One-Way ANOVA F-Statistic

Compare 2–4 group means from summary stats (n, mean, s) — get F, mean squares, and degrees of freedom for the table lookup.

InputF = MSB / MSW, MSB = Σ nᵢ(x̄ᵢ − x̄)² / (k − 1), MSW = Σ (nᵢ − 1)sᵢ² / (N − k)

Group 1 — sample size nGroup 1 — meanGroup 1 — std dev sGroup 2 — sample size nGroup 2 — meanGroup 2 — std dev sGroup 3 — sample size n — optionalGroup 3 — mean — optionalGroup 3 — std dev s — optionalGroup 4 — sample size n — optionalGroup 4 — mean — optionalGroup 4 — std dev s — optional

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The engineering

One-way ANOVA asks whether several group means differ by more than their internal scatter can explain — vendor lots, machine cells, test stations, oven zones. It splits total variation into a between-groups piece (how far the group means sit from the grand mean) and a within-groups piece (pooled sample variance), and F is the ratio of their mean squares. This card works from summary statistics, so you can run it straight off a data-sheet table without the raw readings.

F near 1 means the groups look like random draws from one population; the larger F gets, the less plausible that is. Compare against the F critical value at your α using the two df rows here (e.g., F₀.₀₅ for 2 and 12 df is 3.89 — the default-style example below at F = 5 rejects). Gotcha: the s fields are sample standard deviations, not variances — entering variance inflates SSW by s² and buries a real effect. Assumptions to keep honest: roughly equal variances (check the largest s isn't more than ~2× the smallest) and approximately normal groups.

With exactly two groups, F is the square of the pooled two-sample t statistic — a handy cross-check against your t-test result.

Where this math comes from

Analysis of variance came out of the mud at Rothamsted Experimental Station, where R. A. Fisher arrived in 1919 to make sense of decades of agricultural field trials. Comparing crop yields across plots meant separating treatment effects from soil noise, and Fisher's answer — partitioning the total sum of squares and comparing mean squares — appeared in his 1925 book Statistical Methods for Research Workers, written explicitly for working experimenters rather than mathematicians.

Fisher tabulated the test using z = ½ln(F); it was George Snedecor at Iowa State who, in 1934, published tables of the variance ratio directly and named the distribution F in Fisher's honor — Fisher reportedly wasn't consulted. The method became the backbone of industrial design of experiments, and Douglas Montgomery's Design and Analysis of Experiments (1976, now in its 10th edition) is the standard reference on most test engineers' shelves — including the notation this card uses.

1918R. A. FisherCoins the term 'variance' and its partitioning in his paper on Mendelian inheritance.
1925R. A. FisherPublishes Statistical Methods for Research Workers — ANOVA for experimenters.
1934George SnedecorTabulates the variance ratio directly and names the F distribution for Fisher.
1935R. A. FisherThe Design of Experiments formalizes randomization and replication behind the F test.
1976Douglas MontgomeryDesign and Analysis of Experiments becomes the standard industrial DOE reference.

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