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The F Distribution and the F-Ratio

Interpretation of the significance

The F Distribution and the F-Ratio

The distribution used for the hypothesis test is a new one. It is called the distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator.

For example, if  F follows an F distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F4,10.


The F distribution is derived from the Student’s t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution. One-Way ANOVA expands the t-test for comparing more than two groups. The scope of that derivation is beyond the level of this course.

To calculate the F ratio, two estimates of the variance are made.

  1. Variance between samples: An estimate of σ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation.
  2. Variance within samples: An estimate of σ2 that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation.
  • SSbetween = the sum of squares that represents the variation among the different samples
  • SSwithin = the sum of squares that represents the variation within samples that is due to chance.

To find a “sum of squares” means to add together squared quantities that, in some cases, may be weighted.

MS means “mean square.” MSbetween is the variance between groups, and MSwithin is the variance within groups.

Calculation of Sum of Squares and Mean Square

k = the number of different groups

nj = the size of the jth group

sj = the sum of the values in the jth group

n = total number of all the values combined (total sample size: ∑nj)

x = one value: ∑x = ∑sj

Sum of squares of all values from every group combined: ∑