Glossary

Every key term introduced across the fifteen weekly units, gathered here with one concise definition and organized alphabetically. Symbols are defined where they appear; the Course Formula Sheet collects the matching formulas.

1A¶

• addition rule (disjoint) — For disjoint events, $P(A \text{ or } B) = P(A) + P(B)$.

• addition rule (general) — For any two events, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$; the last term removes double-counted outcomes.

• alternative hypothesis ($H_A$) — The claim you suspect might be true, stated with $<$, $>$, or $\neq$.

• approximately normal — Close enough to bell-shaped and symmetric (mean near median, no long one-sided tail) that the Normal model gives trustworthy probabilities.

• association — Two categorical variables are associated when the conditional proportions of one change noticeably across categories of the other.

2B¶

• bar chart — A graph of categorical frequencies or relative frequencies as bar heights; the y-axis must start at 0 to be read honestly.

• bias — A sampling method’s systematic tendency to miss the true population value; it does not shrink with a bigger sample.

3C¶

• case (observation) — One row of a data table; one individual or unit.

• categorical (qualitative) variable — A variable that sorts cases into groups: nominal (no natural order) or ordinal (a natural order).

• Central Limit Theorem (CLT) — For a large enough sample, the sampling distribution of a sample proportion or mean is approximately Normal, centered at the true parameter, regardless of the population’s shape.

• cluster sample — Randomly selected whole groups (clusters), with every member of a chosen cluster included.

• complement ($A^c$) — The event “$A$ does not happen”; $P(A^c) = 1 - P(A)$.

• conditional proportion — The proportion of one variable’s categories within a single row or column of the other.

• confidence interval — A range of plausible values for a parameter, built as point estimate ± margin of error; for a mean, $\bar{x} \pm t^{*}\, s/\sqrt{n}$.

• confidence level — The long-run success rate of the interval-building procedure (e.g., about 95% of 95% intervals capture the true parameter).

• confounding variable — A variable tangled up with both the explanatory and response variables, clouding a causal read.

• continuous random variable — A random variable that can take any value along a continuum, so its values cannot be listed in a table.

• critical value ($z^{*}$ or $t^{*}$) — The number of standard errors reached on each side of a point estimate for a stated confidence level; $z^{*}$ = 1.645 / 1.960 / 2.576 for 90% / 95% / 99%, while $t^{*}$ is read from a t-table by degrees of freedom.

• cutoff / threshold — The specific data value marking a probability boundary (e.g., “the score for the top 15%”).

4D¶

• degrees of freedom ($df$) — For one-mean inference, $df = n - 1$; it fixes the exact shape of the t-distribution.

• density curve — A smooth curve for a continuous random variable where area under the curve equals probability; it is never negative and its total area is 1.

• deviation — How far a value sits from the mean, $x - \mu$; can be positive or negative.

• discrete probability distribution — A table of every possible value $x$ of a discrete random variable with its probability $P(x)$; always $\sum P(x) = 1$.

• disjoint (mutually exclusive) events — Events that cannot both happen on the same outcome, so $P(A \text{ and } B) = 0$.

5E¶

• empirical rule (68–95–99.7 rule) — For a Normal model, about 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD.

• event — A subset of the sample space; something that either happens or not on a given trial.

• expected value ($E(X)$, $\mu$) — $\sum x \cdot P(x)$; the probability-weighted, long-run average of a random variable.

• experiment — A study in which researchers deliberately assign subjects to treatments.

• explanatory variable / response variable — The possible cause, and the outcome it might explain.

6F¶

• five-number summary — Minimum, $Q_1$, median, $Q_3$, maximum.

• frequency — A raw count of observations in a category.

7I¶

• independent events — Events for which knowing whether one happened gives no information about the other; then $P(A \text{ and } B) = P(A)P(B)$.

• interquartile range (IQR) — $Q_3 - Q_1$; the spread of the middle 50% of the data.

• invNorm(area, μ, σ) — TI-83/84 function returning the value with a given area to its left (the inverse-normal / percentile problem); matches xqnorm() in R.

8J¶

• joint probability ($P(A \text{ and } B)$) — The probability that both $A$ and $B$ happen.

