Glossary

This glossary defines the core vocabulary of MATH 2200. Terms are listed alphabetically, each with a plain-language definition and, where one exists, the symbol we use for it in this book. The first time a term appears in a chapter it is defined there too; this page is the single place to look it up again later. The chapter where a term is introduced is shown in parentheses.

1Conventions for symbols¶

We follow the standard convention that Greek letters denote population parameters (fixed, usually unknown) and Roman letters denote sample statistics (computed from data, used to estimate the parameter).

2A¶

Addition rule — ($P(A \cup B) = P(A) + P(B) - P(A \cap B)$) The probability that at least one of two events occurs; the overlap is subtracted once so it is not double-counted. (Ch. 4)

Alternative hypothesis — ($H_A$) The claim a study is trying to find evidence for: that there is an effect, a difference, or a relationship. It is the logical opposite of the null hypothesis and is chosen before looking at the data. (Ch. 8)

ANOVA (analysis of variance) — A procedure that compares the means of three or more groups with a single test, by comparing the variation between the group means to the variation within the groups using an $F$-statistic. (Ch. 12)

Association — Two variables are associated when knowing the value of one tells you something about the likely value of the other (their conditional distributions differ). Association is not the same as causation. (Ch. 3, Ch. 13)

3B¶

Bar chart — A plot with one bar per category whose height encodes the category’s count or proportion; bars have gaps and may be reordered. (Ch. 3)

Between-group sum of squares (SSB) — ($\text{SSB} = \sum_j n_j(\bar{x}_j-\bar{x})^2$) The total squared distance of each group mean from the grand mean, weighted by group size; the “signal” piece of variation in ANOVA. (Ch. 12)

Bias — A systematic tendency for a study or estimate to miss the truth in a particular direction (for example, a survey that over-counts one group). Bias is about direction, not random luck, and more data cannot fix it. (Ch. 1)

Binomial model — (Binomial$(n, p)$) The model for the number of successes in a fixed number $n$ of independent trials, each with the same success probability $p$. (Ch. 5)

BINS conditions — The four requirements for a binomial model: Binary trials, Independent trials, fixed Number of trials, and the Same success probability on each trial. (Ch. 5)

Bonferroni correction — A multiple-comparisons fix that tests each of $m$ comparisons at the stricter level $\alpha/m$. (Ch. 12)

Bootstrap — A method for estimating how much a statistic would vary by resampling, with replacement, from the data you already have, thousands of times; the middle 95% of the resampled statistics is a 95% bootstrap confidence interval. (Ch. 7)

Boxplot — A plot of the five-number summary: a box from $Q_1$ to $Q_3$ with a median line, whiskers to the extreme non-outlier values, and points for outliers. (Ch. 2)

4C¶

Categorical variable — A variable that records which category an observation falls into (for example, crop type or county region), as opposed to a number you can meaningfully average. (Ch. 1)

Causation — A cause-and-effect relationship in which changing one variable produces a change in another; established by a well-run randomized experiment, not by observational data alone. (Ch. 1)

Census — A study that measures every unit in the population rather than a sample. (Ch. 1)

Central Limit Theorem (CLT) — The result that the sampling distribution of the sample mean (or proportion) becomes approximately Normal, centered at the parameter with spread $\sigma/\sqrt{n}$, as the sample size grows — whatever the shape of the population, provided observations are independent. (Ch. 6)

Chi-square statistic — ($\chi^2$) A measure of how far a table of observed category counts falls from the counts expected under a null hypothesis, computed as $\sum (O-E)^2/E$ over all cells; larger values give stronger evidence against the null. (Ch. 11)

Cluster sampling — A method that divides the population into clusters, randomly selects whole clusters, and measures every unit within the chosen clusters. (Ch. 1)

Coefficient of determination — ($R^2$) The fraction of the variation in the response explained by the regression line; for one predictor it equals $r$ squared. (Ch. 13)

Coefficient of variation (CV) — ($\text{CV}=s/\bar{x}$) A unitless measure of relative spread, the standard deviation divided by the mean (often reported as a percent). (Ch. 2)

