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,


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.

7 Responses to Standardized Effect Size

  1. fita says:

    Thank you so much Mr. All of your explanation so clear and good. That’s very helpful for my thesis. And I want to say, again. Many thanks. I don’t have many word to say because I’m very happy I get what I want from this web.

  2. Helen says:

    Do you only need to calculate effect size on those who are significantly different to each other?

    Also on calculating some effect size a few of my answers were negative i.e. -3.065 and -0.385 is that ok? and if so then how do you interpret it?

    Thank you

    • Charles says:

      In my view, you should calculate an effect size in any case, but it is probably most useful when you have a significant result.
      Depending on the effect size measure that you use, you could get a negative value. This just indicates the direction of the effect. E.g. in calculating the effect size for the difference between the means of sample A and sample B where A has a higher mean, you will get a positive value if you subtract A from B and a negative value if you do the subtraction in the opposite order. Often it is the absolute value that is used and so the negative sign goes away.

  3. Jonathan Bechtel says:

    Hi Charles,

    Is there any tool in the Statistics Resource Pack that you can use to calculate Effect Size using Cohen’s D w/ excel?

    Thank you

    • Charles says:

      Hi Jonathan,
      The effect size (Cohen’s d) is included in a number of data analysis tools. E.g. see T Tests and Non-parametric Equivalents data analysis tool.

  4. Marleen says:

    Is it allowed to calculate Cohen’s d for nonparametric tests like the Mann-Whitney U?

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