📊

AP Statistics — Cheatsheet

Formulas, exam-day tips, and key terms on one page.

Formulas & relationships

Sample standard deviation

s = sqrt( Σ(x_i − x-bar)² / (n − 1) )

Find each deviation from the mean, square it, average the squares using n − 1 (not n), then take the square root. Dividing by n − 1 corrects the tendency of a sample to underestimate the population spread.

Mean vs. median under skew

skewed right: mean > median · skewed left: mean < median · symmetric: mean ≈ median

The mean chases the tail. Whichever side the long tail is on, that is the side the mean sits relative to the median.

z-score (standardizing)

z = (x − μ) / σ

Subtract the mean, divide by the standard deviation. Positive z is above the mean, negative z is below. z has no units.

Empirical (68–95–99.7) rule

μ ± 1σ ≈ 68% · μ ± 2σ ≈ 95% · μ ± 3σ ≈ 99.7%

For a Normal distribution, about 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

Outlier fences (1.5 × IQR rule)

lower fence = Q1 − 1.5·IQR · upper fence = Q3 + 1.5·IQR · IQR = Q3 − Q1

A data value is an outlier if it is below the lower fence or above the upper fence. The fences themselves are not necessarily data values.

Linear transformation y = a·x + b

mean_y = a·mean_x + b median_y = a·median_x + b SD_y = |a|·SD_x IQR_y = |a|·IQR_x variance_y = a²·variance_x

The additive constant b appears in the centers and never in the spreads. This one line answers every transformation question on the exam.

Correlation coefficient

r = (1/(n−1)) · Σ [ (x_i − x-bar)/s_x ] · [ (y_i − y-bar)/s_y ]

r is the average product of the z-scores of x and y. Because it uses standardized values, r has no units and is unaffected by changes of scale.

Regression line and slope

y-hat = a + bx · b = r · (s_y / s_x) · a = y-bar − b·x-bar

The slope equals r times the ratio of the standard deviations. The line passes through (x-bar, y-bar), which is how the intercept formula is derived.

Coefficient of determination

r² = (fraction of variation in y explained by the linear model on x)

r² ranges from 0 to 1. If r² = 0.64, then 64% of the variation in the response is accounted for by the linear relationship with x. Square r to get it.

Residual

residual = y − y-hat = observed − predicted

A positive residual means the actual value lies above the line (model underpredicted); a negative residual means it lies below (model overpredicted).

Completely randomized design

experimental units → RANDOM assignment → treatment groups → compare responses

The defining feature is that chance alone decides which unit gets which treatment. Random assignment is what creates comparable groups and permits a causal conclusion.

Scope of conclusions grid

random assign → causation · random select → generalize · both → causal + general · neither → describe only

Two independent design choices, two independent permissions. Never claim causation without random assignment or generalization beyond the sampled population without random selection.

Core probability rules

P(not A) = 1 − P(A) · P(A or B) = P(A) + P(B) − P(A and B) · P(A and B) = P(A)·P(B|A)

The general addition rule reduces to P(A)+P(B) when A and B are disjoint. The multiplication rule reduces to P(A)·P(B) when A and B are independent.

Mean and variance of a discrete random variable

μ_X = Σ x_i · P(x_i) · σ²_X = Σ (x_i − μ_X)² · P(x_i) · σ_X = sqrt(σ²_X)

Multiply each value by its probability and sum to get the mean; for variance, use the squared deviations weighted by probability. Standard deviation is the square root.

Combining random variables

μ_(X±Y) = μ_X ± μ_Y · for INDEPENDENT X, Y: σ²_(X±Y) = σ²_X + σ²_Y

Means add or subtract directly. Variances always ADD (never subtract) when combining independent variables — you add the variances even for X − Y, then square-root for the standard deviation.

Binomial probability, mean, and standard deviation

P(X = k) = C(n,k) · p^k · (1−p)^(n−k) · μ_X = np · σ_X = sqrt(np(1−p))

C(n,k) is the number of ways to choose which k of the n trials are successes. The mean is np; the standard deviation is the square root of np(1−p).