Complement — ($A^c$, with $P(A^c) = 1 - P(A)$) The event that a given event does not occur. (Ch. 4)

Conditional probability — ($P(A \mid B) = P(A \cap B)/P(B)$) The probability of one event given that another has occurred. (Ch. 4)

Conditional proportion — ($p_{j\mid i} = O_{ij}/R_i$) The share within a given row or column of a two-way table, using that margin’s total as the denominator. (Ch. 3)

Confidence interval (CI) — A range of plausible values for an unknown parameter, of the form $\text{estimate} \pm \text{margin of error}$, built so that a stated percentage (the confidence level, e.g. 95%) of intervals built this way would capture the true parameter over many samples. (Ch. 7)

Confidence level — The stated percentage (e.g. 90%, 95%, 99%) of confidence intervals, built by a given procedure, that would contain the true parameter over many samples; higher levels give wider intervals. (Ch. 7)

Confounding (confounding variable) — A third variable, tied to both the explanatory and the response variable, that offers an alternative explanation for an observed association; the main reason observational studies cannot prove causation. (Ch. 1)

Contingency table — See two-way table: a table of counts cross-classifying observations by two categorical variables. (Ch. 3, Ch. 11)

Continuous random variable — ($X$) A random variable that can take any value in an interval, such as a measured length or weight. (Ch. 5)

Continuous variable — A numerical variable that can take any value in a range, such as a measurement like PM2.5 concentration. (Ch. 1)

Convenience sample — A sample of whoever is easiest to reach; fast and cheap but the most bias-prone, never a sound basis for conclusions about a population. (Ch. 1)

Correlation coefficient — ($r$) A unit-free number between -1 and +1 measuring the strength and direction of a linear relationship between two numerical variables. Values near 0 mean little linear association; $r$ says nothing about non-linear patterns or causation. (Ch. 13)

Count — ($n_i$) The number of observations falling in a category. (Ch. 3)

Coverage — The long-run proportion of confidence intervals built by a procedure that actually contain the true parameter; for a correct 95% method the coverage is about 95%. (Ch. 7)

Cramér’s V — ($V = \sqrt{\chi^2 / (n \cdot \min(r-1, c-1))}$) An effect-size measure for a chi-square test reporting the strength of an association on a 0 (none) to 1 (perfect) scale. (Ch. 11)

Critical value — ($t^\star$, $z^\star$) The multiplier in a confidence interval ($t^\star$ from the $t$-distribution for a mean, $z^\star$ from the Normal for a proportion) chosen so the interval is wide enough for the desired confidence level (e.g. $z^\star = 1.96$ for 95%). (Ch. 7, Ch. 9, Ch. 10)

Cumulative percent — The running total of percents down a (usually sorted) frequency table, reaching 100% at the last row. (Ch. 3)

5D¶

Dataset — An organized table of data in which each row is an observation and each column is a variable. (Ch. 1)

Degrees of freedom — ($df$) The parameter that controls the shape (tail-heaviness) of a $t$- or chi-square distribution. For a one-sample or paired $t$-test, $df = n-1$; for chi-square goodness-of-fit, $df = k-1$; for chi-square independence, $df = (r-1)(c-1)$. (Ch. 10, Ch. 11)

Density plot — A smoothed curve that traces the shape of a numerical distribution without depending on bin edges. (Ch. 2)

Deviation — ($x_i - \bar{x}$) The signed distance of one observation from the mean. (Ch. 2)

Discrete random variable — ($X$) A random variable that takes separated, listable values, such as a count. (Ch. 5)

Discrete variable — A numerical variable that takes countable, separated values, such as a count of monitors or people. (Ch. 1)

Disjoint (mutually exclusive) events — ($P(A \cap B) = 0$) Two events that cannot both occur. (Ch. 4)

Distribution — The pattern of values a variable takes: which values occur, and how often. We describe it by its shape, center, and spread. (Ch. 2)

