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Apologies if this is a really simple question; I'm sure if only I knew what to google I'd be able to find the answer myself, but it's been driving me mad.

I have two datasets with approximately gaussian distributions. Both are measurements of the same background distribution, taken for reproducibility of some optical instrumentation I've developed. I need to prove this using the two measurements.

My understanding is that to achieve this, I integrate common area that's underneath both measured distributions. However...

In my case, gaussian 1 has a mean of 41.3 and a standard deviation of 1.0. Gaussian 2 has a mean of 41.7 and a standard deviation of 1.6. This means that the two gaussians intersect twice.

When I integrate the common area, I get 0.76, which I interpret to mean there's a 0.76 probability that the two measurements are of the same background distribution. This sounds way too low to me.

I had a look at KL divergence, but this is asymmetric and assumes that one of the measured distributions is the 'true' distribution - this is not the case for my measurements.

I have some more similar comparisons with more than two measured distributions to worry about, but I'd like to walk before trying to run...

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    $\begingroup$ One can never actually measure an entire distribution--that's physically impossible. How, then, do you obtain these means and SDs? This is fundamentally important, for otherwise there is no disciplined correct way to answer your question. $\endgroup$
    – whuber ♦
    Jan 19, 2017 at 0:36
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    $\begingroup$ I'm measuring size distributions of particles. A computer algorithm interrogates the scattered light from single particles and uses this to infer a size of that particle. Each of my distributions is composed of a large number (thousands) of such measurements. The two distributions are measuring the same particles; as such I am confident of the same background distribution. $\endgroup$
    – dr_who_99
    Jan 19, 2017 at 8:57
  • $\begingroup$ Following @whuber's comments, it seems you use distribution in an un-statistical sense, which makes your question confusing. $\endgroup$
    – Xi'an
    Jan 19, 2017 at 10:09
  • $\begingroup$ @ Xi'an I've made an edit which hopefully clarifies the question. $\endgroup$
    – dr_who_99
    Jan 19, 2017 at 13:20
  • $\begingroup$ It's unclear what you are asking. Your question seems to be "I need to prove this," but what exactly does "this" refer to? That the optical measurements are reproducible? That the background particle size distributions are indeed the same? That they are indeed approximately Gaussian? (That, by the way, would be unusual for a particle size distribution: typically it's the cube roots of the sizes that tend to have Gaussian distributions.) And precisely what data--and how much of them--do you have to make your "proof"? $\endgroup$
    – whuber ♦
    Jan 19, 2017 at 19:00

2 Answers 2

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What you are looking for is a two-sample test for equality of distribution. There are a number of known tests of this kind, including the Wald-Wolfowitz two-sample runs test, the Friedman-Rafsky two-sample runs test, the Kolmogorov-Smirnov two-sample test, the Henze nearest neighbour test, and the Zeck-Aslan minimum energy test. There are probably many others, but these will get you started. The Kolmogorov-Smirnov two-sample test is a particularly common test which is easy to implement and which has an explicit formula to estimate the p-value.

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Concerning your comments on KL divergence:

I had a look at KL divergence, but this is asymmetric and assumes that one of the measured distributions is the 'true' distribution - this is not the case for my measurements.

I would suggest you use the symmetrized KL divergence: $$ KL_{sym}(P, Q) = (KL(P||Q) + KL(Q||P)) / 2 $$

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