Your FREE AP Statistics Flashcards 2026 – 200+ Cards

How many standard deviations a value is from the mean: $z = \dfrac{x - \mu}{\sigma}$.

In a normal distribution, about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.

Records which group or category an individual falls into (e.g., color, brand). Summarized with counts and proportions.

Takes numerical values for which arithmetic makes sense (e.g., height, time). Summarized with center and spread.

The checklist for describing a distribution: Shape, Outliers (unusual features), Center, Spread — always stated in context.

The arithmetic average and balance point of a distribution. Pulled toward the tail of a skew and toward outliers.

The middle value of an ordered data set. Resists outliers, so it is preferred for skewed distributions.

The most frequent value(s) in a data set; a distribution can be unimodal, bimodal, or multimodal.

The difference between the maximum and minimum values: a simple, non-resistant measure of spread.

The typical distance of values from the mean: $s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}$. Larger means more spread.

The square of the standard deviation; the average squared deviation from the mean.

The spread of the middle 50% of the data: $\text{IQR} = Q_3 - Q_1$. Resistant to outliers.

Flags outliers: a value below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$ is an outlier.

Minimum, Q1, median, Q3, maximum — the basis of a boxplot.

The percentage of values in a distribution at or below a given value.

A long tail on the right (high) side; the mean is typically greater than the median.

A long tail on the left (low) side; the mean is typically less than the median.

A statistic not strongly affected by outliers (median, IQR), unlike the mean and standard deviation.

A graph of a quantitative variable's distribution using bars over equal-width intervals (bins) of values.

A display of the five-number summary; a modified boxplot also marks outliers separately.

A symmetric, bell-shaped distribution described by its mean $\mu$ and standard deviation $\sigma$.

Converting values to z-scores so distributions with different units can be compared on a common scale.

The running total of relative frequencies; an ogive plots it to read percentiles.

Adding a constant shifts center but not spread; multiplying by a constant scales both center and spread.

A graph of two quantitative variables, with the explanatory variable on the x-axis and the response on the y-axis.

The explanatory variable (x) may explain or predict changes in the response variable (y).

Measures the strength and direction of a LINEAR relationship; $-1 \le r \le 1$. The sign matches the slope.

How to describe a scatterplot association: positive/negative direction, linear/curved form, and weak/strong strength.

A strong r shows association, not cause and effect; only a randomized experiment supports a causal claim.

The line $\hat{y} = a + bx$ that minimizes the sum of squared residuals.

The predicted change in y for each one-unit increase in x; interpret it in context with units.

The predicted value of y when x = 0; sometimes not meaningful in context.

Observed minus predicted: $y - \hat{y}$. A positive residual means the model under-predicted.

A plot of residuals vs x; random scatter supports a linear model, while a curved pattern means a line is the wrong fit.

$r^2$ — the fraction of variation in y that is explained by the linear regression model.

$b = r \cdot \dfrac{s_y}{s_x}$ — slope equals the correlation times the ratio of the standard deviations.

Predicting beyond the range of the observed x-values; unreliable because the pattern may not hold there.

A point (often with extreme x) that, if removed, markedly changes the regression line or correlation.

An observation with an x-value far from the mean of x; it can strongly pull the regression line.

The typical size of a prediction error from the regression line.

A single outlier can substantially raise or lower the correlation, since r is not resistant.

A table of counts for two categorical variables; used to find joint, marginal, and conditional distributions.

The distribution of one categorical variable for a fixed category of the other ('given that').

The totals in the margins of a two-way table — the distribution of one variable ignoring the other.

The entire group of individuals about which we want information.

The subset of the population from which we actually collect data.

An attempt to collect data from every individual in the population.

A sample chosen so that every possible group of n individuals has an equal chance of being selected.

Divide the population into homogeneous strata, then take an SRS within each stratum.

Divide the population into clusters, randomly select whole clusters, and sample everyone within them.

Choose a random starting point in an ordered list, then take every kth individual after it.

