Mean squares are obtained by...

• A. Multiplying the sum of squares by the sample size.
• B. Dividing the sum of squares by the number of groups.
• C. Dividing a sum of squares by its degrees of freedom.
• D. Taking the square root of the sum of squares.

Answer: (C)

Mean squares come from turning a sum of squares into an average by adjusting for how much information you had to estimate it. In ANOVA you compute a sum of squares that captures the total squared deviation for a source (like between groups or within groups). To turn that into a mean square, you divide that sum by its degrees of freedom—the number of independent pieces of information used to estimate that source. This gives MS = SS/df, an estimate of the variability associated with that source, and it’s what you use in the F statistic to test significance. Taking a square root would give a standard deviation, not a mean square, and multiplying by the sample size or dividing by the number of groups are not how a mean square is formed.