13.2 The F Distribution and the F Ratio

$S{S}_{\text{between}}={{\displaystyle \sum }}^{\text{}}\left[\frac{{(s_j)}^2}{n_j}\right]-\frac{{\left({{\displaystyle \sum }}^{\text{}}{s}_j\right)}^2}{n }$

$S{S}_{\text{total}}={{\displaystyle \sum }}^{\text{}}{x}^2-\frac{{\left({{\displaystyle \sum }}^{\text{}}x\right)}^2}{n}$

$S{S}_{\text{within}}=S{S}_{\text{total}}-S{S}_{\text{between}}$

df_between = df(num) = k – 1

df_within = df(denom) = n – k

MS_between = $\frac{S{S}_{\text{between}}}{d{f}_{\text{between}}}$

MS_within = $\frac{S{S}_{\text{within}}}{d{f}_{\text{within}}}$

F = $\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}$

F ratio when the groups are the same size: F = $\frac{n{s}_{\overline{x}}{}^2}{s^{\text{2}}{}_{pooled}}$

Mean of the F distribution: µ = $\frac{df(num)}{df(denom) \; -\mathrm{ \; 1}}$

where

• k = the number of groups
• n_j = the size of the j^th group
• s_j = the sum of the values in the j^th group
• n = the total number of all values (observations) combined
• x = one value (one observation) from the data
• $s_{\overline{x}}{}^2$ = the variance of the sample means
• $s^2{}_{pooled}$ = the mean of the sample variances (pooled variance)

13.4 Test of Two Variances

F has the distribution F ~ F (n₁ – 1, n₂ – 1)

F = $\frac{\frac{s_1^2}{{\sigma }_1^2}}{\frac{s_2^2}{{\sigma }_2^2}}$

If σ₁ = σ₂, then F = $\frac{s_1{}^2}{s_2{}^2}$