Interpretation of the Standardized Mean Difference Effect Size When Distributions Are Not Normal or Homoscedastic

Area of Science:

• Statistics
• Psychometrics
• Data Analysis

Background:

• The standardized mean difference (Cohen's d) is a prevalent effect size metric in experimental research.
• It quantifies the difference between two group means relative to their variability.
• Cohen's d is particularly intuitive for normally distributed data with equal variances.

Purpose of the Study:

• To examine the reliability of Cohen's d as a measure of distribution overlap.
• To investigate the impact of non-normality and unequal variances on Cohen's d interpretation.
• To assess the conditions under which Cohen's d interpretations remain valid.

Main Methods:

• The study theoretically analyzes the relationship between Cohen's d and distribution overlap.
• It considers scenarios with non-normally distributed data.
• It evaluates data with substantially unequal standard deviations.

Main Results:

• The mathematical relationship between Cohen's d and distribution overlap is straightforward for normal distributions with equal variances.
• Deviations from normality or equality of variances significantly alter the relationship between Cohen's d and distribution overlap.
• Standard interpretations of Cohen's d become unreliable under these conditions.

Conclusions:

• The interpretation of Cohen's d as an index of distribution overlap is contingent upon data meeting specific assumptions of normality and equal variances.
• Researchers must exercise caution when interpreting Cohen's d in the presence of non-normal data or unequal variances.
• Alternative effect size measures or interpretive frameworks may be necessary when standard assumptions are violated.