AP Statistics cheat sheet

Shape, center, and spread describe a distribution; outliers and gaps are noted separately from these three features.

Mean and standard deviation are sensitive to skew and outliers; median and IQR are resistant, so use them for skewed data.

A point is a potential outlier under the 1.5 × IQR rule if it falls below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).

Standard deviation of a sample: s = sqrt( sum (x_i − xbar)^2 / (n − 1) ); z-score = (x − mean)/SD gives position in standard-deviation units.

In a simple random sample every set of n individuals has an equal chance of selection; random assignment to treatments supports causal inference.

Stratified sampling divides the population into homogeneous strata and samples each; cluster sampling selects whole heterogeneous groups.

Bias sources include voluntary response, convenience sampling, undercoverage, nonresponse, and question wording; random sampling reduces bias but not sampling variability.

In an experiment, control, randomization, and replication support causation; blinding and placebos reduce bias from subjects and evaluators.

Complement rule: P(A^c) = 1 − P(A); addition rule: P(A or B) = P(A) + P(B) − P(A and B).

Conditional probability: P(A|B) = P(A and B)/P(B); events are independent when P(A and B) = P(A)P(B), equivalently P(A|B) = P(A).

For disjoint events P(A and B) = 0, so P(A or B) = P(A) + P(B); disjoint events with positive probability cannot be independent.

Mean and variance of a sum: E(X + Y) = E(X) + E(Y) always; Var(X + Y) = Var(X) + Var(Y) only when X and Y are independent.

For a linear transformation: E(aX + b) = aE(X) + b and SD(aX + b) = |a|SD(X).

Binomial setting: fixed n independent trials, two outcomes, constant success probability p; mean np, SD sqrt(np(1 − p)).

Geometric setting: independent trials, constant p, X counts trials until first success; P(X = k) = (1 − p)^(k−1)p, mean 1/p.

A probability distribution is valid when every probability is between 0 and 1 and the probabilities sum to 1; the mean is sum of x·P(x).

One-proportion z-interval: phat ± z* sqrt(phat(1 − phat)/n); one-proportion z-test statistic z = (phat − p0)/sqrt(p0(1 − p0)/n).

Large-sample conditions: for a test, np0 ≥ 10 and n(1 − p0) ≥ 10; for a confidence interval, use phat counts of at least 10 successes and 10 failures.

For samples, the 10% condition requires n ≤ 10% of the population when sampling without replacement.

Two-proportion z-test: z = (phat1 − phat2)/sqrt(phat_pooled(1 − phat_pooled)(1/n1 + 1/n2)), with phat_pooled combining both samples.

Two-proportion confidence interval uses separate phat values in the standard error and does not pool.

Chi-square goodness-of-fit tests whether a categorical variable follows a claimed distribution; expected counts are n times hypothesized proportions.

Chi-square statistic = sum of (observed − expected)^2/expected over all cells; df = (rows − 1)(columns − 1) for independence and k − 1 for goodness of fit.

All expected counts should be at least 5 for chi-square procedures; the test is always right-tailed because large chi-square values indicate poor fit.

One-sample t-interval: xbar ± t* s/sqrt(n), with df = n − 1; t* comes from the t distribution with that df.

One-sample t-test statistic: t = (xbar − mu0)/(s/sqrt(n)); the t distribution is symmetric, bell-shaped, and heavier-tailed than normal.

Conditions: random sampling, independence (10% condition), and normality of the population or a large sample (n ≥ 30) via the central limit theorem.

A matched-pairs design analyzes the differences within pairs with a one-sample t procedure on the difference values.

Two-sample t-test statistic: t = (xbar1 − xbar2)/sqrt(s1^2/n1 + s2^2/n2); df is estimated by technology, not simply n1 + n2 − 2.

A two-sample t-interval uses the same standard error with a t* based on the smaller of n1 − 1 and n2 − 1 as a conservative df.

A p-value is the probability, assuming the null hypothesis is true, of a statistic at least as extreme as observed; it is not the probability the null is true.

Type I error rejects a true null (probability alpha); Type II error fails to reject a false null; power = 1 − P(Type II error) and increases with larger n or larger effect size.

Least-squares regression line: yhat = a + bx, with slope b = r(sy/sx) and intercept a = ybar − b·xbar.

The line passes through (xbar, ybar); the slope estimates the predicted change in y for a one-unit increase in x.

Correlation r measures strength and direction of linear association, is unitless, and lies between −1 and 1; r^2 is the fraction of variation in y explained by the linear model.

Residual = observed − predicted (y − yhat); the residual plot should show no pattern for a linear model to be appropriate.

Inference for slope uses t = (b − beta0)/SE_b with df = n − 2; conditions are linearity, independence, Normal residuals, equal variance, and random selection.

A confidence interval for slope is b ± t* SE_b; if the interval excludes 0, the test of H0: beta = 0 is rejected at the same alpha.

Influential points can change the regression line markedly; high leverage depends on x-values, and an outlier in y with high leverage is especially influential.

Extrapolation predicts outside the observed range of x and is unreliable; association does not imply causation without random assignment or strong design.