Standardized Effect Size

In Effect Size we introduce the notion of effect size, and briefly mention Cohen’s d. We will now explain this concept further.

Definition 1: Cohen’s d, a statistic which is independent of the sample size and is defined as

Cohens d effect size

where m1 and m2 represent two means and σpooled is some combined value for the standard deviation.

The effect size given by d is conventionally viewed as small, medium or large as follows:

  •  d = 0.20 – small effect
  •  d = 0.50 – medium effect
  •  d = 0.80 – large effect

For single sample hypothesis testing of the mean, we use the following value for d

Cohens d one sample

Example 1: National norms for a school mathematics proficiency exam are distributed N(80,20).  A random sample of 60 students from New York City is taken showing a mean proficiency score of 75 (as in Example 1 of Single Sample Hypothesis Testing). Find the effect size for the sample mean.

Per Definition 1,

image413

which indicates a small effect. Note that the effect size is independent of the sample size. We should interpret d to mean that the sample mean is a quarter of a population standard deviation below the population mean.

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