### 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* = $\frac{df(num)}{df(denom)-1}$

where

*k*= the number of groups*n*= the size of the_{j}*j*group^{th}*s*= the sum of the values in the_{j}*j*group^{th}*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*(*n*_{1} – 1, *n*_{2} – 1)

*F* = $\frac{\frac{{s}_{1}^{2}}{{\sigma}_{1}^{2}}}{\frac{{s}_{2}^{2}}{{\sigma}_{2}^{2}}}$

If *σ*_{1} = *σ*_{2}, then *F* = $\frac{{s}_{1}{}^{2}}{{s}_{2}{}^{2}}$