Cohen's *d* is a perennial favorite in statistics, and you'll encounter it throughout your studies. Let's equip you with some helpful knowledge so you can wow your stats lecturer's pants off!

**Cohen's d is an effect size measure **(more on this in a moment)

**that tells you how large an effect is found for mean differences**.

This would apply to research questions like: "Do women and men differ in their shopping persistence?" or "Is FOMO (fear of missing out) higher in Gen Z than in Millennials?"

If you were conducting one of these studies, you'd first run an independent samples

*t*-test and then see if you get a significant result.

You would then want to know how big the difference in the mean between the two groups is and in what direction it is going. Although we would probably have our assumptions here even without data ....

In the following, I'l show you how you can

**make a statement about the practical relevance of an effect**found by calculating Cohen's

*d*after a significant hypothesis test.

Since there are several variants of this effect size measure (people like things to be a bit complicated in statistics),

**this article will be about Cohen's**.

*d*for the difference between two independent sample means"Independent samples" means that the value of interest was measured only once for all subjects and that the two samples are not connected in any way.

## What are effect sizes?

Effect sizes or effect size measures **show how strong an effect observed in a study is**.

Different effect size measures are used depending on the content of the study and the method used.

They fall into **two groups: distance measures and correlation measures.**

The former usually show the size of the distance between two means in standard deviation units, while the latter are usually variations of the Bravais-Pearson correlation coefficient *r*.

It's a nice trick that both groups can be converted into each other!

## Which area of statistics do they belong to?

Effect sizes belong to the world of inferential statistics, specifically the world of hypotheses testing, but they can also be used descriptively—that is, to describe only what is found in certain samples, without wanting to infer the associated population.

In inferential statistics, effect sizes should always be reported when a significance test is significant!

Unfortunately, there are different effect sizes for different significance test, so make sure you know exactly which one to use with which method.

## What is Cohen's *d*?

Cohen's *d* is one of the most commonly used effect size measures in statistics and **describes how much the means of two samples differ.**

Because Cohen's *d* is a standardized distance measure, it does not matter what instruments were used to measure the dependent variable. This is because standardization converts means measured with different instruments to a new common dimensionless unit, making them directly comparable.

Another advantage is that, unlike the *p*-value, Cohen's *d* does not depend on sample size.**Graphically, the difference between the means indicates how far apart the two means are on the x-axis.**

The closer they are, the smaller the *d—*and the further they are apart, the larger Cohen's *d* and thus the effect found.

In this graphic, the means are relatively far apart, so there's probably a small to medium size effect.

However, whether this effect is large enough to be statistically significant needs to be calculated!

## How to calculate Cohen's *d*

Here's the **formula for calculating the standardized mean difference in two independent samples:**

The formula consists of the difference between the two means of the two groups in the numerator, which is then divided by and thus standardized based on the (common) standard deviation in the denominator.

Since sample means ("x-bar") are usually used to infer expected values in the population ("mu"), the second, right-hand part of the formula is the one used in practice.

Because the standard deviations in the two groups should be roughly the same (one of the assumptions of the *t*-test), you can either use one of them or use the below formula to calculate the common or "pooled" standard deviation. Just check what's being done at your university.

**Let's apply this to the above question about the difference in shopping persistence between women and men.**

Shopping persistence is measured on a scale from 0 (= not at all) to 100 (= proud winner of the platinum-colored shopping bag).

Say we calculated the following from our data:

**- for the women (Group 1), a mean of 65.3 with a standard deviation of 4.5**

** - for the men (Group 2), a mean of 59.6 with a standard deviation of 3.9**

We now have the means for the numerator, but we still need the pooled standard deviation for the denominator.

There are two ways to do this:

#### SAME SAMPLE SIZE IN BOTH GROUPS

If the sample size is the same for both groups, use the following formula to calculate the pooled standard deviation:

For example, if we have a **sample size of 60 people in both groups**, the above formula looks like this:

Now we can** insert all the values into the top equation and get Cohen's d:**

**According to Cohen's (1988) conventions (see below), this would be a very strong and practically significant difference in means!**

#### different SAMPLE SIZE IN BOTH GROUPS

If, on the other hand, the sample size is unequal, you can use this beautiful formula for the pooled standard deviation:

**With sample sizes of 58 women (group 1) and 51 men (group 2), it looks like this:**

**Inserted into the original formula:**

**CONCLUSION:**

The result remains the same, as the different sample sizes have not changed anything (but this is not always the case!).

There is a very strong difference in the mean, i.e. men and women are very different in their shopping persistence (who would have thought!):

**Women are 1.35 standard deviations more persistent than men!**

## Conventions for interpretation

Fortunately, Cohen (1988) proposed some conventions for interpreting mean differences, which can be used as a **rough guide:**

Since Cohen's *d* can also be negative, the lines around the *d* indicate that these conventions apply to the absolute values, meaning that the value can be, for example, -0.2 or +0.2—the effect would in both cases be small.

Because these are only conventions or reference points, you should take into account the typical effects in a particular research area when interpreting them.

Thus, before making a final assessment of the strength of the effect you have found, look at different studies on your topic to get a sense of what might be considered a "small," "medium," or "large" effect in the field.

## Cohen's* d* with SPSS, R, & online calculators

SPSS only outputs d from version 27.

If you have a lower version or don't work with SPSS at all, you can of course calculate it with R or with online calculators like Psychometrica, a wonderful and very helpful site for calculating various statistical figures.

**Finally, a summary:**

## Summary Cohen's *d*

You made it—well done!

If you have a sample data set at hand (e.g. from Andy Field), then try calculating Cohen's *d* and check your results in Psychometrica, R or SPSS!

And now: Big rewards!

Statistics isn't the most fluffy and user-friendly subject.

So we should at least pamper ourselves during and after learning...

**references**

Cohen, J. (1988). *Statistical power analysis for the behavioral sciences* (2nd ed). Hillsdale, N.J: L. Erlbaum Associates.

Ellis, P. D. (2010). *The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results*. Cambridge: Cambridge University Press.

Field, A. (2018). *Discovering Statistics using IBM SPSS Statistics*. SAGE.