# Standard deviation of $$$25$$$, $$$27$$$, $$$24$$$, $$$31$$$, $$$30$$$, $$$19$$$

### Your Input

**Find the sample standard deviation of $$$25$$$, $$$27$$$, $$$24$$$, $$$31$$$, $$$30$$$, $$$19$$$.**

### 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 $$$\mu = 26$$$ (for calculating it, see mean calculator).

Since we have $$$n$$$ points, $$$n = 6$$$.

The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(25 - 26\right)^{2} + \left(27 - 26\right)^{2} + \left(24 - 26\right)^{2} + \left(31 - 26\right)^{2} + \left(30 - 26\right)^{2} + \left(19 - 26\right)^{2} = 96.$$$

Thus, $$$\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{96}{5}$$$.

Finally, $$$s = \sqrt{\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}} = \sqrt{\frac{96}{5}} = \frac{4 \sqrt{30}}{5}$$$.

### Answer

**The sample standard deviation is $$$s = \frac{4 \sqrt{30}}{5}\approx 4.381780460041329$$$A.**