In analysis of variance (ANOVA), what do the $\mathrm{MS}$ (mean square) values measure?

• A. They measure the variance within and between groups by dividing the $\mathrm{SS}$ (sum of squares) by their respective $\mathrm{df}$ (degrees of freedom).
• B. They measure the standard deviation of the residuals.
• C. They measure the total number of observations in the experiment.
• D. They measure the mean of the data in each group.

Answer: (A)

Verified step by step guidance

1. Understand that in ANOVA, the goal is to compare variability within groups to variability between groups to determine if group means differ significantly.
2. Recall that the Sum of Squares (SS) quantifies the total variation: SS\_Between measures variation between group means, and SS\_Within measures variation within groups.
3. Recognize that Mean Square (MS) values are calculated by dividing each Sum of Squares by their respective degrees of freedom (df), which adjusts for the number of groups or observations.
4. Express the formulas as: \(MS_{Between} = \frac{SS_{Between}}{df_{Between}}\) and \(MS_{Within} = \frac{SS_{Within}}{df_{Within}}\), where \(df_{Between} = k - 1\) and \(df_{Within} = N - k\), with \(k\) being the number of groups and \(N\) the total observations.
5. Interpret MS values as estimates of variance: \(MS_{Between}\) estimates variance due to group differences, and \(MS_{Within}\) estimates variance due to random error or residuals within groups.