# What is the probability (area) of overlap of two normal distributions having EQUAL variance

Can you please provide a math formula or an Excel example (math operations and Excel functions jointly used) to compute the probability (area) of overlap of two normal distributions having EQUAL variance?

The original data are as follows:

Mean 1 = 1145 Mean 2 = 1214 Standard Error of both distributions = 276 Variance = Standard Error^2 * square root of sample size = 276^2 * sqrt(30) = 417 233.14

There is one good example made by "wolfies" here.

He gives the formula to compute C and the formula together with Excel Erf function to compute the probability (area) of overlap of two distributions but it is for UNEQUAL variance.

I wish he could see my post.

• In the case where $\sigma_1=\sigma_2$, the value of $c$ is (obviously) midway between the means $c=(\mu_1+\mu_2)/2$. The overlapping area should be given by the formula in the answer. It's not clear to me why you'd use $\text{erf}$ though, since there's a normal cdf function in Excel, which would be considerably simpler to call (going from the $1-F_1(c)+F_2(c)$ line). – Glen_b -Reinstate Monica Jul 18 '17 at 11:38
• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. – gung - Reinstate Monica Jul 18 '17 at 12:22
• You also need to explain what you mean by "overlap" in this context. The normal distribution goes to infinity in both directions, so in one sense, they overlap completely. Are you asking the proportion of the area under one PDF is also under the other PDF? – gung - Reinstate Monica Jul 18 '17 at 12:24
• Yes @gung, I am asking about the proportion of the area under one PDF is also under the other PDF. – David_D Jul 18 '17 at 21:50
• @Glen_b, thanks for your hint, I think I came to the solution, because the proportions I get makes sense as compared to the chart and what I see there. No need for Erf function in this case, I just mentioned it in relation with wolfies post. Thanks to him as well. – David_D Jul 18 '17 at 22:02

## 1 Answer

In the case where $\sigma_1=\sigma_2$, the value of $c$ is (obviously) midway between the means $c=(\mu_1+\mu_2)/2$ (in the notation of the linked question). The overlapping area should be given by the formula in the answer. It's not clear to me why you'd use $\text{erf}$ though, since there's a normal cdf function in Excel, which would be considerably simpler to call (going from the $1-F_1(c)+F_2(c)$ line).