I’m a tall guy at 6’3″ while my wife is little short at 5′. In fact, men have an average height of 70.00″ while women have an average height of 64.2″. It’s easy to see a difference in the means, but *how* different are they? Enter **effect size**, the measure of difference between two means.

## What is Effect Size in the First Place?

Effect size is the statistics way to say how different two means are. For instance, it’s pretty straightforward to say that men are taller than women because the difference in the means is pretty big. But what about the difference between the means of personalized learning and engagement?

One of the big advantages of effect size is that it can be used to compare two scales that may be different. It works by comparing the means in terms of standard deviation. But before we get too much further, let’s take a look at how effect size is calculated.

Effect size is noted by the letter *d*. It is sometimes referred to simply as Cohen’s d after the statistician that came up with it. The next part involves the actual difference between the means. In our case it is 64.2 – 70.0 = -5.8

*But John, which one goes first?*

That is a great question because it makes you think about the study itself. When determining effect size the control group is usually the X1 and the other group is X2. This gives us a sense of how much bigger or greater the difference between the means is. I decided that women are the control group because I want to put my wife’s group first.

But the standard deviation is a little more… interesting. Instead of being a standard deviation of one group or the other, it is a pooled standard deviation. What I mean is that we calculate a new standard deviation based on the sample sizes and sample deviations. The formula is

Now, don’t be scared. There are a lot of numbers and letters here, but we will cover them.

Each *s* stands for a standard deviation of a group. The *n* stands for the sample size for the groups. Whichever group you designated as X_{1} uses *s*_{1} and *n*_{1}. The same goes for X_{2}.

The nifty thing is that it does not really matter what order you put them in during calculation since it is addition. Another nifty thing is that once you have the standard deviations, both means are on the same scale regardless of whether they measure the same thing! This means that we can compare them to see if they are truly different. Yet another nifty thing is that I have already calculated the pooled standard deviation for our example to be 2.8

So our final calculation is

## But What Does That Mean?

When he came up with Cohen’s d, Cohen provided a nice general interpretation of the d-values.

d = 0.20 represents a small effect. This means that they are not that different.

d = 0.50 represents a moderate effect. that means that they are different.

d = 0.80+ represents a large effect, which means that they are really different.

Since our value is greater than the absolute value of 0.80, we can say that there is a large difference between the height of females and males. We interpret this to mean that females are 2.07 standard deviations shorter than males.

### Caution!

The calculations for effect size look very similar to those for a z-score or *t*-test, but do not be tempted to think that they are the same thing. The key lies in interpretation.

z-scores are based on the concept of determining the difference of a single score inside the distribution of the data. *t*-tests are a method of inferential statistics used to determine the difference of two means that exist on the same scale.

Effect sizes are not as bizarre as they sound. They have fairly simple calculations with a singular goal. The goal of an effect size to determine the *difference* between two means whether they are on the same scale or not. It is a method of comparison that allows you to compare apples to oranges. Post any questions that you may have and happy statistics!

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