AP® Statistics Beta

The reference sheet, explained

The exam gives you a reference sheet: the formulas below and three tables. Knowing what each formula means, and when to use it, is what earns the points. Written in our own words and layout.

Three habits that save points

Show the expression, then the numbers, then the answer. Write the formula's structure, substitute the values, then give the result. A calculator result alone is not work.
Name the distribution or the procedure. " $X$ is binomial with $n = 10$ and $p = 0.3$ ", "a one-sample z test for a proportion", with the boundary and the direction.
Table or technology, both count. A p-value from Table B written as a range and the same p-value from a calculator are both right; keep full precision until the end.

I. Descriptive statistics

Formula	What it is	When you use it	Learn it in
$\bar{x}$ $= \frac{1}{n}\sum x_i$ $= \frac{\sum x_i}{n}$	The sample mean: add the values and divide by how many there are.	The center of quantitative data, when the distribution is roughly symmetric with no outliers.	Unit 1 Summary Statistics for One Quantitative VariablePractice it: Data explorer
$s_x$ $= \sqrt{\frac{1}{n - 1}\sum \left(x_i - \bar{x}\right)^2}$ $= \sqrt{\frac{\sum \left(x_i - \bar{x}\right)^2}{n - 1}}$	The sample standard deviation: a typical distance of the values from their mean. Note $n - 1$ , not $n$ .	The spread of quantitative data, alongside the mean.	Unit 1 Summary Statistics for One Quantitative VariablePractice it: Data explorer
$\hat{y} = a + bx$	A regression line: $a$ is the intercept and $b$ the slope; $\hat{y}$ is a predicted value of the response.	Predicting the response from the explanatory variable, inside the range of the data.	Unit 5 Linear Regression ModelsPractice it: Regression lab

II. Probability and distributions

Formula	What it is	When you use it	Learn it in
$P(A \cup B)$ $= P(A) + P(B) - P(A \cap B)$	The addition rule: the probability that A or B (or both) happens.	Any "or" probability; subtract the overlap so it is not counted twice.	Unit 2 Independent Events and Unions of EventsPractice it: Probability trees
$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$	Conditional probability: the probability of A given that B happened.	A "given" probability, often from one row or column of a two-way table.	Unit 2 Conditional ProbabilityPractice it: Probability trees
$\mu_X$ $= E(X)$ $= \sum x_i \cdot P(x_i)$	The mean (expected value) of a discrete random variable: each value weighted by its probability.	The long-run average value of a random variable.	Unit 2 Parameters of Random Variables
$\sigma_X$ $= \sqrt{\sum \left(x_i - \mu_X\right)^2 \cdot P(x_i)}$	The standard deviation of a discrete random variable.	How much the random variable typically varies from its mean.	Unit 2 Parameters of Random Variables
$P(X = x)$ $= \binom{n}{x} p^x (1 - p)^{n - x}$	The binomial probability of exactly $x$ successes in $n$ independent trials with success probability $p$ , for $x = 0, 1, 2, \ldots, n$ .	A count of successes in a binomial setting. Name the distribution and its $n$ and $p$ first.	Unit 2 The Binomial DistributionPractice it: Distribution areas
$\mu_X$ $= np,$ $\quad \sigma_X$ $= \sqrt{np(1 - p)}$	The mean and standard deviation of a binomial random variable.	The center and spread of a binomial count, without listing the whole distribution.	Unit 2 The Binomial DistributionPractice it: Distribution areas

III. Sampling distributions and inferential statistics

Formula	What it is	When you use it	Learn it in
$\text{standardized test statistic}$ $= \frac{\text{statistic} - \text{parameter}}{\text{standard error of the statistic}}$	The shape of every z and t test statistic in the course. In a test, the parameter is its null value.	Any significance test: write the general form, then substitute.	Unit 3 Carrying Out a Test for a Population ProportionPractice it: p-value explorer
$\text{statistic}$ $\pm (\text{critical value})$ $(\text{standard error of the statistic})$	The shape of every confidence interval in the course. The product is the margin of error.	Any confidence interval: name the procedure, then substitute.	Unit 3 Constructing a Confidence Interval for a Population ProportionPractice it: Interval capture
$\chi^2$ $= \sum \frac{(\text{observed} - \text{expected})^2}{\text{expected}}$	The chi-square statistic, summed over every cell of a two-way table.	Tests for homogeneity or independence. Expected count = row total times column total over the table total.	Unit 3 Carrying Out a Chi-Square Test for Homogeneity or IndependencePractice it: Chi-square lab

