AP Statistics Cheat Sheet 2026
Every formula, condition, and decision rule for AP Statistics in one place. Use this for active review — not passive reading.
Descriptive Statistics
| Concept | Formula / Rule |
|---|---|
| Mean | $\bar{x} = \frac{\sum x_i}{n}$ |
| Median | Middle value when sorted; resistant to outliers |
| Standard deviation (sample) | $s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$ |
| IQR | $Q_3 - Q_1$; resistant measure of spread |
| Outlier rule (boxplot) | Below $Q_1 - 1.5\cdot\text{IQR}$ or above $Q_3 + 1.5\cdot\text{IQR}$ |
| z-score | $z = \frac{x - \mu}{\sigma}$; tells how many SDs from mean |
| Percentile | % of data at or below that value |
Regression
| Concept | Formula / Rule |
|---|---|
| LSRL equation | $\hat{y} = a + bx$ |
| Slope | $b = r \cdot \frac{s_y}{s_x}$ |
| Intercept | $a = \bar{y} - b\bar{x}$ |
| Correlation $r$ | −1 ≤ r ≤ 1; direction + strength of linear relationship |
| $r^2$ | % of variation in $y$ explained by the linear model |
| Residual | $y - \hat{y}$ (actual minus predicted) |
| Residual plot | Should show random scatter — no pattern means linear model is appropriate |
Probability
| Rule | Formula |
|---|---|
| Addition rule (any events) | $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ |
| Addition rule (mutually exclusive) | $P(A \cup B) = P(A) + P(B)$ |
| Multiplication rule (any) | $P(A \cap B) = P(A) \cdot P(B|A)$ |
| Independent events | $P(A \cap B) = P(A) \cdot P(B)$; knowing A doesn't change P(B) |
| Conditional probability | $P(A|B) = \frac{P(A \cap B)}{P(B)}$ |
| Expected value | $E(X) = \sum x_i \cdot P(x_i)$ |
| Variance of X + Y (independent) | $\sigma^2_{X+Y} = \sigma^2_X + \sigma^2_Y$ |
Distributions
Binomial Distribution
Conditions: Fixed $n$, two outcomes, constant $p$, independent trials (BINS).
$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$, $\mu = np$, $\sigma = \sqrt{np(1-p)}$
Geometric Distribution
Number of trials until first success. $P(X = k) = (1-p)^{k-1}p$, $\mu = 1/p$
Normal Distribution
Symmetric, bell-shaped. Fully described by $\mu$ and $\sigma$. Use $z = (x-\mu)/\sigma$ then normalcdf on calculator.
Empirical rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ.
Sampling Distributions
| Statistic | Mean | Std Dev (Std Error) | Normal when |
|---|---|---|---|
| $\bar{x}$ (sample mean) | $\mu$ | $\sigma/\sqrt{n}$ | $n \geq 30$ or population normal (CLT) |
| $\hat{p}$ (sample proportion) | $p$ | $\sqrt{p(1-p)/n}$ | $np \geq 10$ and $n(1-p) \geq 10$ |
| $\bar{x}_1 - \bar{x}_2$ | $\mu_1 - \mu_2$ | $\sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}$ | Both samples normal or $n \geq 30$ |
| $\hat{p}_1 - \hat{p}_2$ | $p_1 - p_2$ | $\sqrt{p_1(1-p_1)/n_1 + p_2(1-p_2)/n_2}$ | All four: $n_1p_1, n_1(1-p_1), n_2p_2, n_2(1-p_2) \geq 10$ |
All Inference Tests — Quick Reference
| Test | Use for | Test statistic | Calculator |
|---|---|---|---|
| 1-sample z-test for $p$ | One proportion vs. claim | $z = \frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}$ | 1-PropZTest |
| 2-sample z-test for $p_1-p_2$ | Two proportions | Use pooled $\hat{p}_c$ | 2-PropZTest |
| 1-sample t-test for $\mu$ | One mean vs. claim | $t = \frac{\bar{x}-\mu_0}{s/\sqrt{n}}$, df = $n-1$ | T-Test |
| 2-sample t-test for $\mu_1-\mu_2$ | Two independent means | $t = \frac{\bar{x}_1-\bar{x}_2}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$ | 2-SampTTest |
| Paired t-test | Before/after, matched pairs | $t = \frac{\bar{d}}{s_d/\sqrt{n}}$, use differences | T-Test on diffs |
| Chi-square GOF | One categorical variable vs. claimed distribution | $\chi^2 = \sum\frac{(O-E)^2}{E}$, df = $k-1$ | $\chi^2$ GOF-Test |
| Chi-square homogeneity | Same distribution across groups? | Same formula, df = $(r-1)(c-1)$ | $\chi^2$-Test |
| Chi-square independence | Two variables associated? | Same formula, df = $(r-1)(c-1)$ | $\chi^2$-Test |
| t-test for slope | Linear relationship in population? | $t = b/SE_b$, df = $n-2$ | LinRegTTest |
The Three Conditions (Every Test)
- Random — random sample or randomized experiment
- Normal — for means: $n \geq 30$ or population normal; for proportions: $np \geq 10$ and $n(1-p) \geq 10$
- Independence (10% rule) — $n \leq 0.10 \cdot N$ (sample ≤ 10% of population)
Always state all three conditions explicitly on FRQ — missing one = point deducted.
