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. Marleen says:

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

  2. 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.

  3. 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.

  4. 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.

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