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.

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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 $$$s = \frac{\sqrt{208523}}{35}\approx 13.04694819269461$$$A.