Confidence Intervals
General form: $\text{statistic} \pm z^* \cdot \text{SE}$ or $\text{statistic} \pm t^* \cdot \text{SE}$
| Confidence level | z* |
|---|---|
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
Interpreting CIs: "We are 95% confident that the true [parameter] is between [lower] and [upper]." Never say "95% probability" — the interval either contains the parameter or it doesn't.
Hypothesis Testing Framework
- State hypotheses: $H_0: p = p_0$ vs $H_a: p > p_0$ (or $<$ or $\neq$)
- Check conditions (Random, Normal, Independence)
- Calculate test statistic and p-value
- Make decision: If p-value $< \alpha$, reject $H_0$. If $\geq \alpha$, fail to reject $H_0$.
- Conclude in context: "There is/is not convincing evidence that [Ha in words]."
Common AP Stats Exam Tasks
- Describe a distribution — always mention shape, center, spread, and outliers (SOCS). Include context: don't just say "right-skewed," say "right-skewed, meaning most students scored low but a few scored very high."
- Run a full inference procedure — state hypotheses, check all three conditions with justification, calculate test statistic and p-value, make a decision, write a conclusion in context. Every step must be present for full credit.
- Interpret a confidence interval — use the exact phrasing: "We are 95% confident that the true [parameter] is between [lower] and [upper]." Never say "probability."
- Interpret a p-value — "Assuming H₀ is true, there is a [p-value] probability of getting results this extreme by chance alone." Do not say the p-value is the probability H₀ is true.
- Choose the correct test — identify whether you have means or proportions, one or two samples, and whether samples are paired. Use the inference test table above.
How to Memorize AP Stats (What Actually Works)
- Learn the conditions cold — Random, Normal, Independence must be stated and justified on every FRQ. Practice writing them until they're automatic.
- Drill the conclusion language — there are only a few acceptable phrases for conclusions and CI interpretations. Write them out 10 times, then practice applying them to new scenarios.
- Know which test to pick — create a decision tree: means or proportions? One sample or two? Paired or independent? Practice this with 20 different scenarios before the exam.
- Use your calculator fluently — every inference test has a calculator function. Know 1-PropZTest, T-Test, 2-SampTTest, and χ²-Test by name and location before exam day.
- Read residual plots carefully — random scatter = linear model is appropriate. A curved or fanning pattern = not appropriate. This question appears frequently and is easy points if you know the rule.
Frequently Asked Questions
Do you get a formula sheet on AP Statistics?
Yes. The AP Stats formula sheet includes key formulas and a standard normal table. It does not include conditions, decision rules, or conclusion language — those must be memorized.
What is the most important thing on the AP Stats FRQ?
Writing all three conditions (Random, Normal, Independence) with explicit justification before running any inference test. Missing even one condition costs points — even if all your math is correct.
What is the difference between a t-test and a z-test in AP Stats?
Use a z-test for proportions. Use a t-test for means (when population standard deviation is unknown, which is almost always on the AP exam). In AP Stats you will almost never use a z-test for means.
How hard is it to get a 5 on AP Statistics?
About 16% of students score a 5. A 5 typically requires around 70% of composite points. The most reliable path: perfect the inference procedure language and do well on the investigative task FRQ, which is worth 9 of 50 FRQ points.