Sphericity is an assumption about which of the following?

• A. Sphericity refers to equality of variances of raw scores across groups.
• B. Sphericity requires that the covariance between any two measurements be zero.
• C. Sphericity is a form of normal distribution.
• D. Sphericity is most commonly found in repeated-measures ANOVA but applies only where there are more than two points of data from the same participant.

Answer: (D)

Sphericity is about how consistent the relationships are among repeated measurements within the same person. Specifically, it means the variances of the difference scores between every pair of conditions are equal. This matters in repeated-measures ANOVA because if sphericity is violated, the F-tests can become too liberal, inflating Type I error, and corrections like Greenhouse-Geisser or Huynh-Feldt are used to adjust the degrees of freedom.

This assumption only becomes a concern when there are three or more related measurements for each participant. If there are only two points of data from the same person, there isn’t more than one difference score to compare, so the issue of equal variances of differences doesn’t arise—the condition is automatically satisfied. That’s why sphericity is most commonly discussed in the context of repeated-measures designs with more than two data points per participant.

The other statements describe different ideas: equality of raw-score variances across groups refers to between-subjects variance homogeneity, not the pattern of differences within a person; zero covariance between two measurements would imply independence, which isn’t what sphericity concerns; and normal distribution is about the overall shape of the data, not the specific property of equal variances of difference scores.