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The mean squares between groups ($$MS_B$$) is an estimate of population variance based on the differences among the sample means in an analysis of variance. It is calculated by dividing the sum of squares between groups by the between-groups degrees of freedom, and it serves as the numerator for the $$F$$ statistic.

Mean Squares Between Groups

To illustrate a one-way ANOVA, consider a health psychologist comparing the calorie estimates of three distinct groups: psychology majors, nutrition majors, and dieticians. The sample means are $$187.50$$, $$195.00$$, and $$238.13$$, respectively. Using statistical software or an ANOVA table, the researcher finds the between-groups mean square ($$MS_B$$) is $$5,971.88$$, the within-groups mean square ($$MS_W$$) is $$602.23$$, and the resulting $$F$$ ratio is $$9.92$$. With $$2$$ and $$21$$ degrees of freedom, the computed $$F$$ score ($$9.92$$) exceeds the critical value of $$3.467$$. Correspondingly, the $$p$$-value is $$.0009$$. Because this $$p$$-value is less than the standard $$.05$$ alpha level, the researcher rejects the null hypothesis and concludes that the population mean calorie estimates for the three groups are significantly different.

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