Last Minute AP Statistics Cheat Sheet (WITH FORMULAS)
What You Need to Know
This is the high-yield formula + procedure sheet you use when you’re trying to (1) pick the right method fast, (2) check conditions correctly, and (3) write the minimum-necessary but full-credit inference “story” (parameter → conditions → compute → conclude in context).
Big AP Stats idea: almost every FRQ is either describing data, probability/random variables, or inference (confidence interval or significance test). The fastest way to lose points is skipping conditions or failing to define the parameter.
Golden rule: Your hypotheses and conclusion must be about a population parameter (like p, \mu, \mu_1-\mu_2, \beta), not about sample statistics (like \hat p, \bar x).
Step-by-Step Breakdown
A. Picking the right inference procedure (fast decision tree)
- Is your response variable categorical (yes/no) or quantitative (number)?
- Categorical → proportions, chi-square.
- Quantitative → means, t-procedures, regression.
- How many groups/samples?
- 1 sample → one-proportion z or one-mean t.
- 2 independent samples → two-proportion z or two-sample t.
- Matched pairs → one-sample t on differences.
- Are you comparing distributions of categories across groups?
- One categorical variable vs a claimed model → chi-square GOF.
- Two categorical variables (relationship) → chi-square independence.
- Several populations/treatments and one categorical response → chi-square homogeneity.
- Is it a relationship between two quantitative variables?
- Use linear regression; inference about slope \beta uses a t test/interval with df=n-2.
B. Writing any inference solution (full-credit skeleton)
- Define the parameter (in context).
- Example: p = true proportion of all students at your school who…
- State hypotheses (test only).
- H_0: p=p_0, H_a: p\ne p_0 (or
- Check conditions (name + verify with given info).
- Random, 10% condition, Normal/Large Counts, etc.
- Compute the test statistic or interval (show formula + plug values).
- P-value OR critical value method (usually P-value on AP).
- Conclude in context at level \alpha.
- “Because p\text{-value} < \alpha, reject H_0. There is convincing evidence that …”
C. Mini worked walkthrough (one-proportion z test)
Prompt style: “Is there evidence the true proportion differs from 0.40?”
- Parameter: p = true proportion of (population) who …
- Hypotheses: H_0: p=0.40, H_a: p\ne 0.40
- Conditions:
- Random: stated random sample/assignment.
- 10%: n \le 0.1N (if sampling without replacement).
- Large counts: np_0\ge 10 and n(1-p_0)\ge 10.
- Compute:
- \hat p = \dfrac{x}{n}
- z=\dfrac{\hat p - p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}}
- Get P-value from Normal.
- Conclude in context.
Key Formulas, Rules & Facts
A. Describing data (quick hits)
| Tool | Formula | When to use | Notes |
|---|---|---|---|
| Standard score | z=\dfrac{x-\mu}{\sigma} or z=\dfrac{x-\bar x}{s} | Compare to distribution center/spread | “How many SDs from mean?” |
| Outlier rule | below Q_1-1.5(IQR) or above Q_3+1.5(IQR) | Boxplots/outliers | Not “proof,” just a flag |
| Density/probability | area under curve | Continuous models | Probability = area |
B. Linear transformations & combining variables
| Rule | Formula | Notes |
|---|---|---|
| Add constant a | \mu_{X+a}=\mu_X+a, \sigma_{X+a}=\sigma_X | Shifts center only |
| Multiply by b | \mu_{bX}=b\mu_X, \sigma_{bX}=|b|\sigma_X | Stretch/compress spread |
| Sum (any) | \mu_{X+Y}=\mu_X+\mu_Y | Always true |
| Sum (independent) | \sigma^2_{X+Y}=\sigma_X^2+\sigma_Y^2 | Variances add, not SDs |
| Difference (independent) | \sigma^2_{X-Y}=\sigma_X^2+\sigma_Y^2 | Still add variances |
C. Probability essentials
| Rule | Formula | Use | Notes |
|---|---|---|---|
| Complement | P(A^c)=1-P(A) | “At least one” | Often fastest |
| Addition rule | P(A\cup B)=P(A)+P(B)-P(A\cap B) | Two events | If disjoint, intersection =0 |
| Conditional | P(A\mid B)=\dfrac{P(A\cap B)}{P(B)} | Given info | Restrict sample space |
| Independence | P(A\cap B)=P(A)P(B) | Check independence | Equivalent to P(A\mid B)=P(A) |
| Bayes | P(A\mid B)=\dfrac{P(B\mid A)P(A)}{P(B)} | Reverse condition | Tree diagrams help |
D. Discrete random variables (AP favorites)
