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Statistics

Formula Review

StatisticsFormula Review

13.2 The F Distribution and the F Ratio

 S S between = [ ( s j ) 2 n j ] ( s j ) 2 n   S S between = [ ( s j ) 2 n j ] ( s j ) 2 n 

S S total = x 2 ( x ) 2 n S S total = x 2 ( x ) 2 n

S S within =S S total S S between S S within =S S total S S between

dfbetween = df(num) = k – 1

dfwithin = df(denom) = nk

MSbetween = S S between d f between S S between d f between

MSwithin = S S within d f within S S within d f within

F = M S between M S within M S between M S within

F ratio when the groups are the same size: F = n s x ¯ 2 s 2 pooled n s x ¯ 2 s 2 pooled

Mean of the F distribution: µ = df(num) df(denom)1 df(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
  • s x ¯ 2 s x ¯ 2 = the variance of the sample means
  • s 2 pooled 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 = s 1 2 σ 1 2 s 2 2 σ 2 2 s 1 2 σ 1 2 s 2 2 σ 2 2

If σ1 = σ2, then F = s 1 2 s 2 2 s 1 2 s 2 2

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