Statistics Dictionary

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Sums of Squares

A sum of squares is the sum of squared deviations from a mean score. For example, a one-way analysis of variance makes use of three sums of squares:

  • Between-groups sum of squares. The between-groups sum of squares (SSB) measures variation of group means around the grand mean. It can be computed from the following formula:
    SSB =
    kΣj=1
    n jΣi=1
    X  j - X )2  = 
    kΣj=1
    nj ( X  j - X )2
  • Within-groups sum of squares. The within-groups sum of squares (SSW) measures variation of all scores around their respective group means. It can be computed from the following formula:
    SSW =
    kΣj=1
    n jΣi=1
    ( X i j - X j )2
  • Total sum of squares. The total sum of squares (SST) measures variation of all scores around the grand mean. It can be computed from the following formula:
    SST =
    kΣj=1
    n jΣi=1
    ( X i j - X )2

It turns out that the total sum of squares is equal to the between-groups sum of squares plus the within-groups sum of squares, as shown below:

SST = SSB + SSW

See also:  One-Way Analysis of Variance