Statistics Formulas Cheat Sheet (Free, Printable)

📊 Descriptive Statistics

Meanx̄ = Σxᵢ / n

Sample variances² = Σ(xᵢ − x̄)² / (n − 1)

Sample std deviations = √s²

Population varianceσ² = Σ(xᵢ − μ)² / N

Coefficient of variationCV = s / x̄ × 100%

Interquartile rangeIQR = Q₃ − Q₁

Calculator Mean/Median/Mode guide

🎲 Probability

ComplementP(A′) = 1 − P(A)

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

Intersection, independentP(A∩B) = P(A) · P(B)

ConditionalP(A|B) = P(A∩B) / P(B)

Bayes' theoremP(A|B) = P(B|A)·P(A) / P(B)

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🔢 Counting

Factorialn! = n · (n−1) ··· 1, 0! = 1

Permutations (order matters)nPr = n! / (n − r)!

Combinations (order doesn't)nCr = n! / (r!·(n − r)!)

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🔔 Distributions

Z-score (standardize)z = (x − μ) / σ

BinomialP(X=k) = C(n,k) pᵏ (1−p)ⁿ⁻ᵏ

Binomial mean / varμ = np, σ² = np(1−p)

PoissonP(X=k) = e^(−λ) · λᵏ / k!, μ = σ² = λ

Normal Binomial Poisson Z-Score

📏 Confidence Intervals

Mean (σ unknown)x̄ ± t* · s/√n

Mean (σ known)x̄ ± z* · σ/√n

Proportionp̂ ± z* · √(p̂(1−p̂)/n)

Common z*90% → 1.645 · 95% → 1.96 · 99% → 2.576

Calculator Guide

🧪 Hypothesis Tests

Z-test (mean)z = (x̄ − μ₀) / (σ/√n)

One-sample tt = (x̄ − μ₀) / (s/√n), df = n−1

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

Paired tt = d̄ / (s_d/√n)

Two-proportion zz = (p̂₁ − p̂₂) / √(p̂(1−p̂)(1/n₁ + 1/n₂))

T-Test Two-Proportions P-value guide

🔗 Correlation & Regression

Pearson rr = Σ(xᵢ−x̄)(yᵢ−ȳ) / √[Σ(xᵢ−x̄)²·Σ(yᵢ−ȳ)²]

Regression lineŷ = b₀ + b₁x

Slope / interceptb₁ = r·(s_y/s_x), b₀ = ȳ − b₁x̄

FitR² = fraction of variance explained

Correlation Regression

📐 Effect Size

Cohen's dd = (x̄₁ − x̄₂) / s_pooled

Benchmarks0.2 small · 0.5 medium · 0.8 large

Correlation r0.1 small · 0.3 medium · 0.5 large

ANOVAη² = SS_between / SS_total

Calculator Guide

χ² & ANOVA

Chi-squareχ² = Σ (O − E)² / E

Cramér's VV = √(χ² / (n·(k−1)))

ANOVA F-ratioF = MS_between / MS_within

Chi-Square ANOVA ANOVA vs t-test