# Get overlap between two normally distributed variables

I have three variables ($X_G$,$X_B$, and $X_R$), each of which is normally distributed. Is there a way to show the overlap between each of the pairs using the mean and the standard deviation?

An independent t-test shows that the three groups are very much significantly different, which is good, but I want a way to show that one pair is most different (while two other pairs are equally different)

To show what I mean graphically I've plotted the distributions below;

i.e. red and green are very different, while blue and green are less different. Is there a good way to quantify this difference in overlap? The t-statistic for red and blue is 16.6789, while the t-statistic for green and red is 38.9686. I do find this kind of odd, given how much overlap red and blue seem to have though...

What I'd like is to be able to directly calculate the information presented in the nomogram here. However, I guess I'd thought there would be standard implementations in matlab/R to do this, but couldn't find any.

• Red-green colour blindness is common, meaning that graphs requiring separation of red and green are better avoided. – Nick Cox Jul 8 '13 at 9:38
• Run an ANOVA model and multiple comparisons. – Stéphane Laurent Jul 8 '13 at 11:20
• @NickCox you are of course correct - the plots were an afterthought (hence the lack of labels, titles, etc) – ash Jul 8 '13 at 14:52

Maybe this thread answer your question. It includes some R code to compute the percentage of overlap between 2 normal densities. Since densities add up to one, their proportions (% of overlap) might be directly compared.

Percentage of overlapping regions of two normal distributions

Is this what you were looking for?

• Yeah, basically, which is what I found when I was looking initially. I just thought there may be a more formal way of doing it (i.e. <insert name of statistician>'s test) – ash Jul 8 '13 at 17:26

You are basically asking about the size of the effect. For the difference in means between a pair of variables, Cohen's d is a (standardized) measure of such an effect size and Cohen presented an interpretation in terms of normal distributions' overlap in his book Statistical Power Analysis for the Behavioral Sciences. Note that this is not necessarily the only relevant notion of what “most different” could mean in this context.

Regarding the comment in the last paragraph, one important point is that the t statistic and other tests are based on the sampling distribution for the mean. They tell you how likely it would have been to observe a given difference if the null hypothesis were true, not how big this difference is. As the size of the sample increases, the variance of the sample mean decreases, as reflected in the lower standard error and narrower confidence intervals. Consequently, t can in principle be arbitrarily high and the p value arbitrarily small even if there is a lot of overlap and the difference is tiny in practical terms.

Unlike the standard error of the mean, the variance of the data themselves and the overlap between their distributions obviously does not tend toward 0. Intuitively, adding observations does not make them less different from each other (but it does make their distribution more precisely known, hence the smaller standard error of the mean).

• Not sure the OP is really interested in the overlap (in my opinion he/she's just looking for classical multiple comparisons). So @ash we're waiting for clarification. – Stéphane Laurent Jul 8 '13 at 13:14
• @StéphaneLaurent I thought I was interested in the overlap but (and I mean this in a totally non-sarcastic way!) I'm fully prepared to accept that I'm wrong. I've updated my question slightly. – ash Jul 8 '13 at 14:52
• @StéphaneLaurent I don't think so. I mean with 3 groups, you would often do an ANOVA first and if all pairwise tests allow you to reject equality you would naturally conclude that blue is between red and green. If the groups have the same size, you might even reach the right conclusion by comparing p-values but this is all a roundabout way to talk about effect sizes, which seem to be the real substantive question here. – Gala Jul 8 '13 at 15:03
• @ash Ok. I was under the impression you were just looking for a way to handle the three pairwise mean comparisons. – Stéphane Laurent Jul 8 '13 at 16:57