Statistics Formula Sheet
Every formula from the course, with a plain-English reading next to each one. Print it, fold it, and tape it inside your notebook. Click any course heading to go back to its lessons, where you can try each formula in an interactive.
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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 is 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 varies 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. Only the confidence level, σ and n change it. |
| Sample size for a margin | n = (z* · σ / m)² | Round up. Halving the margin needs 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 the standard error uses p₀, not p̂. |
| Two proportions, pooled p̂ | p̂ = (x₁ + x₂) / (n₁ + n₂), SE = √( p̂(1 − p̂)(1/n₁ + 1/n₂) ) | Under H₀ the two groups share one proportion, so both samples 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 |
|---|---|---|
| Pooled ranking | rank every value 1…N across both groups; tied values share the average of the ranks they cover | The four tests below use these ranks, never the values, so one wild outlier cannot move them. |
| Mann–Whitney U | U₁ = R₁ − n₁(n₁ + 1)/2, U₁ + U₂ = n₁n₂ | R₁ is group 1's rank sum in the pooled ranking. Some books subtract the rank sum from n₁n₂ + n₁(n₁ + 1)/2 instead, which names the same pair the other way round. Test the smaller U either way and reject when U ≤ U*, so a small statistic is the significant one here. |
| Wilcoxon signed ranks | W = the smaller of T₊ and T₋ | Rank the nonzero differences by size, ignoring sign, then total the ranks each sign carries. A zero difference is dropped and n falls with it. Reject when W ≤ W*. |
| Kruskal–Wallis H | H = [12 / (N(N + 1))] · Σ Ri²/ni − 3(N + 1), on χ²(I − 1) | The rank version of the one-way ANOVA. Unlike U and W it is compared with a χ² distribution, so here a large H is the significant one. |
| Spearman's rho | ρs = 1 − 6Σd² / (n(n² − 1)) | d is each pair's rank difference, after ranking the two variables separately. Σd is always 0 and Σd² always even. Reject when |ρs| ≥ ρs*. |
| 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. Because the weights sum to zero, it compares groups. |
| Contrast estimate and SE | ψ̂ = Σ aix̄i, SE = √( MSE · Σ ai²/ni ) | Sample means in place of population means; the error term comes 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 up to the total, so check that sum 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. |
| Repeated-measures SS | SST = SSC + SSS + SSE, with SSC = n Σ (x̄cond − x̄)², SSS = k Σ (x̄person − x̄)² | n people in k conditions. The subjects piece is split off, so it is not counted as error. |
| Repeated-measures error | SSE = Σ (x − x̄person − x̄cond + x̄)² | What is left in a score once its own person and its own condition are both accounted for. It is the person-by-condition interaction. |
| Repeated-measures df | dfC = k − 1, dfS = n − 1, dfE = (n − 1)(k − 1), dfT = nk − 1 | The error df is a product, not a subtraction. Read it backwards and a paper's F(3, 84) gives 4 conditions and 29 people. |
| Repeated-measures F | F = MSC / MSE, on F(k − 1, (n − 1)(k − 1)) | The subjects row gets no F of its own: people are expected to differ, and that is not the question. |
| Variance of a difference | Var(X − Y) = Var(X) + Var(Y) − 2·Cov(X, Y), Cov(X, Y) = r · SD(X) · SD(Y) | Why pairing works. The stronger the correlation between conditions, the more of the between-person variance cancels. |
| Mixed-design df | Between: g − 1 and N − g. Within: k − 1, (g − 1)(k − 1) and (N − g)(k − 1) | g groups, k occasions, N participants. Two error terms, so the group effect and the time effect are judged against different noise. |
| Chi-square | χ² = Σ (O − E)² / E | How far observed counts are from the counts expected under independence. |
| 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. |
| Covariance | cov(x, y) = Σ (x − x̄)(y − ȳ) / (n − 1) | The variance with one copy of x swapped for y. Its units are the two variables’ units multiplied, so it gives the direction and not the size. |
| Pearson r | r = Σ zxzy / (n − 1) | The average product of paired z-scores. |
| Regression slope | b = r · (sy / sx) = cov(x, y) / var(x) | Predicted change in y per one-unit rise in x. The two forms are the same number; use whichever pieces the question gives you. |
| 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. |
| Residual standard error | s = √( Σ (y − ŷ)² / (n − 2) ) | The typical size of a residual. Estimating the slope and the intercept uses up two df, so the tests below run on n − 2. |
| Standard errors of the line | SE(b₁) = s / √Σ (x − x̄)², SE(b₀) = s · √( 1/n + x̄² / Σ (x − x̄)² ) | Small scatter and widely spread x values make both smaller. The x̄² term grows when the intercept is far from the data; center x and it disappears. |
| Slope test and CI | t = b₁ / SE(b₁) on df = n − 2, CI: b₁ ± t* · SE(b₁) | The interval excludes zero exactly when the t clears t*. The intercept row works the same way, and you rarely need to report it. |
| Regression ANOVA | SST = SSR + SSE, df 1, n − 2 and n − 1; F = MSR / MSE | With one predictor F = t² and R² = SSR/SST, so the table repeats what the coefficient row said. It adds information once there are several predictors. |
| Multiple regression | ŷ = b₀ + b₁x₁ + … + bpxp | Each bⱼ is that predictor's effect holding the others constant. With p predictors every t and every s in the model runs on df = n − p − 1. |
| Omitted-variable bias | E[b₁] = β₁ + β₂δ, where δ (delta) is the slope of the omitted x₂ on the included x₁ | Fit y on x₁ alone and this is what you get. The bias is a product, so it is zero unless the omitted variable is tied to both the outcome and the predictor you kept, and its sign is the product of the two signs. |
