Cohen's F Statistic: Definition, Formulas

Cohen’s f statistic is used as a measure of effect size in ANOVA and ANCOVA. The statistic gives an estimate of the proportion of variance explained by the categorical variable.

Cohen’s f is a ratio between two interim sums of squares: the term-sum-of-squares and error-sum-of-squares (or proportion of variance accounted for relative to the proportion of variance unaccounted for). It is similar to omega-squared, with one important difference [2]:

Cohen’s f estimates the proportion of variance in a sample,
Omega-squared estimates the proportion of variance for the population.

Interpreting Cohen’s F and F-Squared

Cohen [1] suggested the following interpretation for f when used in ANOVA / ANCOVA:

.10 = Small effect size,
.25 = Medium effect size,
.40 = Large effect size.

When f = 0, that’s an indication that the population means are all equal. As the means get further and further apart, f will grow indefinitely larger.

For f squared, the suggestions are:

.2 = Small effect size,
.15 = Medium effect size,
.35 = Large effect size.

Formulas for Cohen’s F Statistic

Cohen’s f-squared is defined as:

F-squared can be used as an estimate of effect size for R-squared in regression analysis. In ANOVA / ANCOVA it’s usually calculated by taking the square root, to get Cohen’s f statistic [3]:

You can also get Cohen’s f by transforming eta squared:

Cohen’s F = √(η² / (1 – η²))

You can convert between Cohen’s d and Cohen’s f² with the following formula (k is the number of groups):

When there are only two groups, Cohen’s f is the same as Cohen’s d [4].

References

[1] Cohen. J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum.
[2] Kotrlik, J. et al. Reporting and Interpreting Effect Size in Quantitative Agricultural Education Research. Journal of Agricultural Education. Volume 52, Number 1, pp. 132-142. DOI: 10.5032/jae.2011.01132
[3] UCLA Institute for Digital Research & Education Statistical Consulting. FAQ: How is Effect Size Used in Power Analysis? Retrieved November 16, 2021 from: https://stats.idre.ucla.edu/other/mult-pkg/faq/general/effect-size-power/faqhow-is-effect-size-used-in-power-analysis/
[4] Effect sizes for ANOVAs. Retrieved November 16, 2021 from: https://repo.miserver.it.umich.edu/cran/web/packages/effectsize/vignettes/anovaES.html#cohens-f