• joint proportion — The proportion of all observations in one specific cell of a two-way table (both variables at once).

9L¶

• Law of Large Numbers — As repetitions grow, the observed relative frequency of an event settles toward its true probability.

10M¶

• margin of error (ME) — The distance an interval reaches on either side of the point estimate; $ME = (\text{critical value}) \times SE$.

• marginal proportion — A proportion for one variable read from a table’s row or column total, ignoring the other variable.

• mean ($\bar{x}$) — The arithmetic average, $\sum x_i / n$; pulled toward outliers and skew.

• median — The middle value of sorted data; resistant to outliers and skew.

• misleading graph — A graph (e.g., one with a truncated y-axis) that distorts the visual comparison without changing the underlying numbers.

• mode — The most frequently occurring value.

• multiplication rule (independent events) — $P(A \text{ and } B) = P(A) \times P(B)$, valid only when the events are independent.

11N¶

• normal distribution ($N(\mu, \sigma)$) — A symmetric, unimodal, bell-shaped density curve centered at $\mu$ with spread controlled by $\sigma$.

• normalcdf(lower, upper, μ, σ) — TI-83/84 function returning the area (probability) between two cutoffs on a Normal curve; matches xpnorm() in R.

• null hypothesis ($H_0$) — The skeptical “nothing new” claim about a parameter, stated as an equality and assumed true unless the data give strong evidence otherwise.

• null proportion ($p_0$) / null mean ($\mu_0$) — The specific value claimed under $H_0$; $p_0$ (not $\hat{p}$) is used inside a proportion test’s standard error.

• numerical (quantitative) variable — A variable whose values can be meaningfully averaged: discrete (counted) or continuous (measured on a scale).

12O¶

• observational study — A study in which subjects are measured, not assigned, to groups.

• one-proportion z-interval / z-test — Ch 6 procedures for estimating ($\hat{p} \pm z^{*}SE$) or testing ($z = (\hat{p} - p_0)/\sqrt{p_0(1-p_0)/n}$) a single population proportion.

• one-sample t-test — Tests $H_0: \mu = \mu_0$ using $t = (\bar{x} - \mu_0)/(s/\sqrt{n})$ with $df = n - 1$.

• one-sided / two-sided test — A test whose $H_A$ points in one direction ($<$ or $>$) versus either direction ($\neq$).

• outlier — A value below $Q_1 - 1.5\cdot IQR$ or above $Q_3 + 1.5\cdot IQR$.

13P¶

• parameter — A numerical summary of a population (usually unknown); written with Greek letters, e.g. $\mu$, $p$.

• percentile — The value below which a stated percentage of the data falls.

• placebo / blinding — A placebo is a fake treatment that looks real; blinding keeps subjects (and sometimes researchers) unaware of who received which treatment.

• point estimate — A single statistic used as the best guess at a parameter (e.g., $\hat{p}$ for $p$, $\bar{x}$ for $\mu$).

• population — The entire group a study wants to learn about.

• population proportion ($p$) — The true share of a population that has some trait; a parameter, usually unknown.

• probability ($P(A)$) — A number from 0 (never) to 1 (always) measuring how likely event $A$ is; interpreted here as a long-run relative frequency.

• p-value — The probability of a test statistic at least this extreme, assuming $H_0$ is true.

14Q¶

• quartiles ($Q_1$, $Q_3$) — The 25th and 75th percentiles.

15R¶

• random assignment — Randomly deciding which treatment each subject receives; distinct from random sampling.

• random variable ($X$) — A variable that assigns a number to the outcome of a random process.

• range — Maximum minus minimum.

• relative frequency — A category’s count divided by the total $n$; a proportion or percent.

• resistant (robust) statistic — A statistic that outliers and skew barely change (median, IQR); the mean and SD are not resistant.

16S¶

• sample — The subset of a population actually observed.

• sample mean ($\bar{x}$) — The average of a sample; the point estimate of $\mu$.

• sample proportion ($\hat{p}$) — $x/n$, the share of a sample with a trait; the point estimate of $p$.

• sample space ($S$) — The complete list of possible outcomes of a process.

• sample standard deviation ($s$) — $\sqrt{s^2}$; the typical distance of a value from $\bar{x}$, in original units.

