Statistics Formula Sheet
Every formula from the course, with a plain-English reading next to each one. Print it, fold it, tape it inside your notebook — and click any course heading to revisit the lessons where the formula is a toy you can play with.
Need a critical value? → Tables & calculators
More one-page printables: the which-test cheat sheet, the APA reporting cheat sheet, and the assumption-checks cheat sheet.
Stats 1 — Foundations
| Concept | Formula | Read it as |
|---|---|---|
| Mean | x̄ = Σxᵢ / n | Add everything, divide by the count. |
| Sample variance | s² = Σ(xᵢ − x̄)² / (n − 1) | Average squared distance from the mean (n − 1 because it's a sample). |
| Standard deviation | s = √s² | Typical distance from the mean, back in the original units. |
| Z-score | z = (x − μ) / σ | How many SDs a value sits above or below the mean. |
| Standard error of the mean | SE = s / √n | How much x̄ wobbles from sample to sample. |
| Confidence interval (mean) | x̄ ± t* · SE | Estimate ± margin of error; t* from df = n − 1 (≈ 2 for 95%). |
| One-sample t | t = (x̄ − μ₀) / (s / √n) | Distance from the reference value, in standard-error units. df = n − 1. |
| Paired t | t = d̄ / (sd / √n) | The one-sample t, run on each person's difference score. |
| Independent t | t = (x̄₁ − x̄₂) / SEdiff | The gap between two means over its standard error (Welch: no pooling). |
| Cohen's d | d = (x̄₁ − x̄₂) / spooled | Mean difference in SD units — 0.2 / 0.5 / 0.8 ≈ small / medium / large. |
| Empirical rule | 68% · 95% · 99.7% | Share of a normal distribution within ±1, ±2, ±3 SD. |
Stats 2 — Comparing Groups & Relationships
| Concept | Formula | Read it as |
|---|---|---|
| F-ratio (ANOVA) | F = MSbetween / MSwithin | Signal over noise; each MS = SS / df. |
| ANOVA df | dfb = k − 1, dfw = N − k | k groups, N total observations. |
| Eta-squared | η² = SSbetween / SStotal | Share of outcome variance explained by group membership. |
| Bonferroni correction | αeach = α / m | Split your alpha across m comparisons. |
| Chi-square | χ² = Σ (O − E)² / E | How far observed counts stray from what independence expects. |
| Expected count | E = (row total × column total) / N | Each cell's count if the two variables were unrelated. |
| Chi-square df | df = (r − 1)(c − 1) | Rows and columns of the contingency table. |
| Cramér's V | V = √( χ² / (N · (min(r,c) − 1)) ) | Effect size for χ², from 0 to 1. |
| Pearson r | r = Σ zxzy / (n − 1) | The average product of paired z-scores. |
| Regression slope | b = r · (sy / sx) | Predicted change in y per one-unit rise in x. |
| Intercept | a = ȳ − b·x̄ | The least-squares line always passes through (x̄, ȳ). |
| R² | R² = 1 − SSres / SStot | Share of y's variance the model explains. |
Stats 3 — Advanced Modeling
| Concept | Formula | Read it as |
|---|---|---|
| Multiple regression | ŷ = b₀ + b₁x₁ + … + bkxk | Each bⱼ is that predictor's effect holding the others constant. |
| Adjusted R² | 1 − (1 − R²)(n − 1) / (n − k − 1) | R² with rent charged per predictor. |
| VIF | VIFj = 1 / (1 − R²j) | How much overlap with other predictors inflates bⱼ's variance (worry near 10). |
| Interaction model | ŷ = b₀ + b₁x + b₂m + b₃(x·m) | x's slope becomes b₁ + b₃m — it depends on the moderator. |
| Indirect effect | a × b | X→M path times M→Y path; test it with a bootstrap CI. |
| Effect decomposition | c = c′ + a·b | Total effect = direct + indirect (through the mediator). |
| Logistic model | ln( p / (1 − p) ) = b₀ + b₁x | A straight line — on the log-odds scale. |
| Odds ratio | OR = eb | Multiplies the odds per one-unit rise in the predictor. |
| AIC / BIC | −2·lnL + 2k · −2·lnL + k·ln(n) | Fit plus a complexity penalty; lower wins. BIC punishes harder as n grows. |
Stats 4 — Modern & Advanced
| Concept | Formula | Read it as |
|---|---|---|
| Bayes' theorem | P(θ | data) ∝ P(data | θ) · P(θ) | Posterior ∝ likelihood × prior. |
| Beta–Binomial update | Beta(a, b) → Beta(a + k, b + n − k) | See k successes in n trials: add them straight into the prior. |
| Percentile bootstrap CI | [2.5th, 97.5th percentile] | …of the statistic across thousands of with-replacement resamples. |
| GLM | g(μ) = b₀ + b₁x₁ + … | Link g puts the line on the right scale: logit (binary), log (counts). |
| Intraclass correlation | ICC = τ² / (τ² + σ²) | Share of total variance sitting between clusters. |
| Kaplan–Meier | Ŝ(t) = Π (1 − dᵢ / nᵢ) | Multiply survival through each event time (censored cases stay in nᵢ while observed). |
| Cox model | h(t) = h₀(t) · eb₁x₁ + … | eb is the hazard ratio for that predictor. |
| Meta-analysis weight | wᵢ = 1 / vᵢ (random-effects: 1 / (vᵢ + τ²)) | Precise studies pull the pooled estimate hardest. |
| I² | share of spread beyond chance | ≈ 25 / 50 / 75% — low / moderate / high heterogeneity. |
Formulas are the receipts, not the idea. If any row feels like symbols instead of a picture, that's the lesson to revisit — every one of these has an interactive where you can watch the formula happen.