Geometric probability and mean

P(X = k) = (1 − p)^(k−1) · p · μ_X = 1 / p · σ_X = sqrt(1 − p) / p

X is the trial on which the first success occurs. The k − 1 failures come first, each with probability (1 − p), then one success with probability p.

Conditional probability and independence

P(A | B) = P(A and B)/P(B) independent if P(A | B) = P(A), equivalently P(A and B) = P(A)P(B)

Independence is a claim that conditioning changes nothing. It must be checked, not assumed, and it is a different idea from mutual exclusivity — two mutually exclusive events with nonzero probability are never independent.

Combining independent random variables

μ(X ± Y) = μX ± μY σ²(X + Y) = σ²X + σ²Y σ²(X − Y) = σ²X + σ²Y σ(X ± Y) = √(σ²X + σ²Y)

Variances ADD in both cases, and only if X and Y are independent. Standard deviations never add — you must square, add, and take the root.

Sampling distribution of x-bar

μ_(x-bar) = μ · σ_(x-bar) = σ / sqrt(n) · shape ≈ Normal if population Normal OR n ≥ 30 (CLT)

The standard deviation of x-bar (the "standard error" once σ is estimated) shrinks with the square root of n. Requires the sample be no more than 10% of the population for the formula to hold.

Sampling distribution of p-hat

μ_(p-hat) = p · σ_(p-hat) = sqrt( p(1−p) / n ) · Normal if np ≥ 10 and n(1−p) ≥ 10

Also requires the 10% condition (n ≤ 0.10·N) so trials are approximately independent. The standard deviation uses the true p (or p-hat when p is unknown).

Difference of two independent sample proportions

μ = p₁ − p₂ · σ = sqrt( p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂ )

The mean subtracts, but the variances (the terms under the root) add. The same add-the-variances logic gives σ = sqrt(σ₁²/n₁ + σ₂²/n₂) for a difference of means.

One-sample z-interval for a proportion

p-hat ± z\* · sqrt( p-hat(1−p-hat) / n )

z\* is the critical value (1.96 for 95%, 2.576 for 99%, 1.645 for 90%). The margin of error is everything after the ± sign.

One-sample z test for a proportion

z = (p-hat − p₀) / sqrt( p₀(1−p₀) / n )

Use the null value p₀ (not p-hat) in the standard error, since the p-value is computed assuming H₀ is true. Check Random, 10%, and Large Counts (np₀ ≥ 10, n(1−p₀) ≥ 10).

Errors and power at a glance

P(Type I) = α · P(Type II) = β · power = 1 − β

Type I = reject a TRUE null (false positive). Type II = fail to reject a FALSE null (false negative). Increasing n raises power without raising α.

Two-proportion inference

CI: (p-hat₁ − p-hat₂) ± z\*·sqrt( p-hat₁(1−p-hat₁)/n₁ + p-hat₂(1−p-hat₂)/n₂ ) · TEST SE: sqrt( p-hat_c(1−p-hat_c)(1/n₁ + 1/n₂) )

p-hat_c = (x₁ + x₂)/(n₁ + n₂) is the pooled (combined) proportion, used only in the test. The interval uses unpooled separate proportions.

One-sample t statistic and df

t = (x-bar − μ₀) / (s / sqrt(n)) · df = n − 1

Same structure as the z statistic, but s replaces σ and the reference distribution is t with n − 1 degrees of freedom. Larger n → t looks more like z.

One-sample t-interval for a mean

x-bar ± t\* · ( s / sqrt(n) ) with df = n − 1

t\* depends on the confidence level and df. For the same confidence level, t\* is larger than z\*, making t-intervals wider — the price of not knowing σ.

One-sample t test for a mean

t = (x-bar − μ₀) / ( s / sqrt(n) ) with df = n − 1

Compare the p-value from the t-distribution to α. Reject H₀ if p-value < α; otherwise fail to reject. For a two-sided Hₐ, double the one-tail area.

