# Standard deviation of the mean

The mean value can be calculated by:

$$
\bar x = \frac{1}{N}\sum_{i = 1}^{N}x_i
$$

Standard deviation of the individual measurement from the mean

$$
\sigma_x = \sqrt{\frac{1}{N - 1}\sum_{i = 1}^{N}(x_i - \bar x_i)^2}
$$
Standard deviation of the mean

$$
\sigma_{\bar x} = \sqrt{\frac{1}{N(N - 1)}\sum_{i = 1}^{N}(x_i - \bar x_i)^2}
$$

[source](https://www2.physnet.uni-hamburg.de/TUHH/Versuchsanleitung/Fehlerrechnung.pdf)