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

The calculator will find the standard deviation of $1$, $2$, $3$, $4$, $5$, with steps shown.
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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, $s = \sqrt{\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}$.

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