Can I print the statistics cheat sheet or save it as a PDF?

Yes. A free 2-page PDF is available at https://dackohn.com/stats/statistics-formula-sheet.pdf, or use the Print button: navigation, ads and links are hidden when printing, so you get a clean formula sheet.

Statistics Formula Sheet & Cheat Sheet (Free Printable PDF)

Q: Do I divide by n or n-1 for standard deviation?

Divide by n - 1 when your data is a sample (Bessel's correction), which is the usual case. Divide by N only when you have the entire population.

Q: Are the formulas checked?

Yes. Each formula links to a calculator whose results are verified against SciPy, statsmodels or scikit-learn, and the critical values on this sheet were computed with SciPy.

The five formulas you will use most: the mean x̄ = Σx / n; the sample standard deviation s = √(Σ(x − x̄)² / (n − 1)); the z-score z = (x − μ) / σ; the 95% confidence interval for a mean x̄ ± t*·s/√n; and the one-sample t statistic t = (x̄ − μ₀) / (s/√n). Below are 70+ formulas in 16 topics, from probability to sample size, effect size, inter-rater agreement and machine-learning evaluation metrics, plus a table of common critical values. Each topic links to a calculator that checks your work.

📊 Descriptive Statistics

Meanx̄ = Σxᵢ / n

Medianmiddle value of sorted data (average of the two middle values if n is even)

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

Outlier fencesbelow Q₁ − 1.5·IQR or above Q₃ + 1.5·IQR

CV calculator Percentile calculator Quartile calculator Outlier calculator Skewness calculator Variance step by step Calculator SD step by step 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)

Law of total probabilityP(A) = Σ P(A|Bᵢ)·P(Bᵢ)

Expected value / varianceE[X] = Σ x·P(x), Var(X) = E[X²] − (E[X])²

Expected value calculator Bayes calculator Bayes’ theorem guide Calculator

🔢 Counting

Factorialn! = n · (n−1) ··· 1, 0! = 1

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

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

Calculator

🔔 Distributions

Z-score (standardize)z = (x − μ) / σ

Empirical rule (normal)±1σ: 68.3% · ±2σ: 95.4% · ±3σ: 99.7%

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

Empirical rule calculator 68-95-99.7 guide Normal Binomial Poisson Z-Score

📉 Standard Errors new

MeanSE = s / √n

ProportionSE = √(p̂(1−p̂) / n)

Difference of two means (Welch)SE = √(s₁²/n₁ + s₂²/n₂)

Difference of two proportionsSE = √(p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂)

SD describes the data; SE describes the precision of an estimate.

Standard error calculator SE step by step SD vs SE guide

📏 Confidence Intervals

Mean (σ unknown)x̄ ± t* · s/√n, df = n − 1

Mean (σ known)x̄ ± z* · σ/√n

Proportion (Wald)p̂ ± z* · √(p̂(1−p̂)/n)

Proportion (Wilson, better near 0 or 1)[p̂ + z²/2n ± z√(p̂(1−p̂)/n + z²/4n²)] / (1 + z²/n)

Common z*90% → 1.645 · 95% → 1.960 · 99% → 2.576

Margin of error calculator Proportion CI calculator Calculator Step by step 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), df = n−1

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

Decision rulereject H₀ if p ≤ α

T-test from means and SDs Z-test calculator T-test by hand P-value calculator P-value step by step T-Test Two-Proportions Which test? P-value guide

🎯 Sample Size new

Estimate a mean to ±En = (z* · σ / E)²

Estimate a proportion to ±En = z*² · p̂(1−p̂) / E² (p̂ = 0.5 if unknown)

Compare two means (per group)n = 2(z₁₋α/₂ + z₁₋β)² σ² / Δ²

Power constants80% power → z₁₋β = 0.842 · 90% → 1.282

Classic result: ±5% at 95% confidence needs n = 385 responses.

Power calculator Sample size step by step Survey Two means A/B test

🔀 Nonparametric new

Mann–Whitney UU₁ = R₁ − n₁(n₁+1)/2, U = min(U₁, U₂), U₁ + U₂ = n₁n₂

Spearman's ρ (no ties)ρ = 1 − 6Σdᵢ² / (n(n² − 1))

Use when data is ordinal or clearly non-normal with a small sample.

