AP Stats formula sheet: Every formula explained with a worked example

The AP (Advanced Placement) Statistics formula sheet is a reference packet of formulas and tables you can use on both sections of the exam; from May 2027 it is available both in print and digitally in Bluebook. It has descriptive stats formulas, the two big probability rules, the sampling distributions of means and proportions, every significant test statistic and confidence interval you'll be asked to compute, plus the t, chi-square and z tables. The College Board wants you comparing formulas, not memorizing them.

That's a mixed blessing. Nothing on the sheet does the thinking for you. If you can't tell a one-sample t-interval from a two-sample t-test, or a proportion from a mean, having the formula in front of you doesn't help. This guide walks the whole sheet in the order it's printed, groups each formula by the unit it comes from, and gives you a plain-English what-it-does and when-to-use for every single one. If you'd rather practice the underlying content, Cognito's AP Statistics past papers have every recent released exam with worked solutions.

The current sheet, in the order College Board prints it, has three blocks of formulas plus three probability tables at the back. Block I is descriptive statistics and regression (units 1 and 5 in the revised 2026–27 course). Block II is probability and distributions (unit 2). Block III is sampling distributions and inferential statistics: the standard errors, confidence intervals and test statistics you'll use in units 3 and 4. The tables at the back are the standard normal (z), the t-distribution and chi-square critical values.

The top of page one has the summary statistics you use to describe a data set: the sample mean, the sample standard deviation, and the least-squares regression line with its residuals and correlation coefficient.

Sample mean: x-bar = (sum of xi) / n. This is just the average. It's your point estimate for a population mean whenever you're doing a one-sample t-procedure.

Sample standard deviation: s = sqrt[ (sum of (xi - x-bar)^2) / (n - 1) ]. The n minus 1 is the Bessel correction, and the exam expects you to use it whenever the data are a sample (which they almost always are). Standard deviation tells you the typical distance of a data point from the mean, in the original units.

Regression line: y-hat = b0 + b1 x (the 2020 sheet writes this as y-hat = a + bx), where b1 = r (sy / sx) and b0 = y-bar minus b1 x-bar. Slope tells you how much y changes per unit change in x. Intercept is the predicted y when x equals zero (often meaningless in context, but always calculable).

Correlation coefficient r runs from minus 1 to plus 1 and measures the strength and direction of a linear relationship. r-squared is the proportion of variation in y that's explained by the linear model.

Worked example: 5 students studied for 1, 2, 3, 4 and 5 hours and scored 60, 65, 70, 80 and 85 on a test. x-bar = 3, y-bar = 72, sx = 1.58, sy = 10.37, r = 0.991, so b1 = 0.991 (10.37 / 1.58) = 6.5 and b0 = 72 minus 6.5(3) = 52.5. The regression line is y-hat = 52.5 + 6.5x, and r-squared = 0.983 means 98.3 percent of the variation in test score is explained by hours studied in this sample.

The second block on the sheet covers discrete probability. Two general formulas, then the specifics for the binomial distribution.

Expected value (mean) of a discrete random variable: mu-X = sum of (xi times P(xi)). Multiply each outcome by its probability, sum across all outcomes. This is what an average would settle at over infinite trials.

Variance of a discrete random variable: sigma-squared-X = sum of ((xi minus mu-X)^2 times P(xi)). Standard deviation is the square root.

Worked example: A lottery ticket costs 2 dollars. You win 100 dollars with probability 0.01, and nothing otherwise. Expected profit = (98)(0.01) + (-2)(0.99) = 0.98 - 1.98 = -1.00. On average you lose a dollar per ticket.

Binomial distribution: P(X = k) = (n choose k) p^k (1-p)^(n-k), with mean = np and standard deviation = sqrt(np(1-p)). Use whenever you have a fixed number of independent trials, each with the same probability of success.

Worked example: A basketball player makes 70 percent of free throws. Probability she makes exactly 8 of her next 10 = (10 choose 8) (0.7)^8 (0.3)^2 = 45 times 0.05765 times 0.09 = 0.233.

The next block gives you the standard deviations of the sampling distributions of a sample mean and a sample proportion. These are the workhorses that make every inference procedure in the second half of the sheet work.

Sampling distribution of a sample mean: mean = mu, standard deviation = sigma / sqrt(n). As n grows, the standard deviation shrinks, which is why bigger samples give tighter estimates. When sigma is unknown (almost always), you use s in its place and switch from z to t.

Sampling distribution of a sample proportion: mean = p, standard deviation = sqrt(p(1-p)/n). Same idea. When p is unknown (again, almost always for inference), you use p-hat in its place.

