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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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  • $\begingroup$ Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. $\endgroup$ Apr 30, 2016 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. $\endgroup$ Apr 30, 2016 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, 2016 at 23:57
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    $\begingroup$ And no -- that is not the sample mean. $\endgroup$ May 1, 2016 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, 2016 at 0:35

1 Answer 1

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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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