# Sample/Population Variance Calculator

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

Comma-separated.

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

The sample variance is $s^{2} = \frac{187}{10} = 18.7$A.