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

The calculator will find the standard deviation of $25$, $27$, $24$, $31$, $30$, $19$, with steps shown.
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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, $s = \sqrt{\frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}} = \sqrt{\frac{96}{5}} = \frac{4 \sqrt{30}}{5}$.

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