display of the distribution of a quantitative variable that separates each observation into a stem and a one–digit leaf.
stratified random sample: Sample that first divides the population into groups of similar individuals, called strata. Then a separate SRS is chosen in each stratum and these combined SRSs form the full sample.
subjects: Name given to experimental units who are human beings.
subpopulation: A subset of a population that shares one or more additional properties.
sum of squares: Abbreviated SS, represents variation present in the responses. Calculated by summing squared deviations.
symmetric: Description of a distribution if the values smaller and larger than its midpoint are mirror images of each other.
t distribution: Suppose that an SRS of size n is drawn from an N(µ,σ) population. Then the one–sample t statistic has the t distribution with n — 1 degrees of freedom. There is a t distribution for every positive degrees of freedom k. All are symmetric distributions similar in shape to Normal distributions. The t(k) distribution approaches the N(0,1) distribution as k increases.
t procedures: These procedures are relatively robust against non–Normal populations. They are useful for non–Normal data when 15 ≤ n < 40 unless the data show outliers or strong skewness. When n ≧ 40, these procedures can be used even for clearly skewed distributions.
table of random digits: A list of the digits 0,1,2,3,4,5,6,7,8,9 that has the following properties: 1. The digit in any position in the list has the same chance of being any one of 0,1,2,3,4,5,6,7,8,9. 2. The digits in different positions are independent in the sense that the value of one has no influence on the value of any other.
test of significance: Designed to assess the strength of the evidence against the null hypothesis.
test statistic: Measures compatibility between the null hypothesis and the data in a test of significance. We use it for the probability calculation that we need for our test of significance. It is a random variable with a distribution that we know.
three–way table: Presents data on three categorical variables, printed as separate two–way tables for each level of the third variable.
time plot: Display of the distribution of a variable that plots each observation against the time at which it was measured. Always put time on the horizontal scale of the plot and the variable you are measuring on the vertical scale.
treatment group: In an experiment, the group that receives the treatment that is being evaluated.
treatment: A specific experimental condition applied to experimental units.
tree diagram: A tool that, when drawn, organizes the use of the multiplication and addition rules in problems with several stages.
trimmed mean: Calculated by discarding a certain percent of the observations from each tail and taking the mean of the remaining observations.
two–sample problems: Have the following characteristics: 1. The goal of inference is to compare the responses in two groups. 2. Each group is considered to be a sample from a distinct population. 3. The responses in each group are independent of those in the other group.
two–sample t confidence interval: Suppose that an SRS of size n1 is drawn from a Normal population with unknown mean µ1 and that an independent SRS of size n2 is drawn from another Normal population with unknown mean µ2. The confidence interval for µ1 — µ2 has confidence level at least C no matter what the population standard deviations may be.
two–sample t significance test: To test the hypothesis H0: µ1 = µ2, (where two SRSs are drawn from two Normal populations with unknown means µ1 and µ2), compute the two–sample t statistic and use P–values or critical values for the t(k) distribution, where the value of the degrees of freedom k is approximated by software or by the smaller of n1 — 1 and n2 — 1.
two–sample z statistic: When independent SRSs of sizes n1 and n2 are drawn from two Normal populations with parameters µ1,σ1 and µ2,σ2, this statistic has the N(0,1) sampling distribution.
two–sided alternative: An alternative hypothesis that says there is an effect or difference in either direction.
two–sided significance test: Rejects a hypothesis H0: µ = µ0 exactly when the value µ0 falls outside a level 1 — α confidence interval for µ.
two–way ANOVA: Used to compare population means when populations are classified according to two factors. Assumes that the populations are Normal with possibly different means and the same standard deviation and that independent SRSs are drawn from each population.
two–way table: Organizes data about two categorical variables. Often used to summarize large amounts of data by grouping outcomes into categories.
Type I error: If we reject H0 (accept Ha) when in fact H0 is true.
Type II error: If we fail to reject H0 when in fact Ha is true.
unbiased estimator: A statistic used to estimate a parameter if the mean of its sampling distribution is equal to the true value of the parameter being estimated.
undercoverage: Occurs when some groups in the population are left out of the process of choosing the sample.
uniform distributions: Continuous probability distributions that are very similar to equally likely discrete distributions.
union: The event that at least one of any collection of events occurs.
variability of a statistic: Described by the spread of its sampling distribution. The spread is determined by the sampling design and the sample size n.
variable: A characteristic of a case.
variance: Denoted s2. In a set of observations, this is the average of the squares of the deviations of the observations from their mean.
Venn diagram: A picture that shows the sample space S as a rectangular area and events as areas within S.
voluntary response sample: Consists of people who choose themselves for a sample by responding to a general appeal. This type of sample is biased because people with strong opinions, especially negative opinions, are most likely to respond.
Wald statistic: Another name for the z test statistic that tests the hypothesis H0: β1 = 0, using the fact that z has a distribution that is approximately the standard Normal distribution when the null hypothesis is true. It is denoted b1/SEb1.
Wilcoxon rank sum test: Compares two distributions to assess whether one has systematically larger values than the other. Based on the Wilcoxon rank sum statistic W, which is the sum of the ranks of one of the samples. This test can replace the two–sample t test.
Wilcoxon signed rank statistic: Denoted W+, the sum of the ranks of the positive (or negative) differences when we rank the absolute values of the differences.
Wilcoxon signed rank test: Applies to matched pairs studies. This tests the null hypothesis that there is no systematic difference within pairs against alternatives that assert a systematic difference (either one–sided or two–sided). This test is based on the Wilcoxon signed rank statistic W+.
x–bar chart: Used to measure some quantitative characteristic of a process and monitors variation from sample to sample.
z–score: A standardized value that tells us how many standard deviations the original observation falls away from the mean, and in what direction. Observations larger than the mean are positive when standardized, and observations smaller than the mean are negative.