Cohen's d Statistic

Cohen's d statistic is a type of effect size. An effect size is a specific numerical nonzero value used to represent the extent to which a null hypothesis is false. As an effect size, Cohen's d is typically used to represent the magnitude of differences between two (or more) groups on a given variable, with larger values representing a greater differentiation between the two groups on that variable. When comparing means in a scientific study, the reporting of an effect size such as Cohen's d is considered complementary to the reporting of results from a test of statistical significance. Whereas the test of statistical significance is used to suggest whether a null hypothesis is true (no difference exists between Populations A and B for a specific phenomenon) or false (a difference exists between [Page 181]Populations A and B for a specific phenomenon), the calculation of an effect size estimate is used to represent the degree of difference between the two populations in those instances for which the null hypothesis was deemed false. In cases for which the null hypothesis is false (i.e., rejected), the results of a test of statistical significance imply that reliable differences exist between two populations on the phenomenon of interest, but test outcomes do not provide any value regarding the extent of that difference. The calculation of Cohen's d and its interpretation provide a way to estimate the actual size of observed differences between two groups, namely, whether the differences are small, medium, or large.

Calculation of Cohen's d Statistic

Cohen's d statistic is typically used to estimate between-subjects effects for grouped data, consistent with an analysis of variance framework. Often, it is employed within experimental contexts to estimate the differential impact of the experimental manipulation across conditions on the dependent variable of interest. The dependent variable must represent continuous data; other effect size measures (e.g., Pearson family of correlation coefficients, odds ratios) are appropriate for non-continuous data.

General Formulas

Cohen's d statistic represents the standardized mean differences between groups. Similar to other means of standardization such as z scoring, the effect size is expressed in standard score units. In general, Cohen's d is defined as

where d represents the effect size, μ1 and μ2 represent the two population means, and σ∊ represents the pooled within-group population standard deviation. In practice, these population parameters are typically unknown and estimated by means of sample statistics:

The population means are replaced with sample means (

,j) and the population standard deviation is replaced with Sp, the pooled standard deviation from the sample. The pooled standard deviation is derived by weighing the variance around each sample mean by the respective sample size.

Calculation of the Pooled Standard Deviation

Although computation of the difference in sample means is straightforward in Equation 2, the pooled standard deviation may be calculated in a number of ways. Consistent with the traditional definition of a standard deviation, this statistic may be computed as

where nj represents the sample sizes for j groups and s2j represents the variance (i.e., squared standard deviation) of the / samples. Often, however, the pooled sample standard deviation is corrected for bias in its estimation of the corresponding population parameter, σ∊. Equation 4 denotes this correction of bias in the sample statistic (with the resulting effect size often referred to as Hedge's <>

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