# Sample/Population Variance Calculator

## Calculate sample/population variance step by step

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

### Your Input

**Find the sample variance of $$$2$$$, $$$1$$$, $$$9$$$, $$$-3$$$, $$$\frac{5}{2}$$$.**

### Solution

The sample variance of data is given by the formula $$$s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}$$$, 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 of standard deviation.

The mean of the data is $$$\mu = \frac{23}{10}$$$ (for calculating it, see mean calculator).

Since we have $$$n$$$ points, $$$n = 5$$$.

The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(2 - \frac{23}{10}\right)^{2} + \left(1 - \frac{23}{10}\right)^{2} + \left(9 - \frac{23}{10}\right)^{2} + \left(-3 - \frac{23}{10}\right)^{2} + \left(\frac{5}{2} - \frac{23}{10}\right)^{2} = \frac{374}{5}.$$$

Thus, $$$s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{\frac{374}{5}}{4} = \frac{187}{10}$$$.