AP Statistics cram sheet: every formula and template

How the exam is scored

The AP Statistics exam is 3 hours long and taken fully digitally in Bluebook. Section I is 42 multiple-choice questions in 90 minutes and counts for 50% of your score. Section II is 4 free-response questions worth 10 points each in 90 minutes, the other 50%, so each free-response question is 12.5% of the exam.

The free-response questions come in a fixed order:

• Question 1 uses Practices 1 and 2 (formulating questions and collecting data).
• Question 2 uses Practices 3 and 4 (analyzing data and interpreting results).
• Question 3 is inference: either a hypothesis test or a confidence interval.
• Question 4 uses Practices 2, 3, and 4.

A graphing calculator with statistical capabilities is expected, and a formula sheet and tables are provided in both sections; the sheet is laid out at formulas. The multiple-choice weighting by unit is roughly Unit 1 at 20-30%, Unit 2 at 15-25%, Unit 3 at 15-25%, Unit 4 at 10-20%, and Unit 5 at 10-20%. For the full framework, see AP Statistics.

Unit 1: exploring one-variable data and collecting data

Two formulas anchor this unit: the sample mean $\bar{x} = \frac{\sum x_i}{n}$ (add the values, divide by the sample size $n$) and the sample standard deviation $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$, which divides by $n-1$ because it comes from a sample, not the whole population.

Describe any distribution by shape, center, spread, and unusual features (outliers), always in context. The $1.5 \times \text{IQR}$ rule flags a value as an outlier if it falls below $Q_1 - 1.5(\text{IQR})$ or above $Q_3 + 1.5(\text{IQR})$, where the interquartile range is $\text{IQR} = Q_3 - Q_1$. For roughly normal data, the empirical rule puts about 68% of values within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Collecting data is the other half of this unit. Know simple random, stratified, and cluster sampling, and the bias traps of undercoverage, nonresponse, and voluntary response. An experiment needs comparison, random assignment, and replication, and only a randomized experiment supports a cause-and-effect claim.

More detail: standard deviation by hand, the 1.5 IQR rule, the empirical rule, experiments vs observational studies, and Unit 1.

Unit 2: probability, random variables, and distributions

Probability rules to have cold:

• General addition: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, where $A \cup B$ means "A or B".
• Conditional: $P(A|B) = \frac{P(A \cap B)}{P(B)}$, where $A \cap B$ means "A and B".
• Independent events: $P(A \cap B) = P(A)P(B)$. Mutually exclusive (disjoint) events instead have $P(A \cap B) = 0$, which is not the same as independent.

For a discrete random variable $X$, the mean (expected value) is $\mu_X = \sum x_i \cdot P(x_i)$ and the standard deviation is $\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}$. A binomial variable (a fixed number of independent trials with the same success probability $p$) has $P(X=x) = \binom{n}{x} p^x (1-p)^{n-x}$, mean $\mu_X = np$, and standard deviation $\sigma_X = \sqrt{np(1-p)}$.

For a normal model, standardize with the z-score $z = \frac{x - \mu}{\sigma}$ (how many standard deviations $x$ sits from the mean $\mu$) and read the area from the z-table. The sampling distribution of $\bar{x}$ has mean $\mu$ and standard deviation $\sigma/\sqrt{n}$; the central limit theorem makes it approximately normal for a large sample (about $n \ge 30$), even when the population is not.

More detail: disjoint vs independent events, binomial probability, the central limit theorem, and Unit 2.

Unit 3: inference for proportions

One proportion. The standard error of the sample proportion $\hat{p}$ (read "p-hat") is $SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ for a confidence interval. A significance test instead uses the null value $p_0$ inside the root, $\sqrt{\frac{p_0(1-p_0)}{n}}$, with $z = \frac{\hat{p} - p_0}{SE}$.

Two proportions. The interval standard error is $\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$. A test pools the two samples first: $\hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$, then uses $\hat{p}_c$ in the standard error.

Chi-square (two-way tables). The statistic is $\chi^2 = \sum \frac{(O-E)^2}{E}$, where $O$ is an observed count and $E$ is the expected count $\frac{\text{row total} \times \text{column total}}{\text{grand total}}$. A test of homogeneity compares one categorical variable across several groups; a test of independence checks whether two variables are associated within one sample.

Common critical values from the z-table: 90% uses $z^* = 1.645$, 95% uses $1.96$, and 99% uses $2.576$. More detail: one-proportion vs two-proportion z test, chi-square tests explained, the proportion z-test calculator, and Unit 3.