Selecting individuals who are easiest to reach; prone to bias and not suitable for inference.

People choose themselves to respond; typically biased toward those with strong opinions.

A systematic tendency of a method to favor certain outcomes; a larger sample does not remove it.

When some groups in the population are left out of the sampling process.

When selected individuals cannot be contacted or refuse to participate, differing from those who do.

Inaccurate answers caused by question wording, interviewer effects, or sensitive topics.

Measures variables without imposing a treatment; can show association but not causation.

Deliberately imposes treatments on units; with random assignment it can establish cause and effect.

A lurking variable associated with both the explanatory variable and the response, blocking causal claims.

Using chance to assign units to treatments; balances out lurking variables and enables causal inference.

A baseline group (often given a placebo or standard treatment) for comparison.

A response to a fake treatment driven by belief; controlled with a placebo and blinding.

Keeping subjects (single-blind) and/or those measuring the response (double-blind) unaware of treatment assignments.

Applying each treatment to many units so chance differences average out.

Grouping units by a known variable before random assignment to reduce its effect on the comparison.

Random sampling lets you generalize to a population; random assignment lets you infer causation.

The long-run relative frequency of an outcome over many repetitions of a random process; between 0 and 1.

As the number of trials grows, the proportion of successes approaches the true probability.

$P(A^c) = 1 - P(A)$ — the probability an event does not happen.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Cannot occur at the same time, so $P(A \cap B) = 0$.

One occurring does not change the other's probability, so $P(A \cap B) = P(A)P(B)$.

Disjoint events are usually NOT independent — if one happens, the other can't, so probabilities change.

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ — the probability of A given that B occurred.

$P(A \cap B) = P(A) \cdot P(B \mid A)$.

Events A and B are independent if $P(A \mid B) = P(A)$ (equivalently $P(A \cap B) = P(A)P(B)$).

A variable whose numerical value is determined by the outcome of a random process.

Discrete takes countable values (whole numbers); continuous takes any value in an interval.

$\mu_X = \sum x \cdot P(x)$ — the long-run average outcome.

$\sigma_X^2 = \sum (x - \mu_X)^2 P(x)$; the standard deviation is its square root.

$\mu_{X+Y} = \mu_X + \mu_Y$ — means always add.

For independent variables, variances ADD for both sums and differences: $\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$.

Binary outcomes, Independent trials, fixed Number of trials n, Same probability p each trial.

For $X \sim B(n, p)$: mean $np$, standard deviation $\sqrt{np(1-p)}$.

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$.

Independent trials with the same success probability p, counting the number of trials until the first success.

The mean number of trials until the first success is $\dfrac{1}{p}$.

A binomial may be approximated by a normal model when $np \ge 10$ and $n(1-p) \ge 10$.

A list of all possible values of a random variable with their probabilities, which must sum to 1.

A branching display of sequential outcomes with their probabilities; multiply along branches.

If A and B are disjoint, $P(A \cup B) = P(A) + P(B)$.

For a continuous variable, probability equals area under its density curve; $P(X = x) = 0$ for a single point.

For any discrete random variable, the probabilities of all outcomes sum to exactly 1.

If $Y = a + bX$: $\mu_Y = a + b\mu_X$ and $\sigma_Y = |b|\,\sigma_X$.

For a large enough sample size, the sampling distribution of the sample mean $\bar{x}$ is approximately normal regardless of the population's shape (rule of thumb $n \ge 30$).

A number describing a population (e.g., $\mu, p, \sigma$) — usually unknown.

A number computed from a sample (e.g., $\bar{x}, \hat{p}, s$) used to estimate a parameter.

The distribution of a statistic over all possible samples of a fixed size from the population.

The standard deviation of a sampling distribution; for the sample mean it is $\dfrac{\sigma}{\sqrt{n}}$.

A statistic whose sampling-distribution mean equals the true parameter it estimates.

Mean $\mu$, standard deviation $\dfrac{\sigma}{\sqrt{n}}$; approximately normal by the CLT for large n.

