The AP Statistics formula sheet, annotated

These are the formulas College Board provides on both sections of the AP Statistics exam (from the official CED effective Fall 2026), with a plain-English note on what each one is for. You do not memorize these; you learn to pick the right one fast.

Descriptive statistics

xˉ=1n∑xi=∑xin\bar{x} = \frac{1}{n}\sum x_i = \frac{\sum x_i}{n}

The sample mean: add every value, divide by how many there are. x-bar estimates the population mean mu.

s=∑(xi−xˉ)2n−1s = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n-1}}

The sample standard deviation: typical distance of values from the mean. Note the n minus 1, not n. See the step-by-step guide.

y^=a+bx\hat{y} = a + bx

The least-squares regression line: a is the y-intercept, b is the slope, y-hat is the predicted response.

Probability and distributions

P(A∪B)=P(A)+P(B)−P(A∩B)P(A \cup B) = P(A) + P(B) - P(A \cap B)

The addition rule: subtract the overlap so it is not counted twice.

P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}

Conditional probability: the chance of A given that B happened.

μX=E(X)=∑xi⋅P(xi)\mu_X = E(X) = \sum x_i \cdot P(x_i)

The mean (expected value) of a discrete random variable: each outcome weighted by its probability.

σX=∑(xi−μX)2⋅P(xi)\sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}

The standard deviation of a discrete random variable.

P(X=x)=(nx)px(1−p)n−xP(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

Binomial probability: exactly x successes in n independent trials with success probability p. Valid for x = 0, 1, 2, ..., n.

μX=np\mu_X = np

Mean of a binomial random variable.

σX=np(1−p)\sigma_X = \sqrt{np(1-p)}

Standard deviation of a binomial random variable.

Sampling distributions and inference

standardized test statistic=statistic−parameterstandard error of the statistic\text{standardized test statistic} = \frac{\text{statistic} - \text{parameter}}{\text{standard error of the statistic}}

The universal test-statistic recipe: how many standard errors the sample result sits from the null value.

CI=statistic±(critical value)(standard error of the statistic)\text{CI} = \text{statistic} \pm (\text{critical value})(\text{standard error of the statistic})

The universal confidence-interval recipe. Every interval on the exam is this one sentence.

χ2=∑(Observed−Expected)2Expected\chi^2 = \sum \frac{(\text{Observed} - \text{Expected})^2}{\text{Expected}}

The chi-square statistic: total mismatch between observed and expected counts.

For proportions

μp^=p,σp^=p(1−p)n,SEp^=p^(1−p^)n\mu_{\hat{p}} = p, \qquad \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}, \qquad SE_{\hat{p}} = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

The sampling distribution of a sample proportion: centered at p, with spread shrinking as n grows.

SEp^1−p^2=p^1(1−p^1)n1+p^2(1−p^2)n2SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}

Standard error for the difference between two sample proportions (used for confidence intervals).

SEp^1−p^2=p^c(1−p^c)1n1+1n2,p^c=n1p^1+n2p^2n1+n2SE_{\hat{p}_1 - \hat{p}_2} = \sqrt{\hat{p}_c(1-\hat{p}_c)}\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}, \qquad \hat{p}_c = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}

When a test assumes p1 = p2, pool the samples into p-hat-c and use this standard error instead.

For means

μxˉ=μ,σxˉ=σn,SExˉ=sn\mu_{\bar{x}} = \mu, \qquad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}, \qquad SE_{\bar{x}} = \frac{s}{\sqrt{n}}

The sampling distribution of a sample mean: centered at mu, spread shrinking by the square root of n. This is the Central Limit Theorem's playing field.

SExˉ1−xˉ2=s12n1+s22n2SE_{\bar{x}_1 - \bar{x}_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}

Standard error for the difference between two sample means.

The exam also provides Table A (standard normal probabilities), which we host as a searchable page at /z-table, and t-distribution critical values, at /t-table. To practice choosing formulas, start with which statistical test should I use.