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. 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: Introduction to Statistics
| 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. |
| General addition rule | P(A or B) = P(A) + P(B) − P(A and B) | Add both, then take the overlap back out once. Drop the last term only when the events cannot both happen. |
| General multiplication rule | P(A and B) = P(A) · P(B | A) | The second factor is a plain P(B) only when the two events are independent. |
| Discrete random variable | μ = Σ x·p σ² = Σ (x − μ)²·p | Weight every value by its probability. The SD is √σ², as always. |
| Binomial mean and SD | μ = np σ = √(np(1 − p)) | The shortcut for X ~ B(n, p). The two rows above reach the same answers the long way. |
| Standard error of the mean | SE = s / √n | How much x̄ wobbles from sample to sample. |
| z-interval (σ known) | x̄ ± z* · σ/√n | z* = 1.645 / 1.960 / 2.576 for 90 / 95 / 99% confidence. |
| Margin of error | m = z* · σ/√n | Half the width of the interval. Confidence level, σ and n are the only three things that move it. |
| Sample size for a margin | n = (z* · σ / m)² | Round up. Halving the margin costs four times the data. |
| One-sample z (σ known) | z = (x̄ − μ₀) / (σ/√n) | The gap between sample and claim, measured in standard errors. |
| 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). |
| Two-sample standard error | SEdiff = √( s₁²/n₁ + s₂²/n₂ ) | Also the SE in the interval (x̄₁ − x̄₂) ± t*·SE. By hand use the conservative df = min(n₁ − 1, n₂ − 1); software uses Welch–Satterthwaite. |
| SD of a sample proportion | SD of p̂ = √( p(1 − p) / n ) | The spread of p̂ from sample to sample, centered on the true p. |
| Confidence interval (proportion) | p̂ ± z* · √( p̂(1 − p̂) / n ) | The interval is estimating p, so p̂ goes into the standard error. |
| One-proportion z test | z = (p̂ − p₀) / √( p₀(1 − p₀) / n ) | The test assumes H₀ is true, so p₀ goes in instead of p̂. |
| Two proportions, pooled p̂ | p̂ = (x₁ + x₂) / (n₁ + n₂), SE = √( p̂(1 − p̂)(1/n₁ + 1/n₂) ) | Under H₀ there is only one proportion in the world, so both groups estimate it together. |
| 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: ANOVA & Regression
| Concept | Formula | Read it as |
|---|---|---|
| Between-groups SS | SSG = Σ ni(x̄i − x̄)² | Each group mean’s squared distance from the grand mean, weighted by group size. |
| Within-groups SS | SSE = Σ (x − x̄i)² = Σ (ni − 1)si² | Every score against its own group mean. The right-hand form works straight from group SDs. |
| Total SS | SST = Σ (x − x̄)² = SSG + SSE | Every score against the grand mean; the identity is the arithmetic check. |
| ANOVA df | dfG = I − 1, dfE = N − I, dfT = N − 1 | I groups (some books write k), N observations in total. They add up. |
| Mean square | MS = SS / df | A sum of squares turned into a variance. |
| F-ratio (ANOVA) | F = MSG / MSE, on F(I − 1, N − I) | Signal over noise. With two groups, F = t². |
| Pooled SD | sp = √MSE | The one spread every follow-up comparison reuses. |
| Eta-squared | η² = SSG / SST | Share of outcome variance explained by group membership. |
| Contrast | ψ = Σ aiμi, with Σ ai = 0 | A weighted combination of group means. The zero-sum rule is what makes it a comparison. |
| Contrast estimate and SE | ψ̂ = Σ aix̄i, SE = √( MSE · Σ ai²/ni ) | Sample means in place of population means; the error term is borrowed from the ANOVA table. |
| Contrast test and CI | t = ψ̂ / SE on df = N − I, CI: ψ̂ ± t* · SE | Tested on the error df, not on the groups involved. The interval excludes zero exactly when the t clears t*. |
| Pairwise t after ANOVA | t = (x̄i − x̄j) / ( sp√(1/ni + 1/nj) ), df = N − I | The two-sample t with the pooled SD from every group, so more df than the pair alone would give. |
| Bonferroni correction | αeach = α / m | Split your alpha across m comparisons. |
| Bonferroni critical value | t** = t1−α/(2m) on df = N − I | The stricter bar a corrected comparison must clear. With 6 comparisons on 12 df it is 3.153 against a plain 2.179. |
| Simultaneous CI | (x̄i − x̄j) ± t** · sp√(1/ni + 1/nj) | The whole family of intervals holds at the stated level together, not one at a time. |
| Two-way df | dfA = I − 1, dfB = J − 1, dfAB = (I − 1)(J − 1), dfE = N − IJ, dfT = N − 1 | I levels of A, J of B, IJ cells. The five add up, and the design can be read back out of them. |
| Marginal mean and main effect | x̄i· = mean of row i's cells; main effect of A = difference of the row margins | The margin averages the cells, not the people, so it matches the raw average only when cells are equal. |
| Simple main effect (2 × 2) | A at Bj = x̄2j − x̄1j; interaction = (A at B2) − (A at B1) | Each main effect is the average of its own two simple main effects; the interaction is their difference. |
| Two-way SS (balanced, n per cell) | SSA = nJ Σ (x̄i· − x̄)², SSB = nI Σ (x̄·j − x̄)², SSAB = n ΣΣ (x̄ij − x̄i· − x̄·j + x̄)² | Margins against the grand mean, then the part of each cell its own row and column could not have told you. |
| Two-way error and total | SSE = Σ (x − x̄ij)², SST = SSA + SSB + SSAB + SSE | Every score against its own cell mean. With equal cells the four pieces add back to the total, which is worth a moment on any table you fill in. |
| Two-way F ratios | F = MSA/MSE, MSB/MSE, MSAB/MSE | Three tests, one error term, each on its own df over N − IJ. |
| Partial eta-squared | ηp² = SSeffect / (SSeffect + SSE) | What software prints for a factorial. The denominator changes per row, so the values do not add up. |
| 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 & Elective Topics
| 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 3, continued: modern methods
| Concept | Formula | Read it as |
|---|---|---|
| Bayes' theorem | P(θ | data) ∝ P(data | θ) · P(θ) | Posterior ∝ likelihood × prior. θ (theta) stands for whatever you are estimating. |
| 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). Π is a capital pi and means multiply, the way Σ (sigma) means add. |
| 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% for 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.