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In my statistics book I found an example comparing paired samples. They computed the mean of the difference, $\bar{D} = 10.27$ and the sample standard deviation (or sample standard error) from the difference column as follows:

$s_{\bar{D}}^{2} = \frac{\sum_{i=0}^{11} (D_{i}-\bar{D})^{2}}{11}$

but then they divided $s_{\bar{D}}$ by the sample size again:

$s_{\bar{D}}^{2} = \frac{s_{\bar{D}}^{2}}{11}$

and finally took the square root of $s_{\bar{D}}^{2}$ to find the standard deviation.

My Question: Why does the book divide by the sample size a second time?

Below is the data from the book, and a snippet of R code I used to recreate the example.

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enter image description here

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  • $\begingroup$ They are basically the statistics of a sample mean estimator. $\endgroup$ Commented Apr 24, 2016 at 19:55

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The first division leads to the sample variance, and its square root to the sample standard deviation.

The second division and its square root leads to the sample standard error of the mean and so is suitable for comparing differences in means

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