AP®StatisticsCollege BoardStudy GuidesCourse OverviewCourse OverviewFormula Sheet

Formula Sheet (College Board AP® Statistics): Study Guide

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: ${\bar{x}}_{1} - {\bar{x}}_{2}$	$μ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = μ_{1} - μ_{2}$	$σ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}$	$S E_{{\bar{x}}_{1} - {\bar{x}}_{2}} = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}$

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

I would just like to say a massive thank you for putting together such a brilliant, easy to use website.I really think using this site helped me secure my top gradesin science and maths. You really did save my exams! Thank you.

Beth
IGCSE Student

This website is soooo useful and I can’t ever thank you enough for organising questions by topic like this. Furthermore, the name of the website could not have been more appropriate as it literally did SAVE MY EXAMS!

Fathima
A Level Student

Incredible! SO worth my money, the revision notes have everything I need to know and are so easy to understand. I actually enjoy revising! It makes me feel a lot more confident for my GCSEs in a few months.

Kate
GCSE Student

Absolutely brilliant, both my girls used it for A levels and GCSE. It's saves on paper copies, also beneficial exam questions ranked from easy to hard. It's removed a lot of stress from the exams.

Sameera
Parent

Just to say that your resources are the best I have seen and I have been teaching chemistry at different levels for about 40 years

Mark
Chemistry Teacher

Excellent