AP Statistics formula sheet, explained

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

The AP Statistics exam gives you a formula sheet and tables. Knowing what each formula means, and which one fits a question, is what earns the marks.

What you are given

In the exam you receive a formula sheet with three sections, plus tables for the normal, t and chi-square distributions. You can download the current version from AP Central.

Part 1: Descriptive statistics

• Mean x̄ = Σx/n and standard deviation s_x: centre and spread of one variable.
• Least-squares regression line ŷ = a + bx, with slope b = r(s_y/s_x).
• Correlation r: strength and direction of a linear relationship, from −1 to 1.

Part 2: Probability and distributions

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B): the addition rule.
• P(A | B) = P(A ∩ B)/P(B): conditional probability.
• Expected value μ_X = Σx·P(x): the long-run average.
• Binomial: P(X = x) = C(n, x)p^x(1 − p)^n−x, with mean np and standard deviation √(np(1 − p)).
• Geometric: P(X = x) = (1 − p)^x−1p, the trial on which the first success happens, with mean 1/p.

Part 3: Sampling distributions and inference

• Standardised test statistic = (statistic − parameter) / (standard error of the statistic).
• Confidence interval = statistic ± (critical value)(standard error of the statistic).
• Chi-square statistic = Σ(observed − expected)²/expected.
• A table of standard errors for proportions, means, differences and regression slopes.

Almost every inference question is one of those two general forms. Your job is to choose the right statistic, standard error and critical value.

What the sheet does not do for you

• Check conditions: random sampling, independence (the 10% condition) and normality (large counts or sample size).
• Choose between a z procedure and a t procedure.
• Write the conclusion in context, which is where many marks are won or lost.

How to practise with it

1. Use the sheet for all practice, so you know where each formula sits.
2. For each inference question, name the procedure before calculating.
3. Learn a four-step routine: state, plan, do, conclude.

Worked examples for each part of the sheet

Example 1: least-squares regression line

For a set of data, r = 0.8, s_x = 4, s_y = 10, x̄ = 5 and ȳ = 30. Find the regression line.

• Slope: b = r(s_y/s_x) = 0.8 × 10/4 = 2.
• The line passes through (x̄, ȳ), so a = ȳ − b x̄ = 30 − 10 = 20.
• Equation: ŷ = 20 + 2x. Interpretation: each extra unit of x is associated with a predicted increase of 2 units in y.

Example 2: binomial probability

A player makes 30% of free throws, independently. In 10 shots, what is P(exactly 3 made)?

• X ~ binomial with n = 10, p = 0.3.
• P(X = 3) = C(10, 3)(0.3)³(0.7)⁷ = 120 × 0.027 × 0.08235 ≈ 0.267.
• Mean np = 3 made shots; standard deviation √(np(1 − p)) = √2.1 ≈ 1.45.

Example 3: geometric probability

If p = 0.2 on each independent trial, what is the probability the first success is on the third trial?

• P(X = 3) = (1 − p)²p = 0.8² × 0.2 = 0.128.
• The mean number of trials to the first success is 1/p = 5.

Example 4: a confidence interval for a proportion

In a random sample of 400 students, 248 (62%) said they study at night. Build a 95% confidence interval for the true proportion.

• Conditions: random sample; 400 is less than 10% of all students; successes (248) and failures (152) are both at least 10.
• Standard error: √(p̂(1 − p̂)/n) = √(0.62 × 0.38/400) ≈ 0.0243.
• Interval: 0.62 ± 1.96 × 0.0243 = 0.62 ± 0.048, so (0.572, 0.668).
• Conclusion: we are 95% confident that between 57.2% and 66.8% of all students study at night.

Example 5: a significance test for a proportion

A coin is flipped 200 times and lands heads 112 times (p̂ = 0.56). Is there convincing evidence it favours heads?

• H₀: p = 0.5 and H_a: p > 0.5, with α = 0.05.
• Standard error under H₀: √(0.5 × 0.5/200) ≈ 0.0354.
• Test statistic: z = (0.56 − 0.5)/0.0354 ≈ 1.70, giving a p-value of about 0.045.
• Because 0.045 < 0.05, we reject H₀. There is convincing evidence that the coin lands heads more than half the time.

Choosing the right formula, fast

• Describing one variable? Mean and standard deviation, or median and IQR.
• Relationship between two quantitative variables? Correlation and the regression line.
• Fixed number of independent trials, counting successes? Binomial.
• Trials until the first success? Geometric.
• Estimating a parameter? Confidence interval: statistic ± critical value × standard error.
• Testing a claim? Standardised test statistic and p-value.

Practice questions

Try each one before opening the answer.

Example 6: a chi-square goodness-of-fit test

A die is rolled 60 times with these results for faces 1 to 6: 8, 12, 9, 11, 6, 14. Is there evidence the die is unfair?

• H₀: the die is fair, so each face has probability 1/6. Expected count for each face: 60 × 1/6 = 10.
• Conditions: random rolls, independent, and all expected counts are at least 5.
• χ² = Σ(O − E)²/E = (4 + 4 + 1 + 1 + 16 + 16)/10 = 4.2, with 6 − 1 = 5 degrees of freedom.
• The p-value is about 0.52, far above 0.05. We fail to reject H₀: there is not convincing evidence that the die is unfair.

Common mistakes with the formula sheet

• Using p̂ instead of p₀ in a test. In a significance test for a proportion, the standard error uses the null value p₀. In a confidence interval, it uses p̂.
• Mixing up standard deviation and standard error. Standard deviation describes individual values; standard error describes how a statistic varies from sample to sample.
• Forgetting degrees of freedom for t and chi-square procedures.
• Reporting calculator output only. Graders want the formula or the named procedure, the numbers substituted, and the conclusion.

Frequently asked questions

Can I use my calculator's built-in tests?

Yes, a graphing calculator is allowed. But you still need to name the procedure, check conditions and state the test statistic and p-value in your answer.

Do I need to memorise anything that is not on the sheet?

Yes: the conditions for each procedure, the meaning of key terms, the empirical rule and how to interpret results in context.