The Three Test Distributions — t, χ², and F
Compare the t, chi-squared, and F distributions side by side. Drag degrees of freedom to see how each curve morphs — and why each appears in its own corner of statistical inference.
I.05 / t · χ² · F DISTRIBUTIONS
The Three Test Distributions
Up to now we've tested means assuming σ is known. In practice you must estimate σ too — and the moment you do, Z morphs into t. Test a variance directly: χ². Compare two variances: F. All descendants of N(0,1); the name changes based on what you don't know.
t, χ², F are all derived from the normal. Think of them as "the standard normal, scaled to reflect that we only ever see a sample".
Use them for: t — testing a mean when the population variance is unknown (i.e. nearly every real test of a mean);
χ² — testing a variance, independence, goodness-of-fit for categorical data;
F — ratios of variances (ANOVA, the overall F in regression).
Slide df: t converges to N(0,1) as df→∞, and χ²/F get more symmetric with more df. The CLT is quietly doing the work under the hood.
▶ t distribution
Built from: t = Z / √(χ²ₖ/k), Z~N(0,1).
Use for: testing means with unknown variance, regression t-values.
Flavor: heavier tails than N(0,1); matches N(0,1) as df→∞.
Use for: testing means with unknown variance, regression t-values.
Flavor: heavier tails than N(0,1); matches N(0,1) as df→∞.
↔ Drag the graph horizontally to change df
▶ χ² distribution
Built from: χ²ₖ = Z₁² + Z₂² + ... + Zₖ² (sum of k squared standard normals).
Use for: variance tests, chi-square tests of independence / goodness-of-fit.
Flavor: non-negative, right-skewed. Mean = k, variance = 2k. Goes bell-shaped with large df.
Use for: variance tests, chi-square tests of independence / goodness-of-fit.
Flavor: non-negative, right-skewed. Mean = k, variance = 2k. Goes bell-shaped with large df.
↔ Drag the graph horizontally to change df
▶ F distribution
Built from: F = (χ²ₘ/m) / (χ²ₙ/n) (ratio of two independent χ² / df).
Use for: ANOVA, overall F-test in regression.
Flavor: non-negative, right-skewed. Shape depends on both df.
Use for: ANOVA, overall F-test in regression.
Flavor: non-negative, right-skewed. Shape depends on both df.
↔ Drag horizontally to change df₁ · df₂ is set via slider