Sign test calculator Friedman calculator Mann–Whitney U table

🔗 Correlation & Regression

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

Test of rt = r√(n−2) / √(1−r²), df = n−2

Regression lineŷ = b₀ + b₁x

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

FitR² = 1 − SS_res / SS_tot

r step by step Correlation Regression r table

📐 Effect Size

Cohen's dd = (x̄₁ − x̄₂) / s_pooled

Benchmarks (d)0.2 small · 0.5 medium · 0.8 large

Benchmarks (r)0.1 small · 0.3 medium · 0.5 large

ANOVAη² = SS_between / SS_total

Odds ratio (2×2: a b / c d)OR = (a·d) / (b·c)

Relative riskRR = [a/(a+b)] / [c/(c+d)]

Calculator Odds ratio Guide

χ² & ANOVA

Chi-squareχ² = Σ (O − E)² / E

Expected count (independence)E = (row total × column total) / n

Degrees of freedomdf = (rows − 1)(columns − 1)

Cramér's VV = √(χ² / (n·(k−1))), k = min(rows, cols)

ANOVA F-ratioF = MS_between / MS_within

Two-way ANOVA calculator Chi-square by hand Chi-Square ANOVA Tukey q table

🤝 Agreement & Reliability new

Cohen's κ (2 raters)κ = (pₒ − pₑ) / (1 − pₑ)

Fleiss' κ (3+ raters)κ = (P̄ − P̄ₑ) / (1 − P̄ₑ)

Krippendorff's αα = 1 − D_observed / D_expected

Rough guide (Landis & Koch)0.41–0.60 moderate · 0.61–0.80 substantial · >0.80 almost perfect

Cohen's κ Fleiss' κ Krippendorff's α

🧮 Multiple Comparisons new

Chance of ≥ 1 false positive in m tests1 − (1 − α)ᵐ (20 tests at 0.05 → 64%)

Bonferronireject if p ≤ α / m

Holm (step-down)sort p; reject p₍ᵢ₎ ≤ α / (m − i + 1) until the first failure

Benjamini–Hochberg (FDR)largest i with p₍ᵢ₎ ≤ (i/m)·α; reject 1…i

False positive risk P-hacking explained Calculator

🤖 ML & AI Evaluation new

Precision / recallP = TP/(TP+FP), R = TP/(TP+FN)

F1 scoreF1 = 2PR/(P+R) = 2TP / (2TP + FP + FN)

McNemar (two models, same items)χ² = (|b − c| − 1)² / (b + c)

pass@k (n samples, c correct)pass@k = 1 − C(n−c, k) / C(n, k)

Expected calibration errorECE = Σ (n_b/N) · |acc_b − conf_b|

Elo win probabilityP(A beats B) = 1 / (1 + 10^((R_B − R_A)/400))

Eval CI F1 CI McNemar pass@k ECE All AI eval tools

🧾 Common Critical Values new

z*, two-sided90%: 1.645 · 95%: 1.960 · 99%: 2.576

z*, one-sided 95%1.645

t*, 95% two-sideddf 5: 2.571 · 10: 2.228 · 20: 2.086 · 30: 2.042

χ², α = 0.05df 1: 3.841 · 2: 5.991 · 3: 7.815 · 4: 9.488

Computed with SciPy. Full tables below.

Critical value calculator Critical values guide Z table T table χ² table F table

Frequently asked questions

Most introductory exams rely on the mean, sample standard deviation, z-score, standard error, confidence intervals for a mean and a proportion, one- and two-sample t-tests, the chi-square statistic, Pearson correlation and the regression line. All of them are on this sheet; the which-test wizard helps you pick the right test.

Divide by n − 1 when your data is a sample (Bessel’s correction), which is the usual case. Divide by N only when you have the entire population. See how to calculate standard deviation.

Yes. Download the 2-page PDF directly, or use the Print button — navigation, ads, links and this FAQ are hidden when printing, so you get a clean two-column formula sheet.

Yes. Each topic links to a calculator whose results are verified against SciPy, statsmodels or scikit-learn, and the critical values above were computed with SciPy. See the methodology page.

Need critical values too? Browse the statistical tables → · Step-by-step workings? See the how-to guides → · New to a topic? Read the plain-English guides →

Statistics Formula Sheet (Cheat Sheet)