Formula Review

13.2The F Distribution and the F Ratio

$SStotal=∑​x2−(∑​x)2nSStotal=∑​x2−(∑​x)2n$

$SSwithin=SStotal−SSbetweenSSwithin=SStotal−SSbetween$

dfbetween = df(num) = k – 1

dfwithin = df(denom) = nk

MSbetween = $SSbetweendfbetweenSSbetweendfbetween$

MSwithin = $SSwithindfwithinSSwithindfwithin$

F = $MSbetweenMSwithinMSbetweenMSwithin$

F ratio when the groups are the same size: F = $nsx¯2s2poolednsx¯2s2pooled$

Mean of the F distribution: µ =

where

• k = the number of groups
• nj = the size of the jth group
• sj = the sum of the values in the jth group
• n = the total number of all values (observations) combined
• x = one value (one observation) from the data
• $sx¯2sx¯2$ = the variance of the sample means
• $s2pooleds2pooled$ = the mean of the sample variances (pooled variance)

13.4Test of Two Variances

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

F = $s12σ12s22σ22s12σ12s22σ22$

If σ1 = σ2, then F = $s12s22s12s22$