6E¶

Effect size — A measure of the magnitude of an effect that does not depend on sample size, such as Cohen’s $d$ (the difference in means in pooled-standard- deviation units). (Ch. 8)

Empirical probability — ($\hat{P}(A) = n_A/n$) A data-based estimate of a true probability, equal to the number of times an event occurred divided by the number of observations. (Ch. 4)

Empirical rule — Another name for the 68–95–99.7 rule. (Ch. 5)

Eta-squared — ($\eta^2 = \text{SSB}/\text{SST}$) An ANOVA effect size giving the proportion of total variation explained by the grouping. (Ch. 12)

Event — ($A$) A collection of outcomes of interest; a subset of the sample space. (Ch. 4)

Expected count — ($E$) The number of observations a cell would contain if the null hypothesis were true; for goodness-of-fit $E_i = n p_i$, for independence $E_{ij} = R_i C_j / n$. (Ch. 11)

Expected-count condition — The requirement that every expected count be at least 5 for the chi-square approximation to be trustworthy. (Ch. 11)

Expected value — ($E(X)$, or $\mu$) The probability-weighted (long-run) average value of a random variable — the balancing point of its distribution. (Ch. 5)

Experiment — A study in which the researcher assigns units to treatment conditions, ideally at random; random assignment is what lets an experiment support a causal conclusion. (Ch. 1)

Explanatory variable — ($x$) The variable suspected of influencing another (also called the predictor or independent variable, plotted on the $x$-axis). Its partner is the response variable. (Ch. 1, Ch. 13)

Extrapolation — Using a regression line to predict beyond the observed range of the explanatory variable, where the linear pattern may not hold. (Ch. 13)

7F¶

Family-wise error rate — The probability of at least one false positive across a whole family of tests; approximately $1-(1-\alpha)^m$ for $m$ independent tests. (Ch. 12)

F-distribution — The right-skewed sampling distribution of the $F$-statistic under the null hypothesis, indexed by two degrees of freedom $(k-1,\,N-k)$. (Ch. 12)

Fitted (predicted) value — ($\hat{y}$) The value the regression line gives for a chosen $x$; the “hat” denotes an estimate, not an observation. (Ch. 13)

Five-number summary — The minimum, $Q_1$, median, $Q_3$, and maximum of a numerical variable. (Ch. 2)

Frequency table — A table listing each category of a categorical variable with its count, proportion, and percent. (Ch. 3)

F-statistic — ($F = \text{MSB}/\text{MSW}$) The ratio of between-group to within-group mean squares used in ANOVA; large values are evidence against equal means. (Ch. 12)

8G¶

Goodness-of-fit test — A chi-square test of whether the counts of one categorical variable match a hypothesized set of proportions, using $df = k-1$ for $k$ categories. (Ch. 11)

Grand mean — ($\bar{x}$) The mean of all observations pooled across every group. (Ch. 12)

9H¶

High-leverage point — An observation with an extreme explanatory ($x$) value, giving it the potential to pull the regression line toward itself. (Ch. 13)

Histogram — A graph that shows the distribution of a numerical variable by grouping values into equal-width bins and drawing a bar for the count in each bin. (Ch. 2)

Homogeneity of variance — The ANOVA condition that the groups share roughly equal population variances; checked by the max÷min group-SD ratio (rule of thumb $< 2$). (Ch. 12)

Hypothesis test — A formal procedure for deciding whether sample data give convincing evidence against a null hypothesis, by asking how surprising the data would be if the null were true (the p-value). (Ch. 8)

10I¶

Independence — Two events are independent when the occurrence of one does not change the probability of the other ($P(A\mid B) = P(A)$, equivalently $P(A\cap B) = P(A)P(B)$). Observations are independent when one unit’s value gives no information about another’s — a core condition for almost every inference procedure. (Ch. 4)

Independence (categorical) — The absence of association: every conditional distribution in a two-way table is identical to the marginal distribution. (Ch. 3)

Influential point — A point whose removal noticeably changes the slope or intercept of a regression; typically a high-leverage point that is also off the trend. (Ch. 13)

