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The sample mean is $\bar X = \frac{1}{n}\sum_{i=1}^n X_i$ and the sample variance is $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar X)^2$

Can someone please explain how the sample mean and sample variance are independent?

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closed as unclear what you're asking by Silverfish, John, kjetil b halvorsen, Nick Cox, Xi'an May 1 '16 at 11:46

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ – gung Apr 30 '16 at 23:27
  • $\begingroup$ Are you sure you're not missing an assumption? Under the appropriate additional assumption(s), a few seconds with Google will reveal many proofs on the internet, including on this site. $\endgroup$ – Mark L. Stone Apr 30 '16 at 23:28
  • $\begingroup$ Please follow @gung's advice and let us know whether this is a self-study question. It's somewhat unclear what you are asking at the moment - are you asking what conditions are needed for the sample mean and variance to be independent, or are you asking for a proof that they are independent, but did not realise this required additional assumptions to hold? $\endgroup$ – Silverfish Apr 30 '16 at 23:57
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    $\begingroup$ And no -- that is not the sample mean. $\endgroup$ – StatsStudent May 1 '16 at 0:15
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    $\begingroup$ @Analyst1 I made a typo adding the Latex - it was actually correct in the original picture. Fixed now. $\endgroup$ – Silverfish May 1 '16 at 0:35
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The premise of the question is false - they aren't independent, in general.

Here's an example:

enter image description here

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