| Overall F (regression) | F = (R²/p) / ((1 − R²)/(n − p − 1)), on F(p, n − p − 1) | One test of the whole model against no model at all. Rejecting says at least one predictor has a nonzero coefficient, never which one. |
| Multiple correlation | R = √R² | The plain correlation between y and the model's own predictions ŷ, so it can never be negative. |
| Adjusted R² | 1 − (1 − R²)(n − 1) / (n − p − 1) | R² with a penalty for each predictor. Adding a predictor lowers it when that predictor's |t| is below 1, and plain R² never falls. |
| Interaction model | ŷ = b₀ + b₁x + b₂m + b₃(x·m) | x's slope becomes b₁ + b₃m, so it depends on the moderator. |
| VIF | VIFj = 1 / (1 − R²j) | R²j comes from regressing predictor j on all the others. Worry near 10. |
| AIC / BIC | −2·lnL + 2p · −2·lnL + p·ln(n) | Fit plus a complexity penalty; lower wins. BIC's penalty grows with n. |
| Logistic model | ln( π / (1 − π) ) = b₀ + b₁x | A straight line on the log-odds scale. π (pi) is the probability of the outcome, written Greek here so that p can go on meaning the number of predictors. |
| Odds ratio | OR = eb | Multiplies the odds per one-unit rise in the predictor. b = 0 gives OR = 1, which is the null. |
| Wald test | z = b / SE(b), and SPSS prints z² as Wald χ²(1) | The logistic counterpart of the regression t. It is unreliable for very large coefficients, where a likelihood-ratio test is the safer one. |
Stats 3: Advanced & Elective Topics
| Concept | Formula | Read it as |
|---|---|---|
| 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). |
| One-factor model | yin = μi + βi ξn + εin | Item i for person n is its own mean, plus a loading times the one factor, plus what is left over. ξ (xi) is standardized to mean 0 and variance 1, so every row below is in correlation units. |
| Model-implied correlation and variance | σij = βiβj σii = βi² + ψii | Two items correlate only because they share the factor, so multiply their loadings and stop. An item's own variance splits into the part the factor explains and the part it does not. With m factors the first becomes a sum of m products. |
| Communality and uniqueness | hi² = Σj βij² ψii = 1 − hi² | Read along a row: the share of a standardized item the factors account for, and the share left to itself. Observed minus implied is a residual, and one past about .05 is a reason to read those two items side by side. |
| Eigenvalue | λj = Σi βij² | Read down a column instead: a factor's eigenvalue is the total of its squared loadings, so Σλ = Σh² and Σλ / p is the variance explained. An eigenvalue of 1 equals the variance of one standardized item. The Kaiser criterion comes from this. |
| Cronbach's α | α = (k / (k − 1))(1 − Σσii / σ²total) = k r̄ / (1 + (k − 1)r̄) | How much of a total score's variance the items share. The second form is the caveat: α (alpha) climbs with the number of items at a fixed average correlation, so a long scale reads as reliable while its items barely agree. Covariances enter with their signs, so a reverse-worded item that was not recoded drives α toward zero. |
| EFA fit-test df | df = ((p − m)² − (p + m)) / 2 | p(p + 1)/2 numbers in the matrix, less the pm + p parameters, plus m(m − 1)/2, because rotation leaves that many of them undetermined. Zero degrees of freedom means the model has as many parameters as numbers to fit, so it cannot be tested at all. |
| Percentile bootstrap CI | [2.5th, 97.5th percentile] | …of the statistic across thousands of with-replacement resamples. |
| Bayes' theorem | P(θ | data) ∝ P(data | θ) · P(θ) | Posterior ∝ likelihood × prior. θ (theta) stands for whatever you are estimating. |
| Bayes as a grid | P(Hi | data) = P(Hi)P(data | Hi) / Σj P(Hj)P(data | Hj) | The theorem as a table: a prior column times a likelihood column, divided by its own total, and that total is the evidence. Any constant shared by every row, including a binomial coefficient, cancels in the division. So updating batch by batch gives the same posterior as updating on the pooled count. |
| Beta–Binomial update | Beta(a, b) → Beta(a + k, b + n − k) | See k successes in n trials: add them straight into the prior. This is the conjugate pair for a binomial likelihood, and a + b is the number of trials the prior is worth. Normalize the grid above over the same hypotheses and the two agree exactly. |
| 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 that is between clusters. |
| Wald ratio (IV) | bIV = r₁ / a₁ | Reduced form over first stage: the instrument's effect on the outcome, divided by its effect on the regressor. Equals two-stage least squares when there is one instrument. |
| First-stage F | F = (a₁ / SE(a₁))² | Relevance as a number, and the only one of the three instrument conditions the data can check. Below about 10 the instrument is weak, and the estimate becomes both imprecise and biased toward ordinary least squares. |
| Regression discontinuity | y = a + b(x − c) + d·D + e(x − c)·D, D = 1 when x ≥ c | Center the running variable at the cutoff c so that d reads as the jump exactly there, and keep the interaction so each side gets its own slope. Forcing one slope across the cutoff is a common way to find a jump that is not there. |
| Difference-in-differences | y = b₀ + b₁·Treated + b₂·Post + b₃·(Treated × Post) | b₃ is the whole estimate. b₁ is the standing gap between the groups before anything happened and b₂ the change both would have seen anyway, so neither one is an effect. With two groups and two periods this is identical to unit and time fixed effects. |
| 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 get the most weight in the pooled estimate. |
| I² | share of spread beyond chance | ≈ 25 / 50 / 75% for low / moderate / high heterogeneity. |
If a row is only symbols to you, revisit its lesson. Every formula here has an interactive in its lesson where you can see it at work.