| Model | Probability | Mean | SD | Conditions/Notes |
|---|---|---|---|---|
| Binomial X\sim Bin(n,p) | P(X=k)=\binom{n}{k}p^k(1-p)^{n-k} | \mu=np | \sigma=\sqrt{np(1-p)} | BINS: Binary, Independent, Number fixed, Same p |
| Geometric X\sim Geom(p) | P(X=k)=(1-p)^{k-1}p | \mu=\dfrac{1}{p} | \sigma=\sqrt{\dfrac{1-p}{p^2}} | Counts trials until first success |
| Expected value | \mu_X=E(X)=\sum x\,P(x) | Any discrete RV | Use for “long-run average” |
E. Normal + sampling distributions
| Idea | Formula | When it applies | Notes |
|---|---|---|---|
| Normal model | X\sim N(\mu,\sigma) | Given approx Normal | Standardize to use Normal CDF |
| Sample mean | \mu_{\bar x}=\mu, \sigma_{\bar x}=\dfrac{\sigma}{\sqrt{n}} | SRS; Normal pop or large n | CLT: large n makes \bar x approx Normal |
| Sample proportion | \mu_{\hat p}=p, \sigma_{\hat p}=\sqrt{\dfrac{p(1-p)}{n}} | Large counts | For inference, check large counts |
| Large counts (one prop) | np\ge 10 and n(1-p)\ge 10 | Normal approx for \hat p | For tests use p_0 in check |
| Large counts (two prop) | n_1p_1\ge 10, n_1(1-p_1)\ge 10, n_2p_2\ge 10, n_2(1-p_2)\ge 10 | Two-prop intervals | For tests often use pooled \hat p |
F. Confidence intervals (CI) and test statistics (most-used)
One proportion
| Task | Formula | Notes |
|---|---|---|
| CI for p | \hat p \pm z^*\sqrt{\dfrac{\hat p(1-\hat p)}{n}} | Use \hat p in SE |
| Test for p | z=\dfrac{\hat p-p_0}{\sqrt{\dfrac{p_0(1-p_0)}{n}}} | Use p_0 in SE |
Two proportions (independent)
| Task | Formula | Notes |
|---|---|---|
| CI for p_1-p_2 | (\hat p_1-\hat p_2) \pm z^*\sqrt{\dfrac{\hat p_1(1-\hat p_1)}{n_1}+\dfrac{\hat p_2(1-\hat p_2)}{n_2}} | Don’t pool for CI |
| Test for p_1-p_2 | z=\dfrac{(\hat p_1-\hat p_2)-0}{\sqrt{\hat p(1-\hat p)\left(\dfrac{1}{n_1}+\dfrac{1}{n_2}\right)}} where \hat p=\dfrac{x_1+x_2}{n_1+n_2} | Pool only in the test under H_0: p_1=p_2 |
One mean (quantitative)
| Task | Formula | Notes |
|---|---|---|
| CI for \mu | \bar x \pm t^*\dfrac{s}{\sqrt{n}} with df=n-1 | Use when \sigma unknown (usual) |
| Test for \mu | t=\dfrac{\bar x-\mu_0}{s/\sqrt{n}} with df=n-1 | Check approx Normal / no strong skew+outliers |
Two means (independent samples)
| Task | Formula | Notes |
|---|---|---|
| CI for \mu_1-\mu_2 | (\bar x_1-\bar x_2) \pm t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} | Calculator uses df approximation |
| Test for \mu_1-\mu_2 | t=\dfrac{(\bar x_1-\bar x_2)-0}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}} | Don’t pool SDs in AP Stats |
Matched pairs (paired data)
- Compute differences d_i = x_{1i}-x_{2i}.
- Then do one-sample t on differences:
- \bar d \pm t^*\dfrac{s_d}{\sqrt{n}} and t=\dfrac{\bar d-\mu_{d,0}}{s_d/\sqrt{n}} with df=n-1.
G. Chi-square procedures
| Procedure | Statistic | df | Conditions | Notes |
|---|---|---|---|---|
| GOF | \chi^2=\sum \dfrac{(O-E)^2}{E} | k-1 | Random; expected counts typically \ge 5 | E=n\times p_{model} |
| Independence/Homogeneity | \chi^2=\sum \dfrac{(O-E)^2}{E} | (r-1)(c-1) | Random; expected counts typically \ge 5 | E=\dfrac{(row\ total)(col\ total)}{n} |
H. Regression (least squares + inference)
| Quantity | Formula | Notes |
|---|---|---|
| LSRL | \hat y=a+bx | Predict y from x |
| Slope | b=r\dfrac{s_y}{s_x} | Sign matches r |
| Intercept | a=\bar y-b\bar x | Line goes through \left(\bar x,\bar y\right) |
| Residual | e=y-\hat y | Positive residual = point above line |
| Correlation | -1\le r\le 1 | No units; linear strength only |
| Coef. of determination | r^2 | % variability in y explained by linear model with x |
| Slope test | t=\dfrac{b-0}{SE_b}, df=n-2 | Test H_0: \beta=0 |
| CI for slope | b\pm t^*SE_b | Interpret change in mean response |
Regression conditions (LINER): Linear pattern, Independent, Normal residuals, Equal variance, Random.
I. Inference vocabulary (quick definitions)
- P-value: probability (assuming H_0 true) of getting a statistic as extreme or more extreme than observed.
- Type I error: reject true H_0 (false positive). Probability =\alpha.
- Type II error: fail to reject false H_0 (false negative). Probability =\beta.
- Power: 1-\beta.
Examples & Applications
Example 1: Two-proportion z interval (wording trap)
Situation: Compare vaccination rates in School A vs School B.
