For example, given: $\sum_{i=1}^n x_i^2 = 10$, $n=10$, and $mean = 10$, how would I go about calculating the standard deviation? The formula for standard deviation requires me knowing what $\sum_{i=1}^n (x_i-mean)^2$ is, but that information is not provided to me in this problem.


1 Answer 1


The sample standard deviation is given by

$$\begin{align*} s &=\sqrt{\frac{\sum\left(x_i-\bar{x}\right)^2}{n-1}}\\\\ &=\sqrt{\frac{\sum\left(x_i^2-2x_i\bar{x}+\bar{x}^2\right)}{n-1}}\\\\ &=\sqrt{\frac{n\bar{x}^2+\sum x_i^2-2\bar{x}\sum x_i}{n-1}}\\\\ &=\sqrt{\frac{n\bar{x}^2+\sum x_i^2-2n\bar{x}^2}{n-1}}\\\\ &=\sqrt{\frac{\sum x_i^2-n\bar{x}^2}{n-1}}\\\\ \end{align*}$$

so having $\bar{x}, n,\text{ and}\sum x_i^2$ suffices for calculating the sample standard deviation.

  • $\begingroup$ For the population standard deviation, the formula would be the same, except with n at the denominator instead of (n - 1) right? $\endgroup$
    – Henry Zhu
    Commented Oct 20, 2019 at 8:05
  • $\begingroup$ That is correct $\endgroup$
    – Remy
    Commented Oct 20, 2019 at 8:12

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