### 12.1 Test of Two Variances

if δ_{0}=1 then

Test statistic is :

### 12.3 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* = $\frac{M{S}_{\text{between}}}{M{S}_{\text{within}}}$

*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- ${s}_{\stackrel{\u2013}{x}}{}^{2}$ = the variance of the sample means
- ${s}^{2}{}_{pooled}$ = the mean of the sample variances (pooled variance)