AP®StatisticsCollege BoardRevision NotesCourse OverviewCourse OverviewFormula Sheet

Formula Sheet (College Board AP® Statistics): Revision Note

Written by: Dan Finlay

Reviewed by: Roger B

Updated on 4 June 2026

Formulas given in the exam

I. Descriptive statistics

Mean
- $\bar{x} = \frac{1}{n} \sum x_{i} = \frac{\sum x_{i}}{n}$
Standard deviation
- $s = \sqrt{\frac{1}{n - 1} \sum {(x_{i} - \bar{x})}^{2}} = \sqrt{\frac{\sum {(x_{i} - \bar{x})}^{2}}{n - 1}}$
Linear regression model
- $\hat{y} = a + b x$

II. Probability and distributions

Probability rules
- $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
- $P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
Probability distributions

Probability Distribution	Mean	Standard Deviation
Discrete random variable, $X$	$μ_{X} = E (X) = \sum x_{i} \cdot P (x_{i})$	$σ_{X} = \sqrt{\sum {(x_{i} - μ_{X})}^{2} \cdot P (x_{i})}$
If $X$ has a binomial distribution with parameters $n$ and $p$ , then: $P (X = x) = (\binom{n}{x}) p^{x} {(1 - p)}^{n - x}$ where $x = 0, 1, 2, 3, \dots, n$	$μ_{X} = n p$	$σ_{X} = \sqrt{n p (1 - p)}$

Probability Distribution

Mean

Standard Deviation

Discrete random variable, $X$

$μ_{X} = E (X) = \sum x_{i} \cdot P (x_{i})$

$σ_{X} = \sqrt{\sum {(x_{i} - μ_{X})}^{2} \cdot P (x_{i})}$

If $X$ has a binomial distribution with parameters $n$ and $p$ , then:

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

where $x = 0, 1, 2, 3, \dots, n$

$μ_{X} = n p$

$σ_{X} = \sqrt{n p (1 - p)}$

III. Sampling distributions and inferential statistics

Standardized test statistic
- $\frac{statistic - parameter}{standard error of the statistic}$
Confidence interval
- $statistic \pm (critical value) (standard error of the statistic)$
Chi-square statistic
- $χ^{2} = \sum \frac{{(Observed Count - Expected Count)}^{2}}{Expected Count}$

Sampling distributions for proportions

Sample Statistic	Mean	Standard Deviation	Standard Error
For one population: $\hat{p}$	$μ_{\hat{p}} = p$	$σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}}$	$S E_{\hat{p}} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$
For two populations: ${\hat{p}}_{1} - {\hat{p}}_{2}$	$μ_{{\hat{p}}_{1} - {\hat{p}}_{2}} = p_{1} - p_{2}$	$σ_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{\frac{p_{1} (1 - p_{1})}{n_{1}} + \frac{p_{2} (1 - p_{2})}{n_{2}}}$	$S E_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$ When $p_{1} = p_{2}$ is assumed: $S E_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{{\hat{p}}_{c} (1 - {\hat{p}}_{c}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}$ where ${\hat{p}}_{c} = \frac{n_{1} {\hat{p}}_{1} + n_{2} {\hat{p}}_{2}}{n_{1} + n_{2}}$

Sample Statistic

Mean

Standard Deviation

Standard Error

For one population: $\hat{p}$

$μ_{\hat{p}} = p$

$σ_{\hat{p}} = \sqrt{\frac{p (1 - p)}{n}}$

$S E_{\hat{p}} = \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$

For two populations: ${\hat{p}}_{1} - {\hat{p}}_{2}$

$μ_{{\hat{p}}_{1} - {\hat{p}}_{2}} = p_{1} - p_{2}$

$σ_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{\frac{p_{1} (1 - p_{1})}{n_{1}} + \frac{p_{2} (1 - p_{2})}{n_{2}}}$

$S E_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{\frac{{\hat{p}}_{1} (1 - {\hat{p}}_{1})}{n_{1}} + \frac{{\hat{p}}_{2} (1 - {\hat{p}}_{2})}{n_{2}}}$ When $p_{1} = p_{2}$ is assumed:

$S E_{{\hat{p}}_{1} - {\hat{p}}_{2}} = \sqrt{{\hat{p}}_{c} (1 - {\hat{p}}_{c}) (\frac{1}{n_{1}} + \frac{1}{n_{2}})}$ where ${\hat{p}}_{c} = \frac{n_{1} {\hat{p}}_{1} + n_{2} {\hat{p}}_{2}}{n_{1} + n_{2}}$

Sampling distributions for means

Sample Statistic	Mean	Standard Deviation	Standard Error
For one population: $\bar{x}$	$μ_{\bar{x}} = μ$	$σ_{\bar{x}} = \frac{σ}{\sqrt{n}}$	$S E_{\bar{x}} = \frac{s}{\sqrt{n}}$
For two populations: ${\bar{x}}_{1} - {\bar{x}}_{2}$	$μ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = μ_{1} - μ_{2}$	$σ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}$	$S E_{{\bar{x}}_{1} - {\bar{x}}_{2}} = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

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