Q:

What is the probability that two samples represent the same normal distribution?

Yes, it's a basic question.  But, I have searched about 25 web pages for this and found only things that were irrelevant or incomprehensible.  So I have indeed tried.
My question is: I have two samples.  I know their sizes, means, and standard devs.  I just need a simple formula or procedure for determining the probability that they represent the same normal distribution.  I know it's out there.  In the distant past, I've used it myself.  But can't remember it now.  Can anyone point me to it?  Thanks.

A:

You're asking the wrong question.  This one can't be answered with the information you have.  What you can answer in classical statistics is something like this.  To test 
whether two samples might come from the same normal distribution, you can compute a certain
statistic $S$.  You can then ask: "If the two samples were from the same normal distribution, what is the probability that this statistic would be at least (or at most) the observed value?"  If that probability is small, it is evidence that the samples are not from the same normal distribution.  If it is not small, the test is inconclusive.
In this case you could, for example, use a Kolmogorov-Smirnov test.
Alternatively, you could use a Bayesian approach, but then you'd need more information (some prior probability model for what distributions the two samples could be from).

A:

Since both samples are normally distributed, you can use a t test to see if they are from the same distribution.
You can do this in Excel - T.TEST(array1,array2,tails,type), where tails can be 1 or 2 and type is either 1 - paired samples, 2 - assumes equal variance between the two, 3 - different variances between the two samples. In your case, it's probably safest to use 3.
This will give you a p-value, so the lower the value, the more likely it is that the two samples are from different populations.