Paired and two-sample t procedures

paired: t = (x-bar_diff − 0) / (s_diff / sqrt(n)), df = n − 1 · two-sample: t = (x-bar₁ − x-bar₂) / sqrt( s₁²/n₁ + s₂²/n₂ )

Paired reduces to a one-sample t on the differences. The two-sample standard error adds the two variance contributions (independence). Use technology for the two-sample df.

Required sample size for a proportion

n = (z*/ME)² · p*(1 − p*) with p* = 0.5 when no prior estimate exists always round UP

For a mean the analogous expression is n = (z*σ/ME)², which requires an estimate of σ from a pilot study or prior data.

Chi-square statistic and GOF degrees of freedom

χ² = Σ (observed − expected)² / expected · expected = n·p₀ · df = (number of categories) − 1

For goodness of fit, df is the number of categories minus 1. The p-value is the area to the right of χ² under the chi-square distribution.

Expected count in a two-way table

expected = (row total × column total) / grand total

Computed for every cell under the assumption of no association. The Large Counts condition requires every expected count ≥ 5.

Chi-square test for a two-way table

χ² = Σ (observed − expected)² / expected · df = (r − 1)(c − 1)

Here r is the number of rows and c the number of columns. A 3×4 table has df = (3−1)(4−1) = 6. Always upper-tailed.

Choosing the test

one sample, two variables → INDEPENDENCE · multiple samples/groups, one variable → HOMOGENEITY

Both use χ² = Σ(O−E)²/E with df = (r−1)(c−1). Only the design and the wording of the hypotheses change.

Standardized slope and df

t = (b − β₀) / SE_b · df = n − 2

b is the sample slope, β₀ the hypothesized slope (usually 0), and SE_b the standard error of the slope (read from computer output). Two parameters estimated → df = n − 2.

Confidence interval for the slope

b ± t\* · SE_b with df = n − 2

t\* comes from the t-distribution with n − 2 degrees of freedom at the chosen confidence level. An interval that excludes 0 signals a statistically significant linear relationship.

t test for the slope

t = b / SE_b · df = n − 2 · (printed p-value is two-sided)

For H₀: β = 0. The standard computer output p-value tests β ≠ 0; for a one-sided alternative, divide it by 2 if the slope goes the predicted way.

LINER checklist

L linear · I independent · N Normal · E equal variance · R random

Linearity and Equal variance are judged from the residual plot; Normality from a graph of the residuals; Independence from the 10% condition; Random from the study design.