Mean p, standard deviation $\sqrt{\dfrac{p(1-p)}{n}}$; approximately normal when Large Counts holds.

For p-hat to be approximately normal: $np \ge 10$ and $n(1-p) \ge 10$.

When sampling without replacement, the sample must be no more than 10% of the population for independence.

The standard error shrinks as n grows — proportional to $\dfrac{1}{\sqrt{n}}$, not $\dfrac{1}{n}$.

It lets us use normal-based inference for means even when the population is skewed, given a large enough sample.

The natural variation in a statistic from one random sample to another.

Bias = center of the sampling distribution off target; variability = how spread out the estimates are.

The distribution of $\bar{x}_1 - \bar{x}_2$ or $\hat{p}_1 - \hat{p}_2$ across all pairs of samples.

$\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$ (variances add for independent samples).

$\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$.

A sampling distribution of x-bar is approximately normal if the population is normal OR n is large (CLT).

Repeatedly drawing random samples and recording the statistic builds an approximate sampling distribution.

Rejecting a true null hypothesis — a false positive. Its probability equals the significance level $\alpha$.

Failing to reject a false null hypothesis — a false negative. Its probability is $\beta$.

The probability of getting a result at least as extreme as the one observed, assuming the null hypothesis is true.

An interval estimate, statistic plus or minus margin of error, for an unknown population parameter.

Critical value times standard error; the half-width of a confidence interval.

'We are C% confident the true parameter lies between the bounds' — C% describes the method, not one interval.

In repeated sampling, about C% of the intervals built this way would capture the true parameter.

$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.

A higher confidence level uses a larger critical value, producing a wider interval.

A larger sample lowers the standard error, producing a narrower interval.

The default claim of no effect, no difference, or no relationship that a test seeks evidence against.

The claim being investigated — that an effect, difference, or relationship exists (one- or two-sided).

The p-value cutoff for rejecting H0; it also equals the probability of a Type I error.

If $p \le \alpha$, reject H0; if $p > \alpha$, fail to reject H0 — you never 'accept' H0.

$z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$, using the hypothesized $p_0$ for the standard error.

The probability a test correctly rejects a false null hypothesis: $1 - \beta$.

Increase the sample size, the true effect size, or the significance level $\alpha$.

Lowering $\alpha$ reduces Type I errors but raises the chance of Type II errors, all else equal.

It is not the probability that H0 is true; it is the probability of the data assuming H0 is true.

A tiny effect can be statistically significant with a huge sample yet have little real-world importance.

$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

In a two-proportion z-test, the combined sample proportion $\hat{p}_c$ used to estimate the common standard error under H0.

Random sample/assignment, the 10% condition, and Large Counts (expected successes and failures at least 10).

The number of standard errors for a given confidence level (e.g., $z^* = 1.96$ for 95%).

A one-sided Ha tests a single direction (>, <); a two-sided Ha tests for any difference and doubles the tail area.

A value outside a C% two-sided interval would be rejected by the matching test at $\alpha = 1 - C$.

A normal distribution with mean 0 and standard deviation 1; z-scores follow it.

Reject or fail to reject H0, then translate it into a statement about the real-world question and variable.

State (parameter and hypotheses), Plan (procedure and conditions), Do (statistic and p-value), Conclude (in context).

The population standard deviation $\sigma$ is unknown, so we estimate it with s, requiring the t-distribution.

Bell-shaped like the normal but with heavier tails, set by its degrees of freedom; approaches normal as df grows.

$df = n - 1$ for a one-sample t procedure.

$\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}$.

$t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$.

Random sample, 10% condition for independence, and an approximately normal population or large n (CLT).

Analyzes the differences within naturally linked pairs (before/after, twins) with a one-sample t on $\mu_d$.

Two measurements tied to the same individual or matched unit signal a paired (not two-sample) design.

Compares the means of two INDEPENDENT groups; the null is that the two population means are equal.