- Parameter: p_A-p_B = true difference in vaccination proportions.
- Use CI:
- \left(\hat p_A-\hat p_B\right) \pm z^*\sqrt{\dfrac{\hat p_A(1-\hat p_A)}{n_A}+\dfrac{\hat p_B(1-\hat p_B)}{n_B}}
Key insight: If CI contains 0, a “difference” claim isn’t supported.
- \left(\hat p_A-\hat p_B\right) \pm z^*\sqrt{\dfrac{\hat p_A(1-\hat p_A)}{n_A}+\dfrac{\hat p_B(1-\hat p_B)}{n_B}}
Example 2: Matched pairs vs two-sample (super common)
Situation: Same students take a pretest and posttest.
- Don’t do two-sample t.
- Compute d_i=post-pre, then one-sample t on \mu_d.
- Test statistic: t=\dfrac{\bar d-0}{s_d/\sqrt{n}}.
Key insight: Pairing reduces variability; ignoring pairing can hide effects.
Example 3: Chi-square independence (interpretation)
Situation: Is seat location (front/middle/back) related to passing (yes/no)?
- Parameter: whether the two categorical variables are independent in the population.
- Expected count: E=\dfrac{(row\ total)(col\ total)}{n}.
- Statistic: \chi^2=\sum \dfrac{(O-E)^2}{E}, df=(r-1)(c-1).
Key insight: A significant result says “associated,” not “causes.”
Example 4: Regression slope inference (what you conclude)
Situation: Predict exam score from hours studied.
- Test H_0: \beta=0 vs H_a: \beta>0.
- Compute t=\dfrac{b}{SE_b} with df=n-2.
- Conclusion in context: “There is convincing evidence of a positive linear relationship between hours studied and mean exam score.”
Key insight: You’re making a claim about mean response changing with x, not about individual predictions being perfect.
Common Mistakes & Traps
Mistake: Hypotheses about \hat p or \bar x instead of p or \mu.
- Why wrong: sample stats are random; parameters are fixed truths.
- Fix: define parameter first, then write H_0 and H_a about it.
Mistake: Using t vs z incorrectly.
- Why wrong: means with unknown \sigma require t; proportions use z.
- Fix: quantitative → t, categorical → z.
Mistake: Pooling in a two-proportion CI.
- Why wrong: pooling assumes p_1=p_2, which is exactly what you’re estimating in a CI.
- Fix: Pool only for the hypothesis test of p_1-p_2=0.
Mistake: Skipping or mis-checking large counts.
- Why wrong: Normal approximation can fail badly with small expected successes/failures.
- Fix: For one-prop tests use p_0; for intervals use \hat p.
Mistake: Treating matched pairs as independent samples.
- Why wrong: within-person pairing creates dependence; you must analyze differences.
- Fix: If the same subject is measured twice (or paired units), do one-sample t on d.
Mistake: Wrong chi-square df / expected counts.
- Why wrong: df controls the reference distribution; wrong df → wrong P-value.
- Fix: GOF df=k-1; two-way tables df=(r-1)(c-1); compute E using row/col totals.
Mistake: Regression conclusion implies causation.
- Why wrong: observational studies can have confounding.
- Fix: Only randomized experiments justify cause-and-effect.
Mistake: “No significance” = “proved equal.”
- Why wrong: failing to reject H_0 means insufficient evidence, not proof.
- Fix: say “not enough evidence to conclude…”
Memory Aids & Quick Tricks
| Trick / mnemonic | Helps you remember | When to use |
|---|---|---|
| SOCS | Shape, Outliers, Center, Spread | Describing distributions fast |
| BINS | Binomial conditions: Binary, Independent, Number fixed, Same p | Decide binomial vs not |
| 10% condition | Independence when sampling without replacement | Any sampling inference |
| PLAN | Parameter, Label (hypotheses), Assumptions/conditions, Name test/interval | Any inference FRQ write-up |
| “Pool for test, not for CI” | Two-proportion pooling rule | Two-proportion inference |
| LINER | Linear, Independent, Normal residuals, Equal variance, Random | Regression inference |
| “df = n-1, n-2, (r-1)(c-1)” | df for one-sample t, regression slope, chi-square table | Don’t lose df points |
| CUSS | Chi-square: Counts, Use expected, Sum \dfrac{(O-E)^2}{E}, Shape is right-skew | Chi-square setup + interpretation |
Quick Review Checklist
- [ ] You defined the parameter (with population + context).
- [ ] Your H_0 and H_a are about the parameter, and direction matches the prompt.
- [ ] You checked Random, 10%, and the correct Normal/Large Counts condition.
- [ ] Proportions: z procedures; Means: t procedures; Paired: analyze differences.
- [ ] Two-prop test uses pooled \hat p; two-prop CI does not.
- [ ] You used the correct df: n-1 (one-sample/paired t), n-2 (regression slope), (r-1)(c-1) (chi-square table).
- [ ] Your conclusion is in context and matches the decision: reject vs fail to reject.
- [ ] You didn’t claim causation unless it was a randomized experiment.
You’ve got this—run the checklist on every inference question and you’ll avoid the biggest point leaks.