Intercept — ($b_0$) The predicted value of the response when the explanatory variable equals zero; meaningful only if $x=0$ lies within the data range. (Ch. 13)

Interpolation — Using a regression line to predict for an $x$ value that lies inside the observed range of the data, where the fitted line is on firmer ground (contrast extrapolation). (Ch. 13)

Interquartile range (IQR) — ($\text{IQR} = Q_3 - Q_1$) The spread of the middle 50% of the data: the third quartile minus the first quartile. Resistant to outliers. (Ch. 2)

Intersection — ($A \cap B$) The event that both of two events occur (“A and B”). (Ch. 4)

11J¶

Joint proportion — ($p_{ij} = O_{ij}/n$) The count in one cell of a two-way table divided by the grand total; the share of everything in that combination. (Ch. 3)

12L¶

Law of Large Numbers — The principle that an empirical probability settles down toward the true probability as the number of observations grows. (Ch. 4)

Least-squares regression line — ($\hat{y}=b_0+b_1x$) The unique straight line that minimizes the sum of squared residuals; it always passes through the point of averages $(\bar{x},\bar{y})$. (Ch. 13)

LINE conditions — The four assumptions of regression inference: Linearity, Independence, Normal residuals, and Equal spread of residuals. (Ch. 13)

Long-run-frequency probability — ($\hat{P}(A) = n_A/n$) A probability viewed as the fraction of repetitions in which an event occurs over many repeats of a process; estimated from data. (Ch. 4)

13M¶

Margin of error (ME) — ($\text{ME} = (\text{critical value}) \times SE$) The “±” part of a confidence interval: how far the interval reaches on each side of the estimate. (Ch. 7, Ch. 9)

Marginal proportion — ($p_{i\cdot}=R_i/n$, $p_{\cdot j}=C_j/n$) The share in one category of a single variable, ignoring the other, read from a row or column total of a two-way table. (Ch. 3)

Mean — ($\mu$ for a population, $\bar{x}$ for a sample) The arithmetic average: add the values and divide by how many there are, $\bar{x} = \frac{1}{n}\sum x_i$. Sensitive to outliers. (Ch. 2)

Mean difference — ($\bar{d}$) The average of the within-pair differences in a paired design, the quantity a paired $t$-test tests against zero. (Ch. 10)

Mean square between (MSB) — ($\text{MSB}=\text{SSB}/(k-1)$) SSB divided by its degrees of freedom; an estimate of variance that includes any real mean differences. (Ch. 12)

Mean square within (MSW) — ($\text{MSW}=\text{SSW}/(N-k)$) SSW divided by its degrees of freedom; an estimate of the common within-group variance. (Ch. 12)

Median — ($M$) The middle value when the data are sorted (the 50th percentile); half the observations fall below it and half above. Resistant to outliers, so it is often preferred for skewed data. (Ch. 2)

Missing value — A value that was not recorded for an observation, written NA (“not available”) in R; how and why a value is missing can itself bias results. (Ch. 1)

Monte Carlo simulation — Estimating a distribution or quantity by drawing many random samples on a computer and summarizing the results. (Ch. 6)

Mosaic plot — A display of a two-way table in which rectangle areas are proportional to cell counts. (Ch. 3)

Multiple-comparisons problem — The inflation of the overall false-positive rate that occurs when many hypothesis tests are run on the same data. (Ch. 12)

Multiplication rule — ($P(A \cap B) = P(B)\,P(A \mid B)$) The probability that both events occur; for independent events this simplifies to $P(A)\,P(B)$. (Ch. 4)

14N¶

Nominal variable — A categorical variable whose categories have no inherent order, such as major or crop type. (Ch. 1)

Non-response bias — Bias that arises when the people who decline to participate differ systematically from those who respond. (Ch. 1)

Normal distribution (Normal model) — ($N(\mu, \sigma)$) A symmetric, bell-shaped distribution described entirely by its mean $\mu$ and standard deviation $\sigma$. The 68–95–99.7 rule says about 68%, 95%, and 99.7% of values lie within 1, 2, and 3 standard deviations of the mean. (Ch. 5)

