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What is the shape of the distribution when the sample mean is greater than the population mean?

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closed as unclear what you're asking by Xi'an, John, Nick Cox, Andy, Sean Easter Sep 18 '16 at 17:31

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    $\begingroup$ How does this question arise? $\endgroup$ – Glen_b Sep 18 '16 at 1:23
  • $\begingroup$ How would you know that the sample mean is greater than the population mean, without knowing the population mean itself? It is difficult to imagine a situation in which you know the population mean without also knowing the shape of the distribution $\endgroup$ – Silverfish Sep 18 '16 at 12:09
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Observing a sample mean larger than the population mean doesn't really restrict the shape of the distribution enough to collectively describe it.

The only restrictions that immediately come to mind are that the population mean must be finite and the variance must not be 0. Essentially any distribution with a mean and positive variance will have some chance to observe a sample mean greater than the population mean (perhaps there could be some odd exceptions but none occur at present). That really doesn't give us enough information to characterize the shape. It could have essentially any shape.

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The sample mean isn't fixed to the distribution, just to the sample. Given a particular distribution, you can evaluate a new sample mean for an arbitrary number of samples of the same or different sizes as the first sample and get a different sample mean each time. The law of large numbers tells us that as the size of the sample grows, the sample mean converges to the population mean.

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