On the exam

When a prompt says "skewed" or mentions outliers, default to **median and IQR**. When it says "roughly symmetric" or "bell-shaped," the **mean and standard deviation** are appropriate. Matching the summary to the shape is a routine AP decision point.
Show the fence arithmetic explicitly on free response: state IQR = Q3 − Q1, multiply by 1.5, then add to Q3 and subtract from Q1. Graders want to see the boundary values, not just the word "outlier."
A reliable checklist: **shape, center, spread, unusual — in context, comparatively**. Four elements, two requirements. Say the variable name and units at least once. If the distributions are skewed, say so *and* switch to median and IQR in the same breath.
Keep three quantities distinct. The **raw score** is the measurement in its own units. The **z-score** says how many standard deviations from the mean it lies and is unitless. The **percentile** says what proportion of the distribution lies below it. They are three different answers to three different questions, and a question asking for one earns nothing for another.
Slope interpretations need three parts: the words "predicted" or "on average," the units of y, and the units of x. "For each additional *hour*, the *predicted score* increases by 6 *points*" scores; "the score goes up 6" does not.
Keep two "randoms" distinct: random **selection** (sampling) lets you *generalize* to a population; random **assignment** (experiments) lets you conclude *causation*. A study can have one, both, or neither — and that determines exactly what conclusions are allowed.
When asked to identify bias, name the *type* (undercoverage, nonresponse, response, or wording) **and** state its likely *direction* — will the estimate be too high or too low? AP rubrics reward explaining how the flaw pushes the result.
A classic free-response ending: "Can we conclude the treatment *caused* the difference, and can we generalize to all ___?" Answer both parts using the design — cite random *assignment* for causation and random *selection* for generalization. Missing either random feature limits the claim.
When a question asks whether a larger sample would fix a described problem, identify the problem first. If it is bias, the answer is no, and you must say **why** — the systematic tilt is present in every observation, so collecting more of them collects more of the same tilt.
The diagnostic question is simple: **is there a natural reason to link one observation in the first group with one specific observation in the second?** Before-and-after on the same person, left hand and right hand, twins, plots of land split in two — all paired. Two separate randomly assigned groups of different people — not paired.
When combining independent random variables, **variances add even when you subtract the variables**. To find the standard deviation of a difference X − Y, compute σ²_X + σ²_Y first, then take the square root. Never subtract standard deviations, and never add standard deviations directly.
The quickest way to tell binomial from geometric: is the *number of trials fixed*? Fixed n, count successes → **binomial**. Keep going until the first success, count the trials → **geometric**. The phrase "until" almost always means geometric.
Before finding a probability for x-bar, justify the Normal shape: either state the *population* is Normal, or invoke the **CLT with n ≥ 30**. Free-response answers that skip this justification lose the condition point even if the arithmetic is perfect.
Every two-sample standard error you will meet is built by the same move: compute each group’s variance contribution, **add** them, then square-root. Memorize the pattern once and it powers two-proportion and two-mean intervals and tests alike.
A question that says "the distribution of sample means" or "if we repeated this many times" is about the **sampling** distribution. A question that says "the heights of the 40 students in the sample" is about the **sample**. Reading which one is being asked about is often the whole item.
Always describe errors *in context*. On free response, spell out what rejecting or failing to reject means for the real situation, and state a consequence. "Type I error: concluding the water is unsafe when it is actually safe, causing a needless costly shutdown" earns full credit.
Match the standard error to the procedure: a two-proportion **test** pools into p-hat_c because H₀ assumes equality; a two-proportion **confidence interval** keeps the proportions separate because it makes no equality assumption. Using the wrong one is a common lost point.
The template: "Assuming [H₀ in context] is true, there is a p-value probability of observing a sample result at least as extreme as the one obtained purely by chance." Then the conclusion: "Because p < α, we reject H₀ and conclude there is convincing evidence that [H_a in context]." Contextual wording is required for the point in both sentences.
For small samples, you must address the shape condition explicitly: state that a dotplot/boxplot/stemplot of the data shows no strong skew or outliers, so it is plausible the population is approximately Normal. Skipping this on free response loses the conditions point.
Spot paired data by asking: is there a *natural one-to-one link* between the two measurements (same subject twice, twins, matched pairs)? If yes, take differences and run a **one-sample t**. Only use a two-sample t when the two groups are genuinely independent.
Free-response inference is scored in four parts: **state** the hypotheses or parameter, **plan** by naming the procedure and checking conditions, **do** the calculation, **conclude** in context and linked to the p-value or interval. A perfect calculation with no conditions and no context typically earns about half the available credit.
Expected counts are almost always decimals (like 18.4) — do not round them to whole numbers before computing χ². And check Large Counts on the **expected** counts, not the observed ones: every expected count must be at least 5.
Decide independence vs. homogeneity by counting samples: **one** random sample with two variables → independence; **several** samples/groups compared on one variable → homogeneity. State the matching hypotheses — "independent" vs. "same distribution across groups" — to earn the setup point.
A slope interval interpretation must name the *context and units*: "average change in [y-variable] per one-unit increase in [x-variable]." And always check whether **0 is inside** — that single fact tells you whether a linear relationship is plausible.
On free response, name each condition and cite the *specific evidence* you used: "the residual plot shows no leftover curve (Linear) and roughly constant spread (Equal variance)." Vague statements like "the conditions are met" without evidence do not earn the point.