III (continued). Sampling distributions for proportions

Formula	What it is	When you use it	Learn it in
$\mu_{\hat{p}}$ $= p,$ $\quad \sigma_{\hat{p}}$ $= \sqrt{\frac{p(1 - p)}{n}}$	The center and spread of the sampling distribution of a sample proportion.	Probabilities about $\hat{p}$ when $p$ is known, and in a test, where $p$ is the null value $p_0$ .	Unit 3 Sampling Distributions for Sample ProportionsPractice it: Sampling distributions
$\text{SE}_{\hat{p}}$ $= \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$	The standard error: the same formula with $\hat{p}$ in place of the unknown $p$ .	A one-proportion confidence interval (a test uses $p_0$ instead).	Unit 3 Constructing a Confidence Interval for a Population ProportionPractice it: Interval capture
$\mu_{\hat{p}_1 - \hat{p}_2}$ $= p_1 - p_2,$ $\quad \sigma_{\hat{p}_1 - \hat{p}_2}$ $= \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}}$	The center and spread of the sampling distribution of the difference of two independent sample proportions.	Probabilities about a difference of sample proportions.	Unit 3 Sampling Distributions for the Difference Between Sample ProportionsPractice it: Sampling distributions
$\text{SE}_{\hat{p}_1 - \hat{p}_2}$ $= \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$	The standard error of a difference of sample proportions.	A two-proportion confidence interval.	Unit 3 Constructing a Confidence Interval for the Difference Between Two Population ProportionsPractice it: Interval capture
$\text{SE}$ $= \sqrt{p_c(1 - p_c)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)},$ $\quad p_c$ $= \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}$	The standard error when $p_1 = p_2$ is assumed, with $p_c$ the combined proportion of successes in both samples.	A two-proportion test, where the null hypothesis says $p_1 = p_2$ .	Unit 3 Setting Up a Test for the Difference Between Two Population ProportionsPractice it: p-value explorer

III (continued). Sampling distributions for means

Formula	What it is	When you use it	Learn it in
$\mu_{\bar{x}}$ $= \mu,$ $\quad \sigma_{\bar{x}}$ $= \frac{\sigma}{\sqrt{n}}$	The center and spread of the sampling distribution of a sample mean.	Probabilities about $\bar{x}$ when $\mu$ and $\sigma$ are known.	Unit 4 Sampling Distributions for Sample MeansPractice it: Sampling distributions
$\text{SE}_{\bar{x}} = \frac{s}{\sqrt{n}}$	The standard error of a sample mean, with $s$ in place of the unknown $\sigma$ .	One-sample and paired t intervals and tests (for paired data, $s$ is the SD of the differences).	Unit 4 Constructing a Confidence Interval for a Population Mean or Population Mean Difference
$\mu_{\bar{x}_1 - \bar{x}_2}$ $= \mu_1 - \mu_2,$ $\quad \sigma_{\bar{x}_1 - \bar{x}_2}$ $= \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$	The center and spread of the sampling distribution of the difference of two independent sample means.	Probabilities about a difference of sample means.	Unit 4 Sampling Distributions for the Difference Between Two Sample MeansPractice it: Sampling distributions
$\text{SE}_{\bar{x}_1 - \bar{x}_2}$ $= \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$	The standard error of a difference of sample means. It is never pooled.	Two-sample t intervals and tests.	Unit 4 Constructing a Confidence Interval for the Difference Between Two Population Means

Reading the tables

Table A: standard normal probabilities. The area to the left of $z$ , for $z$ from $-3.4$ to $3.4$ in steps of 0.01. Round $z$ to two decimal places, find the row for the first decimal and the column for the second. For an area to the right, subtract from 1.
Table B: t critical values. One row per degrees of freedom (1 to 30, then 40, 50, 60, 80, 100, 1000 and infinity); columns are upper-tail probabilities from 0.25 to 0.0005, with confidence levels from 50% to 99.9% along the bottom. When your df is not a row, use the next lower row. A t statistic between two columns gives a range for the p-value.
Table C: chi-square critical values. One row per degrees of freedom (1 to 30, then 40, 50, 60, 80 and 100); columns are the same upper-tail probabilities as Table B, from 0.25 to 0.0005. A statistic between two columns gives a range for the p-value, such as "between 0.025 and 0.05".

Not on the sheet: know these

$z = \frac{x - \mu}{\sigma}$ . A z-score: how many standard deviations a value is from the mean. Comparisons of the Distributions for One Quantitative Variable
$1.5 \times \text{IQR}$ . The outlier rule: a value more than $1.5 \times \text{IQR}$ below $Q_1$ or above $Q_3$ is an outlier. Summary Statistics for One Quantitative Variable
The empirical rule. In a normal distribution, about 68%, 95% and 99.7% of values fall within 1, 2 and 3 standard deviations of the mean. The Normal Distribution
Conditions. Random sample or random assignment; the 10% condition when sampling without replacement; large counts (at least 10 successes and 10 failures) for proportions; a normal population or a large sample for means; expected counts at least 5 for chi-square. Sampling Distributions for Sample Proportions
Degrees of freedom. $n - 1$ for one-sample and paired t; technology df (or the smaller $n - 1$ ) for two-sample t; $(\text{rows} - 1)(\text{columns} - 1)$ for chi-square. Carrying Out a Chi-Square Test for Homogeneity or Independence
Slope and correlation. The 2026 sheet gives only $\hat{y} = a + bx$ : read $a$ , $b$ , $r$ , $r^2$ and $s$ from computer output, and interpret each in context. Least-Squares Regression

Practice with it

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