AP Statistics Formula Sheet 2026 — Complete Reference Guide
The AP Statistics exam provides a formula sheet during the test. But knowing which formulas are given vs. which you must memorize — and how to use them quickly — is what separates a 3 from a 5.
What's on the AP Stats Formula Sheet (Provided on Exam Day)
The College Board provides a reference sheet with formulas organized into three sections: Descriptive Statistics, Probability, and Inferential Statistics.
Section I: Descriptive Statistics
Sample mean: $$\bar{x} = \frac{\sum x_i}{n}$$
Sample standard deviation: $$s_x = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$
Standardized score (z-score): $$z = \frac{x - \mu}{\sigma}$$
Simple linear regression line: $$\hat{y} = a + bx$$
Slope: $$b = r \cdot \frac{s_y}{s_x}$$
Intercept: $$a = \bar{y} - b\bar{x}$$
Residual: $$\text{residual} = y - \hat{y} = y - (a + bx)$$
Section II: Probability
Addition rule: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$
Multiplication rule: $$P(A \cap B) = P(A) \cdot P(B|A)$$
Conditional probability: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Mean of a discrete random variable: $$\mu_X = \sum x_i \cdot P(x_i)$$
Variance of a discrete random variable: $$\sigma_X^2 = \sum (x_i - \mu_X)^2 \cdot P(x_i)$$
If X has a binomial distribution:
$$\mu_X = np$$
$$\sigma_X = \sqrt{np(1-p)}$$
$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$
If $\bar{x}$ is the mean of a random sample of size n:
$$\mu_{\bar{x}} = \mu$$
$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$
Section III: Inferential Statistics
Standardized test statistic: $$\text{statistic} = \frac{\text{estimate} - \text{parameter}}{\text{standard error of estimate}}$$
Confidence interval: $$\text{estimate} \pm (t^* \text{ or } z^*) \cdot SE$$
Standard Errors
| Parameter | Standard Error |
|---|---|
| $\hat{p}$ (sample proportion) | $\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$ |
| $\bar{x}$ (sample mean) | $\dfrac{s}{\sqrt{n}}$ |
| $\hat{p}_1 - \hat{p}_2$ | $\sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$ |
| $\bar{x}_1 - \bar{x}_2$ | $\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$ |
| $b$ (regression slope) | $\dfrac{s}{\sqrt{\sum(x_i - \bar{x})^2}}$ |
Chi-square statistic: $$\chi^2 = \sum \frac{(O - E)^2}{E}$$
What's NOT on the Formula Sheet (Must Memorize)
These are commonly tested but not provided on the AP Stats exam:
| Formula/Concept | What You Need to Know |
|---|---|
| IQR rule for outliers | Q1 - 1.5×IQR and Q3 + 1.5×IQR |
| Large counts condition | np ≥ 10 and n(1-p) ≥ 10 |
| 10% condition | n ≤ 10% of population |
| Normal/Large sample condition | n ≥ 30 or population is normal |
| Degrees of freedom (t-test) | df = n - 1 (one sample); conservative: min(n₁-1, n₂-1) |
| Chi-square df | (rows - 1)(columns - 1) for two-way table |
| Coefficient of determination | r² = fraction of variation in y explained by x |
Key Distributions Tested
Normal Distribution
- Empirical rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ
- Use z-table or calculator (normalcdf, invNorm)
t-Distribution
- Used when σ unknown (which is always in practice)
- Heavier tails than normal; approaches normal as df → ∞
- Use t-table or calculator (tcdf, invT)
Binomial Distribution
- Conditions: fixed n, two outcomes, constant p, independent trials (FINN)
- Calculator: binompdf(n, p, k) and binomcdf(n, p, k)
Chi-Square Distribution
- Always right-skewed, values ≥ 0
- Used for goodness-of-fit and tests of association
Inference Procedures Quick Reference
| Test | Use When | Formula |
|---|---|---|
| z-test for proportion | 1 proportion, σ known | $z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$ |
| t-test for mean | 1 mean, σ unknown | $t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$ |
| 2-proportion z-test | Compare 2 proportions | Use pooled $\hat{p}$ for H₀: p₁=p₂ |
| 2-sample t-test | Compare 2 means | Use unpooled SE |
| Paired t-test | Matched pairs | t-test on differences $d_i = x_{1i} - x_{2i}$ |
| Chi-square GOF | 1 categorical var vs. model | df = k - 1 |
| Chi-square homogeneity | Compare distributions | df = (r-1)(c-1) |
| Chi-square association | 2 categorical vars | df = (r-1)(c-1) |
| t-test for slope | Linear relationship exists? | $t = b/SE_b$, df = n-2 |
Conditions Checklist (Every Test/Interval)
You must state and verify three conditions for every inference procedure:
1. Random — data from a random sample or randomized experiment
2. Normal / Large Counts
- For means: n ≥ 30 OR population is approximately normal
- For proportions: np ≥ 10 AND n(1-p) ≥ 10
3. Independence (10% condition) — n ≤ 10% of population size
Missing conditions = automatic point deduction on FRQ.
AP Stats Formula Sheet Tips
- The formula sheet is a reference, not a crutch — you need to know which formula to use, not just that it exists
- Calculator functions are more important than the formulas themselves:
normalcdf,invNorm,tcdf,binompdf,linreg,1-PropZTest,2-SampTTest - Always interpret in context — writing "p-value = 0.03" earns zero points without "…so we reject H₀. There is convincing evidence that…"
- Degrees of freedom: when in doubt on 2-sample t, use conservative df = smaller of n₁-1 and n₂-1
Common AP Statistics Exam Tasks
- Complete a full inference procedure — state hypotheses, verify all three conditions (Random, Normal, Independence), calculate the test statistic and p-value, make a decision, write a conclusion in context. Every step must be present.
- Construct and interpret a confidence interval — use the correct formula, check conditions, calculate the interval, and write the interpretation using the exact required language.
- Describe a distribution — SOCS: shape, outliers, center, spread. Always include context ("the distribution of test scores is roughly symmetric...").
- Investigative Task FRQ — a multi-part question worth 9 of 50 FRQ points. Involves sustained statistical reasoning across several connected parts. Time management is critical — don't spend all your time here at the expense of shorter questions.
Frequently Asked Questions
Does AP Statistics provide a formula sheet?
Yes. AP Statistics provides a formula sheet with key inference formulas and tables (z-table, t-table, chi-square table). However, conditions, decision rules, and conclusion language are not provided — those must be memorized.
What is the most common mistake on AP Statistics FRQs?
Failing to state and justify all three conditions (Random, Normal, Independence) before running an inference test. Even if your math is perfect, missing a condition costs points every time.