• sample variance ($s^2$) — The average squared distance from the mean, $\sum(x_i - \bar{x})^2/(n-1)$.

• sampling distribution — The distribution of a statistic’s value across every possible sample of a given size.

• sampling variability — The natural tendency of a statistic to differ from sample to sample.

• segmented (stacked) bar chart / mosaic plot — Graphs that show conditional proportions within each category, useful for spotting association.

• ShadeNorm(lower, upper, μ, σ) — TI-83/84 draw command that shades the requested region under a Normal curve.

• significance level ($\alpha$) — The pre-chosen risk of a wrong rejection; commonly 0.05.

• simple random sample (SRS) — A sample drawn so that every possible sample of size $n$ has an equal chance of selection.

• skewed distribution — An asymmetric distribution with a longer tail on one side (right-skewed: mean > median).

• standard deviation of a random variable ($\sigma$, $SD(X)$) — $\sqrt{\sum(x - \mu)^2 P(x)}$; the typical distance an outcome falls from $\mu$.

• standard error (SE) — The standard deviation of a sampling distribution; $SE_{\hat{p}} = \sqrt{\hat{p}(1-\hat{p})/n}$ and $SE_{\bar{x}} = s/\sqrt{n}$.

• standard error of the mean ($SE_{\bar{x}}$) — $s/\sqrt{n}$; how much sample means typically vary from the true population mean.

• standard normal distribution ($N(0,1)$) — The Normal model with mean 0 and SD 1; what any Normal distribution becomes once standardized.

• standardizing — Converting a value (or a whole distribution) to z-scores so it can be compared on a common scale.

• statistic — A numerical summary of a sample, computed from data; written with Roman letters, e.g. $\bar{x}$, $\hat{p}$.

• statistically significant — The decision reached when the p-value is less than $\alpha$.

• stratified sample — Random sampling done separately within predefined subgroups (strata).

• success–failure condition — The check ($n\hat{p} \ge 10$ and $n(1-\hat{p}) \ge 10$, or $np_0 \ge 10$ and $n(1-p_0) \ge 10$ for a test) that justifies a Normal-based procedure for a proportion.

• symmetric distribution — Roughly a mirror image on both sides of the center.

17T¶

• t-distribution — A bell-shaped curve like the Normal but with heavier tails, used for one-mean inference when $\sigma$ is unknown; indexed by its degrees of freedom.

• test statistic — A standardized measure (here $z$ or $t$) of how far a sample statistic is from the value $H_0$ claims, in standard-error units.

• TInterval / T-Test — TI-83/84 menu items (under STAT ▸ TESTS) for a one-mean confidence interval and hypothesis test.

• t.test() / tsum.test() / zsum.test() — R functions for one-mean inference: t.test() from raw data, tsum.test() from summary statistics, and zsum.test() as the large-sample z equivalent.

• treatment / control group — The condition applied to a group; the control group gets no treatment (or a standard one) for comparison.

• two-way (contingency) table — A table cross-tabulating two categorical variables.

• Type I error — Rejecting a true $H_0$; occurs with probability $\alpha$ by design.

• Type II error — Failing to reject a false $H_0$.

18U¶

• unbiased — Describes a statistic whose sampling distribution is centered on the true parameter (it does not systematically run high or low).

• union (“or”) — The event that at least one of $A$ or $B$ happens.

• unimodal / bimodal — A distribution with one peak, or with two peaks.

19V¶

• variable — One column of a data table; a characteristic that can vary from case to case.

• variance ($\sigma^2$, $s^2$) — The (probability-)weighted average of squared deviations; its square root is the standard deviation.

20W¶

• weighted average — An average in which some values count more because they are multiplied by a weight (e.g., a probability) before summing.

• which-test decision language — The habit of reading a scenario for its variable type (categorical / numerical) and its verb (estimate / test) before choosing a procedure.

21Z¶

• z-score — $z = (x - \mu)/\sigma$; how many standard deviations a value is above (positive) or below (negative) the mean.

• z-table — A printed table of cumulative areas $P(Z < z)$ under the standard Normal curve, indexed by $z$ to two decimals.