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

The sample standard deviation is $s = \frac{\sqrt{208523}}{35}\approx 13.04694819269461$A.