Statistics
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AP Statistics — Formulas & Tables

7 sections · 44 entries · print it and keep it beside your practice sets

AP Statistics supplies three pages: descriptive and probability formulas, inferential formulas, and the z / t / χ² tables. Because everything is given, the exam grades whether you can pick the right procedure, check its conditions, and interpret the output in context.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

Descriptive statistics

x̄
x̄ = (1/n) Σ xᵢ

Sample mean

s_x
s_x = √( (1/(n−1)) Σ (xᵢ − x̄)² )

Sample standard deviation

z
z = (x − μ) / σ

Standardized value (z-score)

How many standard deviations above the mean a value sits. Works for any distribution; only converts to a percentile if the shape is normal.

ŷ
ŷ = a + b x

Least-squares regression line

b
b = r ( s_y / s_x )

Slope from the correlation and the two standard deviations

a
a = ȳ − b x̄

Intercept — the line always passes through (x̄, ȳ)

r
r = (1/(n−1)) Σ ( (xᵢ − x̄)/s_x )( (yᵢ − ȳ)/s_y )

Correlation coefficient

−1 ≤ r ≤ 1, unitless, and unchanged if you swap x and y. r² is the proportion of variation in y explained by the model.

residual
residual = y − ŷ

Observed minus predicted

s
s = √( Σ (yᵢ − ŷᵢ)² / (n − 2) )

Standard deviation of the residuals

Probability & random variables

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

General addition rule

P(A | B)
P(A | B) = P(A ∩ B) / P(B)

Conditional probability

μ_X
μ_X = E(X) = Σ xᵢ pᵢ

Expected value of a discrete random variable

σ_X
σ_X = √( Σ (xᵢ − μ_X)² pᵢ )

Standard deviation of a discrete random variable

Combining
μ_(X±Y) = μ_X ± μ_Y; σ²_(X±Y) = σ_X² + σ_Y²

Sums and differences of independent random variables

VARIANCES add — even for a difference. Never add standard deviations.

Binomial
P(X = x) = C(n, x) p^x (1 − p)^(n − x)

n independent trials, probability p of success

Binomial μ, σ
μ = n p, σ = √( n p (1 − p) )

Mean and standard deviation

Geometric
P(X = x) = (1 − p)^(x − 1) p

Trials until the first success

Geometric μ, σ
μ = 1/p, σ = √(1 − p) / p

Mean and standard deviation

Sampling distributions (standard deviations)

x̄
μ_x̄ = μ, σ_x̄ = σ / √n

Sample mean of n observations

Use s/√n as the standard ERROR when σ is unknown, and switch to t.

p̂
μ_p̂ = p, σ_p̂ = √( p(1 − p) / n )

Sample proportion

x̄₁ − x̄₂
σ = √( σ₁²/n₁ + σ₂²/n₂ ), SE = √( s₁²/n₁ + s₂²/n₂ )

Difference of two independent sample means

p̂₁ − p̂₂
σ = √( p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂ )

Difference of two independent sample proportions

b (slope)
SE_b = s / ( s_x √(n − 1) )

Standard error of the regression slope

Inference — the two master templates

Test statistic
statistic = (point estimate − null value) / (standard error of the estimate)

Every z and t test on the exam has this shape

Confidence interval
point estimate ± (critical value) × (standard error of the estimate)

Every interval on the exam has this shape

Interpretation

What a confidence level means

A 95% confidence level means that in 95% of repeated samples the interval produced would capture the true parameter — not that there is a 95% chance the parameter is in this one interval.

p-value

What a p-value means

The probability of getting a result at least this extreme IF the null hypothesis were true. Small p ⇒ reject H₀. It is never the probability that H₀ is true.

Inference — specific procedures

One-sample z (proportion)
z = (p̂ − p₀) / √( p₀(1 − p₀)/n )

Testing H₀: p = p₀

Use p₀ in the standard error for the TEST, but p̂ for the confidence INTERVAL.

Two-proportion z
p̂_c = (X₁ + X₂)/(n₁ + n₂); z = (p̂₁ − p̂₂) / √( p̂_c(1−p̂_c)(1/n₁ + 1/n₂) )

Pooled (combined) proportion for the test

One-sample t (mean)
t = (x̄ − μ₀) / ( s / √n )

Testing H₀: μ = μ₀ with df = n − 1

Two-sample t
t = ( (x̄₁ − x̄₂) − (μ₁ − μ₂) ) / √( s₁²/n₁ + s₂²/n₂ )

Difference of means, unpooled

Matched pairs t
t = ( d̄ − μ_d ) / ( s_d / √n )

One-sample t on the differences, df = n − 1

χ²
χ² = Σ ( (observed − expected)² / expected )

Chi-square statistic for counts

χ² df
goodness of fit: df = k − 1; two-way table: df = (r − 1)(c − 1)

Degrees of freedom

Expected count
expected = (row total × column total) / table total

For a two-way table

t for slope
t = (b − β₀) / SE_b

Inference for the regression slope, df = n − 2

Conditions worth memorising (not on the sheet)

Random

Data come from a random sample or randomized experiment

10%

When sampling without replacement, n ≤ 0.10 N so observations are near-independent

Large counts

For proportions: n p̂ ≥ 10 and n(1 − p̂) ≥ 10

For a test use n p₀ ≥ 10 and n(1 − p₀) ≥ 10.

Normal / large sample

For means: the population is normal, or n ≥ 30, or the sample shows no strong skew or outliers

χ² counts

All expected counts ≥ 5

Tables supplied on the exam

Table A

Standard normal cumulative probabilities — area to the LEFT of z

Table B

t distribution critical values by df and confidence level

Common z* values: 1.645 (90%), 1.960 (95%), 2.576 (99%).

Table C

χ² critical values by df and tail probability

Practise with the sheet openExam-skill drillsCheatsheet