AP Statistics Reference

AP Statistics provides the formula sheet and tables in both sections of the exam, so the memorization burden is unlike most AP math: the sheet hands you the formulas, and the points come from knowing what each one is for, which conditions it needs, and how to interpret the result. This page follows the sheet's own organization and flags what is worth knowing cold anyway. It covers the redesigned course (first exam May 2027); confirm the exact provided materials on AP Central.

Section 1

Descriptive Statistics

Center, spread, and the least-squares line. All of these are printed on the provided sheet; what to know cold is what each one measures and how to say so.

$$ \bar{x} = \dfrac{\sum x_i}{n} $$

Sample mean

On the sheet. The balance point of the data. It is not resistant, so a single outlier drags it; the median is the resistant measure of center.

Unit 1 · One-variable data

$$ s_x = \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n - 1}} $$

Sample standard deviation

On the sheet, dividing by n - 1, not n. Know cold: s carries the same units as the data, is never negative, and equals zero only when every value is identical.

Unit 1 · One-variable data

$$ \hat{y} = b_0 + b_1 x $$

Least-squares regression line

On the sheet, written with a hat. Know cold: the hat means PREDICTED. Writing y without it conflates a prediction with an observed value, and the difference is the residual.

Unit 5 · Regression analysis

$$ b_1 = r\,\dfrac{s_y}{s_x} $$

Slope of the regression line

On the sheet. The interpretation to know cold: the predicted change in y for each 1-unit increase in x, stated in both variables' own units.

Unit 5 · Regression analysis

$$ b_0 = \bar{y} - b_1\,\bar{x} $$

Intercept of the regression line

On the sheet. The equivalent fact worth knowing cold: the least-squares line always passes through the point of averages, (x-bar, y-bar).

Unit 5 · Regression analysis

$$ r = \dfrac{1}{n-1} \sum \left( \dfrac{x_i - \bar{x}}{s_x} \right)\!\left( \dfrac{y_i - \bar{y}}{s_y} \right) $$

Correlation

On the sheet, and never computed by hand on the exam: the calculator gives r. Know cold: r is unitless, runs from -1 to 1, measures LINEAR association only, and correlation is not causation.

Unit 5 · Regression analysis

Section 2

Probability and Random Variables

The sheet prints the rules. The exam tests whether you can tell mutually exclusive from independent, and P(A given B) from P(B given A).

$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$

Addition rule

On the sheet. The subtraction removes the double count of outcomes in both events. Only when A and B are mutually exclusive does that last term vanish.

Unit 2 · Probability

$$ P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} $$

Conditional probability

On the sheet. The condition shrinks the sample space to B. Know cold: P(A given B) and P(B given A) are different questions with different answers.

Unit 2 · Probability

$$ P(A \cap B) = P(A)\,P(B) \;\;\text{(independent)} $$

Multiplication for independent events

The conditional-probability line with P(A given B) = P(A). Know cold: independence must be checked or stated, never assumed, and it is NOT the same as mutually exclusive.

Unit 2 · Probability

$$ \mu_X = \sum x_i\,p_i $$

Mean of a discrete random variable

On the sheet. A probability-weighted average: the long-run average outcome over many, many repetitions, not the most likely single outcome.

Unit 2 · Random variables

$$ \sigma_X = \sqrt{\textstyle\sum (x_i - \mu_X)^2\,p_i} $$

SD of a discrete random variable

On the sheet. Know cold: standard deviations never add. Variances of INDEPENDENT random variables add, and they still add when the variables are subtracted.

Unit 2 · Random variables

$$ \mu = np, \qquad \sigma = \sqrt{np(1-p)} $$

Binomial mean and SD

On the sheet. Know cold the binomial setting itself: a fixed number of trials, two outcomes, constant probability, independent trials. No setting, no binomial.

Unit 2 · Probability distributions

Section 3

Sampling Distributions

Statistics vary from sample to sample; these describe by how much. The most-tested distinction in the course is the spread of the data versus the spread of a statistic.

$$ \sigma_{\bar{x}} = \dfrac{\sigma}{\sqrt{n}} $$

SD of a sample mean

On the sheet. Know cold: this is the spread of x-bar across repeated samples, NOT the spread of the data, and it shrinks with the square root of n: quadruple the sample to halve it.

Unit 4 · Inference for means

$$ \sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}} $$

SD of a sample proportion

On the sheet, written with the population p. When p is unknown, the standard ERROR substitutes p-hat. For fixed n it is largest at p = 0.5.

Unit 3 · Inference for proportions

$$ n \ge 30 \;\Rightarrow\; \bar{x} \text{ approx. normal} $$

Central Limit Theorem

Not a formula on the sheet: know it cold. For large samples the sampling distribution of x-bar is approximately normal REGARDLESS of the population's shape. It says nothing about the data themselves becoming normal.

Unit 4 · Inference for means

Section 4

Inference

Every test statistic and every interval on the sheet follows one of two anatomies. Learn the anatomy and the sheet hands you the rest.

$$ \dfrac{\text{statistic} - \text{parameter}}{\text{standard error}} $$

Standardized test statistic

On the sheet as the template every z and t statistic follows: how many standard errors the sample result sits from the null value. Know cold which procedure fills each slot, and name that procedure in your answer.

Units 3-4 · Inference

$$ \text{statistic} \pm (\text{critical value}) \times (\text{standard error}) $$

Confidence interval

On the sheet as the template for every interval. Know cold the interpretation: 95 percent confidence describes the METHOD capturing the parameter in repeated sampling, not a 95 percent probability for this one interval.

Units 3-4 · Inference

$$ \chi^2 = \sum \dfrac{(\text{observed} - \text{expected})^2}{\text{expected}} $$

Chi-square statistic

On the sheet. Expected counts come from the null hypothesis, not the data, and the condition to verify with numbers is that all expected counts are at least 5.

Unit 3 · Categorical data

Knowing the fact is not the same as not slipping on it

The formula sheet is in the exam booklet; the interpretation is not. Most AP Statistics points are lost to a small set of predictable misconceptions about what these formulas mean, and the lessons target exactly those.

Open the course