Null distribution — The distribution of a test statistic across many simulated worlds in which the null hypothesis is true; the yardstick against which the observed statistic is compared. (Ch. 8)

Null hypothesis — ($H_0$) The “nothing is going on” claim a test assumes true while weighing the evidence: no effect, no difference, no relationship. A test never proves $H_0$; it either rejects it or fails to reject it. (Ch. 8)

Null standard error — ($SE_0 = \sqrt{p_0(1-p_0)/n}$) The standard error of $\hat{p}$ computed under the null hypothesis, using the hypothesized value. (Ch. 9)

Numerical variable — A variable whose values are numbers it makes sense to do arithmetic on (for example, PM2.5 concentration or yield per acre). May be discrete (counts) or continuous (measurements). (Ch. 1)

15O¶

Observation — A single entity you have data on (also called a case or unit); one row of a dataset, such as one Kern census tract. (Ch. 1)

Observational study — A study in which researchers observe and measure without assigning treatments. It can reveal association but, because of possible confounding, generally cannot establish causation. (Ch. 1)

Observed count — ($O$, or $O_{ij}$ for a two-way table) The actual number of observations falling in a category or table cell, taken directly from the data. (Ch. 11)

Okabe–Ito palette — The eight-color, colorblind-safe qualitative palette used for every figure in this book. (Ch. 3)

Omnibus test — A single overall test (such as the ANOVA $F$-test) of whether any difference exists among several groups, prior to locating which groups differ. (Ch. 12)

One-proportion z-test — A hypothesis test comparing a single sample proportion to a hypothesized value using a $z$-statistic. (Ch. 9)

One-sample t-test — (t.test(~x, data=) / tsum.test) A hypothesis test (and matching confidence interval) comparing the mean of one group of measurements to a fixed hypothesized value $\mu_0$, using $t = (\bar{x} - \mu_0)/(s/\sqrt{n})$. (Ch. 10)

One-sided test — ($H_A: \mu_1 - \mu_2 > 0$ or $< 0$) A test whose alternative specifies a direction, legitimate only when the direction is justified before seeing the data. (Ch. 8)

One-way ANOVA — ANOVA with a single categorical grouping variable (one factor) and a numerical response. (Ch. 12)

Ordinal variable — A categorical variable whose categories have a natural order, such as class standing (Freshman < Sophomore < Junior < Senior). (Ch. 1)

Outlier — An observation that lies unusually far from the bulk of the data; the boxplot rule flags points beyond $Q_1 - 1.5\,\text{IQR}$ or $Q_3 + 1.5\,\text{IQR}$. Outliers can heavily distort the mean, standard deviation, and a regression line, so they are always investigated, never quietly deleted. (Ch. 2, Ch. 13)

16P¶

Paired t-test — (t.test(~diff, data=) on the differences) A one-sample $t$-test applied to the within-pair differences of matched data; used when each observation in one group is matched one-to-one with one in the other. (Ch. 10)

Parameter — A fixed (usually unknown) number describing a population, such as $\mu$ or $p$, typically denoted by a Greek letter. Inference uses sample statistics to estimate parameters. (Ch. 6)

Percent — ($100\,p_i$) A proportion expressed out of 100. (Ch. 3)

Percentile — A value below which a stated percentage of the data fall (e.g. the 90th percentile). (Ch. 2)

Permutation test — Another name for a randomization test: the null distribution comes from permuting (shuffling) the observed labels or values. (Ch. 8)

Pooled proportion — ($\hat{p}_{\text{pool}} = (x_1+x_2)/(n_1+n_2)$) The combined success rate across two groups, used in the two-proportion test. (Ch. 9)

Pooled standard error — ($\sqrt{\hat{p}_{\text{pool}}(1-\hat{p}_{\text{pool}})(1/n_1+1/n_2)}$) The standard error for a two-proportion test built from the pooled proportion. (Ch. 9)

Population — The entire collection of individuals or items we want to draw a conclusion about. (Ch. 1)

