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I am trying to understand if the division for n-1 in the calculation of standard deviation related to degrees of freedom, unbiasing of the estimator, or both.

I understand what is a degree of freedom when we talk about t, chi, f distribution and so on.

I was also trying to understand what is a degree of freedom in respect of the standard deviation.

I became aware of another explanation, which relates to the bias of the estimation of $\sigma$.

Now I am confused: why do we divide by $n-1$? Is it because of the degrees of freedom (and in this case, what does it actually mean?) or is it because we want to have an unbiased estimator? Or are the two explanations connected in some way?

(Is it normal that the concept of DOF is so slippery and difficult to understand? Or is that just me?)

--I have read the suggested topic, and I think that it only apparently answered my question. The OP of that topic asked about degrees of freedom, and the answer talked about biased estimator. I don't understand how they relate to each other.

Vaaal
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  • http://stats.stackexchange.com/questions/16008/what-does-unbiasedness-mean – ocram Jul 30 '13 at 14:28
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    Degrees of freedom cause degrees of frustration for almost everyone, but see http://stats.stackexchange.com/questions/16921/how-to-understand-degrees-of-freedom In terms of the question, the usual motivation for n - 1 as divisor is that this makes the estimator of variance unbiased, but this does _not_ make the estimator of SD unbiased, although the bias is usually minor except for small n. – Nick Cox Jul 30 '13 at 14:28
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    Concerning the bias in the SD estimator see http://stats.stackexchange.com/questions/11707/why-is-sample-standard-deviation-a-biased-estimator-of-sigma. For more about this subject generally, please [search our site](http://stats.stackexchange.com/search?tab=relevance&q=standard%20deviation%20unbiased%20estimator). Another [search](http://stats.stackexchange.com/search?q=degrees+freedom+standard+deviation) turns up a hundred posts directly addressing the main question. – whuber Jul 30 '13 at 15:54
  • I understand why we have to use an unbiased estimator, but I don't understand what THIS has to do with degrees of freedom. – Vaaal Jul 30 '13 at 16:04
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    At present, I think this question can remain closed. If you can clarify how your question is distinct from what's in the comments here, in the linked thread, & already covered on the site, maybe we car re-open it for you. – gung - Reinstate Monica Jul 30 '13 at 17:04
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    I agree with @gung: it seems to me that in the duplicate thread James Stanley clearly and plainly answers the question stated here, "why do we divide by $n-1$?". Perhaps it might help to point out that *because* the sample variance is an unbiased estimator of the variance, then dividing it by *any other number* would bias it. That is why $n-1$ and bias are intimately connected. – whuber Jul 30 '13 at 17:51
  • Probably I keep missing something. I perfectly understand (now, thanks to the topic you pointed) why we devide by n - 1. What I still don't understand is how **is it linked with degrees of freedom**. It doesn't seem to me that the linked thread replies to this answer. – Vaaal Jul 31 '13 at 08:53

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