Formula Review

13.2 The F Distribution and the F Ratio

 SSbetween=[(sj)2nj](sj)2n  SSbetween=[(sj)2nj](sj)2n 

SStotal=x2(x)2nSStotal=x2(x)2n

SSwithin=SStotalSSbetweenSSwithin=SStotalSSbetween

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 = df(num)df(denom)  −  1df(num)df(denom)1

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)

13.4 Test of Two Variances

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

F = s12σ12s22σ22s12σ12s22σ22

If σ1 = σ2, then F = s12s22s12s22