How to get a 5

For every test write all four steps: hypotheses (parameters defined), name + conditions, statistic + p-value, and a conclusion linking p to α in context (name the variable).
Interpret every number in context with units — a bare "slope = 6.5" earns nothing without "points per hour."
Never say a result "proves" anything or that you "accept H₀"; say "convincing evidence" / "fail to reject." Causation needs a randomized experiment.
Distinguish the interval interpretation ("95% confident the true parameter is between…") from the confidence-level interpretation (about the method) — the exam tests both.
On every inference FRQ, follow the four-step structure — state hypotheses and define the parameter, check conditions by name, compute the statistic and P-value, conclude in context — skipping conditions is the most common lost point.
Write conclusions with the required phrasing: “convincing evidence for Hₐ” after rejecting, and never “accept H₀” — a large P-value only means the data fail to provide convincing evidence.
Interpretations of slope, r², intervals, and probabilities must include context and units; a formula-only answer earns partial credit at best on the AP rubric.
Distinguish random selection (supports generalizing to a population) from random assignment (supports cause-and-effect) — scope-of-conclusion questions appear on nearly every exam.
Show calculator inputs as statistical notation (e.g. z = (0.52 − 0.60)/√(0.6·0.4/150)) rather than calculator syntax; naked outputs like “2-PropZTest” can lose credit.
When stuck on probability, draw a tree diagram or two-way table first — most conditional probability traps disappear once the counts are laid out.

Key terms

z vs t for a mean — Use t when the population σ is unknown (nearly always); use z only when σ is known.

Interpreting a 95% confidence level — If we repeated the sampling many times, about 95% of resulting intervals would capture the true parameter.

Conditions for inference on a proportion — Random sample; 10% condition (n ≤ 0.10N); Large Counts: np̂ ≥ 10 and n(1−p̂) ≥ 10.

What the p-value measures — P(statistic as extreme or more, assuming H₀ true).

Type I vs Type II error — Type I: reject a true H₀ (prob α). Type II: fail to reject a false H₀ (prob β).

Power of a test — 1 − β = P(reject H₀ | Hₐ true). Increases with larger n, larger effect size, or larger α.

Observational study vs experiment — An experiment imposes a treatment; an observational study only records. Only an experiment can establish causation.

General confidence-interval form — statistic ± (critical value)(standard error)

Binomial mean and SD — μ = np; σ = √(np(1−p))

Unbiased statistic — Its sampling distribution is centered at the true parameter.

Central Limit Theorem — For large n, the sampling distribution of x̄ is approximately Normal regardless of population shape (n ≥ 30 rule of thumb).

Chi-square goodness-of-fit — One sample, one categorical variable; df = (number of categories − 1).

Describing a distribution (SOCS) — Shape, Outliers, Center, Spread — always in the context of the variable, with units.

When to prefer median and IQR — For skewed distributions or data with outliers; mean and standard deviation suit roughly symmetric data.

1.5 × IQR outlier rule — A point is an outlier if it lies below Q1 − 1.5(IQR) or above Q3 + 1.5(IQR).

z-score — z = (value − mean)/SD: how many standard deviations a value sits from the mean; unitless, so it compares across distributions.

Correlation r — Strength and direction of a linear relationship, between −1 and 1; unitless, not resistant to outliers, and never proof of causation.

Least-squares slope interpretation — For each 1-unit increase in x, the predicted y changes by the slope — always say “predicted” (or “on average”).

Residual — Actual y minus predicted ŷ. Positive means the point lies above the line; a curved residual plot signals a poor linear fit.

r² (coefficient of determination) — The fraction of variation in y explained by the least-squares regression on x.

Stratified vs. cluster sampling — Stratified: sample some units from every group. Cluster: randomly choose whole groups and survey everyone in them.

Common sampling biases — Undercoverage (frame misses part of the population), nonresponse (chosen people don’t answer), voluntary response (self-selection), and response bias (question wording or dishonesty).

Experiment vs. observational study — Only an experiment imposes treatments with random assignment, so only experiments support cause-and-effect conclusions.

Blocking — Grouping units that are similar on a known variable, then randomizing treatments within each block to reduce variability — the design analog of stratifying.