Population distribution — The distribution of a variable’s values across every member of the population. (Ch. 6)

Population proportion — ($p$) The true, usually unknown, fraction of the whole population with a given trait. (Ch. 9)

Pearson correlation coefficient — See correlation coefficient ($r$): the full name of the standard correlation measure, distinguishing it from rank-based alternatives. (Ch. 13)

Point of averages — ($(\bar{x}, \bar{y})$) The point made of the mean of $x$ and the mean of $y$; the least-squares regression line always passes through it. (Ch. 13)

Population slope — ($\beta_1$) The unknown true slope of the linear relationship that the sample slope $b_1$ estimates. (Ch. 13)

Positive / negative association — Two numerical variables are positively associated when large values of one tend to occur with large values of the other, and negatively associated when large values of one occur with small values of the other. (Ch. 13)

Power — The probability of correctly rejecting a false null hypothesis; equals $1 - \beta$ and rises with larger samples and larger true effects. (Ch. 8)

Practical significance — Whether an effect is large enough to matter in the real world, judged by the effect size and confidence interval rather than the p-value. (Ch. 8)

Probability — ($P(A)$) A number between 0 and 1 measuring how likely an event is to occur; 0 means impossible and 1 means certain. (Ch. 4)

Probability distribution — ($P(X = x)$) The list of values a discrete random variable can take together with the probability of each. (Ch. 5)

Proportion (relative frequency) — ($p$ for a population, $\hat{p}$ for a sample; $p_i = n_i/n$) The fraction of a group that has a particular characteristic; a value between 0 and 1. Equivalently the mean of 0/1 (success/failure) data. (Ch. 3, Ch. 9)

p-value — The probability of getting a result at least as extreme as the one observed, if the null hypothesis were true. A small p-value means the data would be surprising under $H_0$, which counts as evidence against it. It is not the probability that $H_0$ is true. (Ch. 8)

17Q¶

Quantile — The value below which a stated fraction of a distribution falls; the inverse of a cumulative probability, found in R with qnorm. (Ch. 5)

Quartiles — ($Q_1$, $Q_2 =$ median, $Q_3$) The three values that split sorted data into four equal-sized parts (the 25th, 50th, and 75th percentiles). (Ch. 2)

18R¶

Random process — Any action or experiment whose outcome is not known in advance, such as rolling a die or measuring a randomly chosen day’s air quality. (Ch. 4)

Random variable — ($X$) A numerical outcome of a random process, before we know its value (for example, the result of rolling a die). Discrete if its values are listable, continuous if it can be any value in an interval. (Ch. 5)

Randomization test — A test that builds the null distribution of a statistic by repeatedly shuffling the group labels and recomputing the statistic, valid when assignment to groups is (under the null) exchangeable. (Ch. 8)

Range — ($\max - \min$) The distance from the smallest to the largest value. (Ch. 2)

Reproducibility — The property that anyone re-running your code on the same data obtains the same result; in R, achieved for random procedures by fixing the seed with set.seed(). (Ch. 1)

Repetition (rep) — One simulated sample (draw a sample, compute the statistic) in building a simulated sampling distribution. (Ch. 6)

Residual — ($e = y - \hat{y}$) The vertical gap between an observed value and the value a model predicts for it; what the model got “wrong” for that point. Least squares minimizes the sum of these squared. (Ch. 13)

Residual standard error — ($s_e=\sqrt{\text{SSE}/(n-2)}$) The typical size of a residual, reported in the units of the response. (Ch. 13)

Resistant statistic — A summary (such as the median or IQR) whose value changes little when a few extreme observations are added or removed. (Ch. 2)

Response variable — ($y$) The outcome variable a study measures, suspected of being influenced by the explanatory variable (plotted on the $y$-axis). (Ch. 1, Ch. 13)

Row total / column total / grand total — ($R_i$, $C_j$, $n$) The sum across a row, down a column, or over the whole two-way table. (Ch. 3)

19S¶

Sample — The subset of the population we actually collect data on. (Ch. 1)

