# Standard deviation of $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$5$$$

### Your Input

**Find the sample standard deviation of $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$, $$$5$$$.**

### 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 $$$\mu = 3$$$ (for calculating it, see mean calculator).

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

The sum of $$$\left(x_{i} - \mu\right)^{2}$$$ is $$$\left(1 - 3\right)^{2} + \left(2 - 3\right)^{2} + \left(3 - 3\right)^{2} + \left(4 - 3\right)^{2} + \left(5 - 3\right)^{2} = 10$$$.

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

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

### Answer

**The sample standard deviation is $$$s = \frac{\sqrt{10}}{2}\approx 1.58113883008419$$$A.**