Central Limit Theorem note: the sampling distribution of x-bar is approximately Normal when n is large enough (roughly 30 or more, or any n if the population is Normal). The sampling distribution of p-hat is approximately Normal when np and n(1-p) are both at least 10. Both conditions have to be checked before you use z or t procedures, and both are worth writing down explicitly on FRQs to earn the conditions point.

Every confidence interval on the sheet has the same shape: point estimate plus or minus (critical value times standard error). The four blocks give you the specific standard error for each parameter.

Worked example (one-mean t-interval): 25 randomly sampled boxes of cereal have a mean weight of 502 grams with s = 8 grams. A 95 percent confidence interval for the true mean weight uses t* = 2.064 (df = 24). Margin of error = 2.064 (8 / sqrt(25)) = 3.30. The interval is 502 plus or minus 3.30, or (498.7, 505.3) grams. Interpretation: we are 95 percent confident that the true mean weight of all boxes lies between 498.7 and 505.3 grams. That interpretation wording matters for the FRQ point.

Worked example (two-proportion z-interval): a survey of 200 men found 120 who owned a car; a survey of 250 women found 130. p-hat_1 = 0.60, p-hat_2 = 0.52. Difference = 0.08. Standard error = sqrt(0.6(0.4)/200 + 0.52(0.48)/250) = sqrt(0.0012 + 0.000998) = 0.0469. 95 percent CI: 0.08 plus or minus 1.96(0.0469) = (-0.012, 0.172). Because zero is inside the interval, we can't conclude the true proportions differ.

Every significance test on the sheet has the same shape: (statistic minus null value) divided by standard error. The only trick: for a proportion test the standard error uses p_0 (the null value), because you're testing under the null hypothesis.

Worked example (one-sample t-test): a company claims its light bulbs last 1,000 hours on average. A sample of 20 bulbs has x-bar = 985 with s = 40. Test H0: mu = 1000 versus Ha: mu < 1000 at alpha = 0.05. t = (985 - 1000) / (40 / sqrt(20)) = -15 / 8.944 = -1.677. With df = 19 the critical t for a one-tailed 0.05 test is -1.729. Because -1.677 is not below -1.729 (or equivalently, p-value = 0.055 > 0.05), we fail to reject H0. There is not convincing evidence that the true mean lifetime of the bulbs is less than 1,000 hours.

Both chi-square tests in the course use the same formula. The difference is in the sampling design.

Test of independence: Two categorical variables from one sample, testing whether they're associated. df = (rows - 1)(columns - 1). Expected count in each cell = (row total)(column total) / grand total.

Test of homogeneity: The same test statistic and df as independence, but the sampling design differs. Two or more separate samples, one categorical response, testing whether the distribution of that response is the same across populations. Same formula, same df formula, different wording of the null hypothesis.

The FRQ almost always signals which version you need by how the data were collected. One sample cross-classified on two variables = independence. Multiple samples with counts of one response = homogeneity.

The last two pages of the sheet are three probability tables: the standard normal cumulative distribution, the t-distribution critical values, and chi-square critical values.

Standard normal table gives P(Z less than z) for z from about -3.4 to 3.4 in steps of 0.01. You look up the row for the first two digits of z and the column for the third. On the exam you can use this or your calculator (normalcdf); either is accepted, but the table is what's provided if your calculator fails.

t-distribution table gives critical values t* for a given confidence level or one-tail alpha, indexed by degrees of freedom. Rows are df from 1 to 30, then 40, 50, 60, 80, 100, 1000 and infinity; columns are upper-tail probabilities (0.25, 0.20, 0.15, 0.10, 0.05, 0.025, 0.02, 0.01, 0.005, 0.0025, 0.001, 0.0005). Matching confidence levels from 50 to 99.9 percent (including 90, 95, 98 and 99 percent) are printed along the bottom.

Chi-square table gives critical values for df from 1 to 30, then 40, 50, 60, 80 and 100 at common alpha levels. If you compute a chi-square that exceeds the critical value at your alpha, reject the null hypothesis. If you're looking for a p-value from a chi-square statistic, most calculators (chi-squared cdf on the TI-84) give it directly, but the table lets you sandwich the p-value between two column values.

The best practice for the formula sheet is timed released-FRQ work with the sheet open on the desk, exactly how you'll have it in the exam. Pull free-response questions from AP Central going back the last five or six years and sort them by which section of the sheet they use. You'll likely find that inference for means and proportions accounts for a large share of the marks.

A useful drill: read a released FRQ stem and, before touching your calculator, say out loud which section of the sheet you'll pull from and why. If you can't do that quickly on 20 sample stems, the calculator work isn't the bottleneck. If you want structured practice on any of the underlying content, Cognito's AP Statistics past papers have every recent released exam with worked solutions.