Sample/Population Standard Deviation Calculator

For the given set of observations, the calculator will find their standard deviation (either sample or population), with steps shown.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Your Input

Find the sample standard deviation of $$$1$$$, $$$37$$$, $$$9$$$, $$$0$$$, $$$- \frac{3}{5}$$$, $$$9$$$, $$$10$$$.

Solution

The sample standard deviation of data is given by the formula $$$s = \sqrt{\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}}$$$, where $$$n$$$ is the number of values, $$$x_i, i=\overline{1..n}$$$ are the values themselves, and $$$\mu$$$ is the mean of the values.

Actually, it is the square root of variance.

The mean of the data is $$$\mu = \frac{327}{35}$$$ (for calculating it, see mean calculator).

Since we have $$$n$$$ points, $$$n = 7$$$.

The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(1 - \frac{327}{35}\right)^{2} + \left(37 - \frac{327}{35}\right)^{2} + \left(9 - \frac{327}{35}\right)^{2} + \left(0 - \frac{327}{35}\right)^{2} + \left(- \frac{3}{5} - \frac{327}{35}\right)^{2} + \left(9 - \frac{327}{35}\right)^{2} + \left(10 - \frac{327}{35}\right)^{2} = \frac{178734}{175}.$$$

Thus, $$$\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{\frac{178734}{175}}{6} = \frac{29789}{175}$$$.

Finally, $$$s = \sqrt{\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}} = \sqrt{\frac{29789}{175}} = \frac{\sqrt{208523}}{35}$$$.

Answer

The sample standard deviation is $$$s = \frac{\sqrt{208523}}{35}\approx 13.0469481926946$$$A.