12.3 The f distribution and the f-ratio

The distribution used for the hypothesis test is a new one. It is called the F distribution , invented by George Snedecor but named in honor of 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 ~ F _4,10 .

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

1. Variance between samples : An estimate of σ ² 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 σ ² 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.

• SS _between = the sum of squares that represents the variation among the different samples
• SS _within = 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. We used sum of squares to calculate the sample variance andthe sample standard deviation in [link] .

MS means " mean square ." MS _between is the variance between groups, and MS _within is the variance within groups.

Calculation of Sum of Squares and Mean Square

• k = the number of different groups
• n _j = the size of the j ^th group
• s _j = the sum of the values in the j ^th group
• n = total number of all the values combined (total sample size: ∑ n _j )
• x = one value: ∑ x = ∑ s _j
• Sum of squares of all values from every group combined: ∑ x ²
• Between group variability: SS _total = ∑ x ² – $\frac{\left({\displaystyle \sum x^2}\right)}{n}$
• Total sum of squares: ∑ x ² – $\frac{{\left(\sum x\right)}^2}{n}$
• Explained variation: sum of squares representing variation among the different samples:
  SS _between = $\sum \left[\frac{{(s_j)}^2}{n_j}\right]-\frac{{(\sum s_j)}^2}{n}$
• Unexplained variation: sum of squares representing variation within samples due to chance: $S{S}_{\text{within}}=S{S}_{\text{total}}–S{S}_{\text{between}}$
• df 's for different groups ( df 's for the numerator): df = k – 1
• Equation for errors within samples ( df 's for the denominator): df _within = n – k
• Mean square (variance estimate) explained by the different groups: MS _between = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$
• Mean square (variance estimate) that is due to chance (unexplained): MS _within = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$

MS _between and MS _within can be written as follows:

• $M{S}_{\text{between}}=\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}=\frac{S{S}_{\text{between}}}{k-1}$
• $M{S}_{within}=\frac{S{S}_{within}}{d{f}_{within}}=\frac{S{S}_{within}}{n-k}$

The one-way ANOVA test depends on the fact that MS _between can be influenced by population differences among means of the several groups. Since MS _within compares values of each group to its own group mean, the fact that group means might be different does not affect MS _within .