Sample distribution — The distribution of the values in a single observed sample of size $n$. (Ch. 6)

Sample mean — ($\bar{x} = \frac{1}{n}\sum x_i$) The arithmetic average of the observations in a sample. (Ch. 6)

Sample proportion — ($\hat{p}$) The fraction of “successes” in a sample of a yes/no variable; used to estimate the population proportion. (Ch. 6, Ch. 9)

Sample space — ($S$) The set of all possible outcomes of a random process. (Ch. 4)

Sampling bias — A systematic tendency for a sampling method to over- or under-represent part of the population, tilting estimates in a fixed direction that more data cannot fix. (Ch. 1)

Sampling distribution — The distribution of a statistic (like $\bar{x}$ or $\hat{p}$) across all possible samples of a given size $n$. Its spread is the standard error; the CLT describes its shape. (Ch. 6)

Scope of inference — The limits on what a study’s conclusions can claim, set by two design choices: random assignment (allows causal claims) and random sampling (allows generalization to a population). (Ch. 1)

Seed — (set.seed(k)) A fixed starting value for a random-number generator that makes a simulation reproducible. (Ch. 1, Ch. 4)

Segmented (stacked) bar chart — A bar chart in which each bar is divided by a second categorical variable; at 100% height it shows the conditional distribution within each group. (Ch. 3)

Significance level — ($\alpha$) The threshold, chosen in advance (commonly 0.05), below which a p-value leads us to reject the null hypothesis; also the probability of a Type I error. (Ch. 8)

Simple random sample — A sample in which every unit of the population has an equal chance of being selected. (Ch. 1)

Simulation — Estimating a probability (or a sampling distribution) by using a computer to generate many repetitions of a random process and recording the fraction in which an event occurs. (Ch. 4)

Skew — Asymmetry in a distribution; right- (positive-) skew has a longer tail toward large values, left- (negative-) skew toward small values. (Ch. 2)

Slope — ($b_1$) The predicted change in the response for a one-unit increase in the explanatory variable; shares the sign of the correlation. (Ch. 13)

Standard deviation — ($\sigma$ for a population, $s$ for a sample) A measure of spread: roughly, the typical distance of a value from the mean; the square root of the variance, in the original units. (Ch. 2)

Standard deviation (of a random variable) — ($\sigma$) The square root of the variance of a random variable, the spread in the original units. (Ch. 5)

Standard error (SE) — The standard deviation of a statistic’s sampling distribution — how much the estimate would bounce around from sample to sample. For a mean, $SE = s/\sqrt{n}$; for a proportion, $\sqrt{p(1-p)/n}$. (Ch. 6)

Standard error of the mean — ($SE = s/\sqrt{n}$) The standard deviation of the sampling distribution of the sample mean. (Ch. 10)

Standard error of the slope — ($\text{SE}(b_1)=s_e/\sqrt{S_{xx}}$) How much the estimated slope would vary from sample to sample. (Ch. 13)

Standard Normal — ($N(0, 1)$) The Normal model of z-scores, with mean 0 and standard deviation 1. (Ch. 5)

Standardized residual — ($r_{ij} = (O_{ij}-E_{ij})/\sqrt{E_{ij}}$) The signed, scaled gap between a cell’s observed and expected counts; values beyond about $\pm 2$ flag cells that drive a significant chi-square result. (Ch. 11)

Statistic — A number computed from a sample (such as $\bar{x}$, $s$, or $\hat{p}$), used to estimate a parameter. (Ch. 6)

Statistical significance — A result is statistically significant when its p-value falls below $\alpha$, meaning it is unlikely to be chance alone. It does not by itself mean the effect is large or important in practice. (Ch. 8)

Stratified sampling — A method that divides the population into groups (strata) and randomly samples some units from every stratum, guaranteeing each group is represented. (Ch. 1)

Student’s t-distribution — ($t$) A bell-shaped distribution, a little heavier in the tails than the Normal, used for inference about a mean when the population standard deviation is unknown and estimated by $s$. Its exact shape is set by its degrees of freedom. (Ch. 10)

