# 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 $$s = \frac{\sqrt{208523}}{35}\approx 13.0469481926946$$\$A.