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AP Statistics Formula Sheet

Every formula you need for the AP Statistics exam in one printable reference. A formula sheet is provided on exam day, but knowing these formulas cold will save you time and reduce errors.

Descriptive Statistics

Sample Mean x̄ = Σx_i / n

Sample Standard Deviation s = √[Σ(x_i − x̄)² / (n − 1)]

Sample Variance s² = Σ(x_i − x̄)² / (n − 1)

Interquartile Range IQR = Q3 − Q1

Outlier Boundaries Q1 − 1.5(IQR) to Q3 + 1.5(IQR)

z-Score z = (x − μ) / σ

Normal Distribution

Empirical Rule 68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ

Standardizing z = (x − μ) / σ

Unstandardizing x = μ + zσ

Linear Regression

Regression Equation ŷ = a + bx

Slope b = r(s_y / s_x)

y-Intercept a = ȳ − bx̄

Correlation Coefficient r = Σz_xz_y / (n − 1)

Coefficient of Determination r² = variation explained / total variation

Residual e = y − ŷ

Probability

Addition Rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Multiplication Rule P(A ∩ B) = P(A) × P(B|A)

Conditional Probability P(A|B) = P(A ∩ B) / P(B)

Independent Events P(A ∩ B) = P(A) × P(B)

Complement P(A') = 1 − P(A)

Random Variables

Expected Value (Mean) μ_X = Σx_i × P(x_i)

Variance of X σ²_X = Σ(x_i − μ)² × P(x_i)

Linear Transform: Mean μ_aX+b = aμ_X + b

Linear Transform: Variance σ²_aX+b = a²σ²_X

Sum of Independent RVs μ_X+Y = μ_X + μ_Y, σ²_X+Y = σ²_X + σ²_Y

Binomial & Geometric Distributions

Binomial Probability P(X = k) = C(n,k) × p^k × (1−p)^n−k

Binomial Mean μ = np

Binomial SD σ = √[np(1 − p)]

Geometric Mean μ = 1 / p

Sampling Distributions

Mean of x̄ μ_x̄ = μ

SD of x̄ σ_x̄ = σ / √n

Mean of p̂ μ_p̂ = p

SD of p̂ σ_p̂ = √[p(1 − p) / n]

Confidence Intervals

General CI Formula statistic ± (critical value)(SE)

CI for Proportion p̂ ± z* × √[p̂(1 − p̂) / n]

CI for Mean x̄ ± t* × (s / √n)

CI for Two Proportions (p̂₁ − p̂₂) ± z* × SE

CI for Two Means (x̄₁ − x̄₂) ± t* × SE

Sample Size (Proportion) n = (z* / ME)² × p̂(1 − p̂)

Hypothesis Tests

General Test Statistic test stat = (statistic − parameter) / SE

1-Proportion z-Test z = (p̂ − p₀) / √[p₀(1 − p₀) / n]

1-Sample t-Test t = (x̄ − μ₀) / (s / √n), df = n − 1

2-Proportion z-Test z = (p̂₁ − p̂₂) / SE, using pooled p̂_c

2-Sample t-Test t = (x̄₁ − x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Chi-Square Statistic χ² = Σ[(O − E)² / E]

Expected Count (Chi-Square) E = (row total × col total) / grand total

Regression Slope t-Test t = b / SE_b, df = n − 2

Decision Rule Reject H₀ if p-value < α

Conditions for Inference (Memorize These!)

Random Data from random sample or randomized experiment

Normal (Proportions) np ≥ 10 and n(1 − p) ≥ 10

Normal (Means) n ≥ 30 (CLT) or population is normal

Independent (10% Rule) n ≤ 10% of population (sampling w/o replacement)

Chi-Square Condition All expected counts ≥ 5

FAQ

Common questions

Does the AP Statistics exam provide a formula sheet?

Yes. The AP Statistics exam provides a formula sheet with key formulas for descriptive statistics, probability, and inference. However, it does not include explanations of when to use each formula. This reference sheet covers both the formulas and when to apply them, so you can study effectively before exam day.

What formulas do I need to memorize for AP Statistics?

While a formula sheet is provided, you should memorize: the conditions for inference (random, normal, independent), the interpretation templates for confidence intervals and hypothesis tests, the meaning of p-value, Type I and Type II errors, and how to read computer output for regression. The exam tests understanding more than formula recall.

What is the most important formula on the AP Statistics exam?

The confidence interval formula (statistic ± critical value × standard error) is arguably the most important because it appears in multiple contexts: proportions, means, and regression slopes. Understanding this template formula helps you tackle a large portion of the inference questions on the exam.

How do I know which hypothesis test to use on the AP Statistics exam?

Identify the parameter (proportion or mean), number of samples (one or two), and whether you're testing or estimating. One proportion uses z-test, one mean uses t-test, two proportions uses two-proportion z-test, two means uses two-sample t-test, and categorical data uses chi-square. Regression slopes use a t-test on the slope coefficient.

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