Success–failure condition — The check that there are at least 10 expected successes and at least 10 expected failures ($np \ge 10$ and $n(1-p) \ge 10$) needed for the Normal approximation to a proportion. (Ch. 6, Ch. 9)

Sum of squared errors (SSE) — The total of the squared residuals, the quantity least squares minimizes. (Ch. 13)

20T¶

t-distribution — See Student’s t-distribution. (Ch. 10)

Test of independence — A chi-square test of whether the two categorical variables of a two-way table are associated, using $df = (r-1)(c-1)$ for an $r \times c$ table. (Ch. 11)

Test statistic — A single number computed from the data that measures how far the sample falls from what the null hypothesis predicts (for example, a difference in group means $D$). (Ch. 8)

Total sum of squares (SST) — ($\text{SST} = \text{SSB} + \text{SSW}$) The total squared variation of all observations about the grand mean. (Ch. 12)

Tukey HSD (Honest Significant Difference) — An ANOVA-specific procedure that adjusts all pairwise group comparisons jointly to control the family-wise error rate. (Ch. 12)

Two-proportion z-test — A hypothesis test for whether two population proportions differ. (Ch. 9)

Two-sample t-test (Welch) — (t.test(y ~ g, data=)) A hypothesis test (and confidence interval) comparing the means of two independent groups without assuming equal variances, using $t = ((\bar{x}_1 - \bar{x}_2) - \mu_0)/\sqrt{s_1^2/n_1 + s_2^2/n_2}$. (Ch. 10)

Two-sided test — ($H_A: \mu_1 - \mu_2 \neq 0$) A test whose alternative allows a difference in either direction. (Ch. 8)

Two-way table (contingency table) — ($O_{ij}$) A table of counts cross-classifying observations by two categorical variables simultaneously. (Ch. 3, Ch. 11)

Type I error — Rejecting a true null hypothesis (a “false alarm”); its probability is $\alpha$. (Ch. 8)

Type II error — Failing to reject a false null hypothesis (a “miss”); its probability is denoted $\beta$. (Ch. 8)

21U¶

Unbiased estimator — A statistic whose sampling distribution is centered on the parameter it estimates (so on average it hits the target). (Ch. 6)

Union — ($A \cup B$) The event that at least one of two events occurs (“A or B”). (Ch. 4)

Unpooled standard error — The standard error for a confidence interval on a difference of proportions, using each group’s own $\hat{p}_i$. (Ch. 9)

22V¶

Variable — Any characteristic recorded for each observation in a dataset. Variables are either numerical or categorical. (Ch. 1)

Variance — ($s^2$ for a sample, $\sigma^2$ for a random variable) The average squared deviation from the mean. For a sample, $s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2$; for a random variable, $E(X^2) - \mu^2$. (Ch. 2, Ch. 5)

23W¶

Wald standard error — ($\sqrt{\hat{p}(1-\hat{p})/n}$) The standard error of $\hat{p}$ computed from the data, used for a confidence interval. (Ch. 9)

Welch ANOVA — A version of one-way ANOVA that does not assume equal group variances, used when the homogeneity condition fails. (Ch. 12)

Within-group sum of squares (SSW) — ($\text{SSW} = \sum_j\sum_i (x_{ij}-\bar{x}_j)^2$) The total squared distance of each observation from its own group mean; the “noise” piece of variation in ANOVA. (Ch. 12)

24Z¶

z-score — ($z = (x - \mu)/\sigma$) How many standard deviations a value lies above (positive) or below (negative) the mean. Standardizing lets us compare values measured on different scales and read Normal probabilities. (Ch. 5)

25Symbol index¶

A quick lookup from notation to term (see the entry for the full definition).

26Glosario bilingüe (English $\leftrightarrow$ Español)¶

Para estudiantes hispanohablantes: los términos clave del curso con su traducción. For Spanish-speaking students: the course’s key terms with their Spanish translation. (CSUB es una institución al servicio de hispanos / Hispanic-Serving Institution.)