F-Statistic Calculators

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The F-statistic is a ratio of two variance estimates used in hypothesis testing to compare group means (ANOVA) or to assess model fit (regression). Named after Ronald Fisher, F = MS_between / MS_within in one-way ANOVA, where MS is mean square (sum of squares divided by degrees of freedom). Under the null hypothesis that all group means are equal, F follows an F-distribution with numerator and denominator degrees of freedom. A large F indicates that between-group variance greatly exceeds within-group variance, providing evidence against the null hypothesis.

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F-Statistic Formula (One-Way ANOVA)

F = MS_between / MS_within

MS_between = SS_between / df_between = variance due to group differences. MS_within = SS_within / df_within = unexplained (error) variance. df_between = k − 1 (k = groups); df_within = N − k (N = total observations). Under H₀ (all means equal): F ≈ 1. Large F → reject H₀.

F in Regression

F = MS_regression / MS_residual = (SS_reg/df_reg) / (SS_res/df_res). Tests whether the regression model explains significantly more variance than a null model. df_reg = number of predictors (p); df_res = N − p − 1.

Interpreting F

  • Compare F to critical value from F-table at α = 0.05 and appropriate df_num, df_den
  • p < 0.05: significant — reject H₀; at least one group mean differs (ANOVA) or at least one predictor is useful (regression)
  • F ≈ 1: no evidence against H₀

Assumptions

  • Normally distributed residuals within groups
  • Equal variances (homoscedasticity — test with Levene's test)
  • Independent observations
  • Use Welch's ANOVA for unequal variances

Glossary

F-Statistic
The ratio of two variance estimates: MS_between/MS_within in ANOVA; F ≈ 1 under H₀; large F indicates group differences; compared to F-distribution for p-value.
Mean Square (MS)
Variance estimate in ANOVA: MS = SS/df; MS_between measures variance between groups; MS_within measures unexplained within-group variance (error).
Post-Hoc Test
Follow-up comparisons after a significant ANOVA to identify which specific group pairs differ; examples: Tukey HSD, Dunnett, Bonferroni; control family-wise Type I error rate.

Frequently Asked Questions

The F-statistic is the ratio of two variance estimates: F = MS_between / MS_within in ANOVA; F = MS_regression / MS_residual in regression. It tests whether group means differ (ANOVA) or whether a regression model has predictive value. Under H₀, both numerator and denominator estimate the same error variance, so F ≈ 1. Large F means between-group (or model) variance greatly exceeds unexplained variance — evidence against H₀.

F = MS_between / MS_within. Step 1: SS_between = Σnⱼ(ȳⱼ − ȳ)² (weighted squared deviations of group means from grand mean). Step 2: SS_within = ΣΣ(yᵢⱼ − ȳⱼ)² (sum of squared deviations within groups). Step 3: MS_between = SS_between/(k−1); MS_within = SS_within/(N−k). Step 4: F = MS_between/MS_within. Step 5: Find p-value from F-distribution with df₁ = k−1, df₂ = N−k.

A significant F (p < 0.05) means at least one group mean differs from others — but not which ones. Follow up with post-hoc tests: Tukey HSD (controls family-wise error rate, compares all pairs); Dunnett (compares all groups to a control); Bonferroni (conservative). Report effect size: η² = SS_between/SS_total (proportion of total variance explained by group). Large F with small effect size in a large sample may be statistically significant but practically meaningless — always report effect size alongside p-value.

ANOVA F: numerator = variance due to group membership (MS_between); denominator = within-group random error (MS_within); tests whether any group means differ. Regression F: numerator = variance explained by the regression model (MS_regression); denominator = unexplained residual variance (MS_residual); tests whether the model has overall predictive value. Both use the same F-distribution logic — comparing explained to unexplained variance — but the context and interpretation differ. In multiple regression, a significant overall F means at least one predictor is useful; individual t-tests identify which specific predictors are significant.