$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$.

For a two-sample t by hand, use $df = \min(n_1 - 1, n_2 - 1)$ — the smaller sample size minus 1.

$t^* \dfrac{s}{\sqrt{n}}$; shrink it by increasing n or lowering the confidence level.

'We are C% confident the true mean lies between the bounds,' stated with the variable and units.

$H_0: \mu_d = 0$ versus an alternative $\mu_d \ne 0$ (or one-sided), where $\mu_d$ is the mean difference.

$H_0: \mu_1 = \mu_2$ — the two population means are equal (no difference).

A larger sample lowers the standard error and narrows the interval, all else equal.

How many estimated standard errors the sample mean is from the hypothesized mean.

t procedures are fairly robust to non-normality, especially for larger samples, but sensitive to strong outliers.

With small n, examine a dotplot/boxplot/normal probability plot for strong skew or outliers before using t.

$SE_{\bar{x}} = \dfrac{s}{\sqrt{n}}$ — the sample standard deviation s divided by $\sqrt{n}$.

Use z procedures for categorical proportions and t procedures for quantitative means.

Paired: one column of differences (linked data). Two-sample: two separate, independent groups.

Pairing controls for individual variation, often producing a more powerful test than two independent samples.

$\chi^2 = \sum \dfrac{(O - E)^2}{E}$, summing over all cells — how far observed counts fall from expected.

Tests whether ONE categorical variable's distribution matches a claimed distribution.

$df = (\text{number of categories}) - 1$.

$E = n \times p_i$ — sample size times the hypothesized proportion for each category.

Uses ONE sample classified by TWO categorical variables to test whether they are associated.

Compares the distribution of one categorical variable across SEVERAL populations or groups.

$df = (r - 1)(c - 1)$, where r and c are the numbers of rows and columns.

$E = \dfrac{(\text{row total})(\text{column total})}{\text{grand total}}$.

Random sample(s), the 10% condition, and ALL expected counts at least 5.

Larger $\chi^2$ means observed counts are farther from expected, giving a smaller (one-tailed) p-value.

H0: the two categorical variables are independent (no association) in the population.

H0: the distribution of the categorical variable is the same across all groups.

Each cell's contribution $\dfrac{(O-E)^2}{E}$; the largest components show where observed and expected differ most.

Independence: one sample, two variables. Homogeneity: multiple samples/groups, one variable.

Reject or fail to reject H0, then state the evidence for/against association or a difference in context.

Use chi-square for COUNTS across categories; use z for a single proportion or comparing two proportions.

Larger samples increase expected counts and the chi-square statistic, making real associations easier to detect.

Compare conditional distributions across groups to see the association a chi-square test formalizes.

Inference about the population slope $\beta$ of the true regression line relating x and y.

$H_0: \beta = 0$ — there is no linear relationship between x and y in the population.

$t = \dfrac{b}{SE_b}$ — the sample slope divided by its standard error.

$df = n - 2$, because two parameters (slope and intercept) are estimated.

$b \pm t^* \cdot SE_b$ — an interval estimate for the true slope $\beta$.

Evidence that a real linear association exists between the explanatory and response variables.

Linear relationship, Independent observations, Normal residuals, Equal variance, Random data collection.

The slope is the 'Coef'/'Estimate' on the predictor's row; 'SE Coef' is its standard error.

Reported as 's'; the typical size of a prediction error from the regression line.

The two-sided probability for the slope's t statistic, testing $H_0: \beta = 0$.

$SE_b$ measures how much the estimated slope would vary across repeated samples.

A residual plot with random scatter (no pattern) supports the linear condition for slope inference.

Residuals should show roughly constant spread across x (no fanning) for slope inference.

'We are C% confident the true slope (predicted change in y per unit x) is between the bounds,' in context.

The population line $\mu_y = \alpha + \beta x$ that the sample LSRL estimates.

A nonzero sample slope could arise from sampling variability; a test checks whether the relationship is real.