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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?

Blueberry
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  • Is this a question from a course or textbook? If so, please add the `[self-study]` tag & read its [wiki](http://stats.stackexchange.com/tags/self-study/info). – gung - Reinstate Monica Apr 30 '16 at 23:27
  • 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. – Mark L. Stone Apr 30 '16 at 23:28
  • 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? – Silverfish Apr 30 '16 at 23:57
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    And no -- that is not the sample mean. – StatsStudent May 01 '16 at 00:15
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    @Analyst1 I made a typo adding the Latex - it was actually correct in the original picture. Fixed now. – Silverfish May 01 '16 at 00:35
  • Its a proof questions, we just have to show why the two are independent – Blueberry May 01 '16 at 01:16
  • Possible duplicate of [What is the most surprising characterization of the Gaussian (normal) distribution?](http://stats.stackexchange.com/questions/4364/what-is-the-most-surprising-characterization-of-the-gaussian-normal-distributi) – Xi'an May 01 '16 at 11:46

1 Answers1

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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

Glen_b
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