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This seems like it should be straightforward but I am at a loss.

How does one test the following hypothesis: the observed data have the same mean (or, alternatively, the observed data have the same variance)?

To provide context, I read the following online: "Here's another way of getting at whether there's a real pattern: How much variation in the standard deviations would you expect if all the players had the same underlying standard deviation (and the observed differences were just a result of sampling)?" It then occurred to me that I did not know what statistical test to run to confirm this. That is, if I have a set of numbers, how do I determine if observed differences between the numbers are the result of sampling uncertainty or if such differences are the result of variance in individual-specific means. Or is this impossible and the question is not well-specified. It would seem to me that if the observed sample were bi-modal or clustered around two points, for example, one could conclude that there are two population means generating the data, with observations clustered around these two points according to sampling uncertainty. But how does one draw such a conclusion more rigorously.

Any thoughts would be greatly appreciated.

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Sounds like you want the two-sample pooled t-test, equal variances.

See also here.

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Alternatively, if you want to test for the variability between those two, you could use F test (parametric test) or Levene's test (non-parametric version)

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