Unit 4: inference for means

Every mean procedure uses $t$, not $z$, because you estimate the population standard deviation with the sample standard deviation $s$. For the one-sample and paired procedures the degrees of freedom are $n-1$; for two independent samples the degrees of freedom come from your calculator, or, conservatively, the smaller of $n_1 - 1$ and $n_2 - 1$.

• One sample: $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$ and $t = \frac{\bar{x} - \mu_0}{SE}$, where $\mu_0$ is the claimed mean.
• Two independent samples: $SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.
• Paired data: compute each difference first, then run a one-sample $t$ on the differences.

If the population standard deviation $\sigma$ is somehow known (rare in practice), the sampling distribution standard deviation is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$. Read the critical value $t^*$ from the t-table using your degrees of freedom.

More detail: t-test vs z-test, how to calculate a confidence interval, the one-sample t-test calculator, and Unit 4.

Unit 5: regression analysis

The least-squares regression line predicts $\hat{y} = a + bx$, where $a$ is the y-intercept and $b$ is the slope. The correlation $r$ runs from $-1$ to $1$ and measures the direction and strength of a linear association, while $r^2$ is the fraction of the variation in $y$ that the linear model with $x$ explains.

A residual is $\text{observed} - \text{predicted}$, that is $y - \hat{y}$. A residual plot with no leftover pattern supports using a straight-line model; a clear curve does not. On the Fall 2026 exam, regression is descriptive only, so you interpret the slope, $r$, $r^2$, and residuals but you do not run a significance test for the slope.

The slope and $r^2$ interpretation templates are in the section below. More detail: least-squares regression line, r vs r-squared, how to interpret residual plots, the regression calculator, and Unit 5.

The inference frame: State, Plan, Do, Conclude

Every inference free-response answer, and many multiple-choice setups, follows four steps.

1. State: name the parameter in words, then write both hypotheses about that parameter, not about the sample. Use $p$ for a population proportion or $\mu$ for a population mean.
2. Plan: name the exact procedure and check its conditions with numbers.
3. Do: show the sample statistic, the test statistic or interval, and the p-value.
4. Conclude: compare the p-value to the significance level $\alpha$ and answer in context, or interpret the interval in context.

Two formula-sheet identities drive the Do step: the standardized test statistic is $\frac{\text{statistic} - \text{parameter}}{\text{standard error}}$, and a confidence interval is $\text{statistic} \pm (\text{critical value})(\text{standard error})$. For point-by-point scoring, see the FRQ guide; to pick the right procedure, see which statistical test to use.

Condition checklists

Almost every inference procedure checks the same families of conditions. Show the numbers, not just the names, or you lose the condition point.

• Random: the data comes from a random sample or a randomized experiment.
• 10%: when sampling without replacement, the sample is at most 10% of the population, so the trials stay roughly independent.
• Large counts (proportions): $np$ and $n(1-p)$ are each at least 10. Use $p_0$ for a test and $\hat{p}$ for an interval.
• Normal or large sample (means): the population is stated to be normal, or the sample size is at least 30 so the central limit theorem applies, or a graph of the sample shows no strong skew and no outliers. This is the condition the $t$ procedures need.

Interpretation templates worth memorizing

Scorers want the result tied back to the context. Fill the brackets with the problem's own words and units.

• Confidence interval: "We are $C\%$ confident that the interval from [low] to [high] captures the true [parameter in context]." The confidence level describes the method, not one interval. See what 95% confidence means.
• P-value: "Assuming $H_0$ is true, the p-value is the probability of getting a statistic at least as extreme as the one observed." If the p-value is at most $\alpha$, reject $H_0$ and state convincing evidence for $H_a$ in context; otherwise fail to reject. See what a p-value means.
• Slope: "For each additional one [x-unit], the predicted [y] changes by $b$ [y-units]."
• r-squared: "About [$r^2$]% of the variation in [y] is explained by the linear model with [x]."

Classic point-losers

These are the mistakes that cost the most points, and every one is avoidable:

• Dividing by $n$ instead of $n-1$ for a sample standard deviation, or mixing up the sample and population formulas.
• Writing hypotheses about a sample statistic like $\hat{p}$ or $\bar{x}$ instead of the parameter $p$ or $\mu$.
• Naming a condition without showing the numbers that make it hold.
• A conclusion with no context, a sentence that could be pasted onto any problem.
• Writing "accept $H_0$"; you fail to reject it, you never accept it.
• Treating mutually exclusive events as independent, or claiming causation from an observational study.
• Confusing $r$ with $r^2$, or reporting a bare calculator command with no values shown.