Statistic analysis is very important for judging the property of a large quantity of experiment data. For example, standard error can be used to calculate error bar.

Basic definition

Suppose we have lots of experiment data with discrete values as a finite series: $x_1, x_2, x_3, ..., x_N$. Then the related statistic values are defined like these.

Mean value

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

Standard deviation (SD)

In some kind of definition, the number $N-1$ is used instead of $N$ [2]. However, as the number of the samples increases $1/N \sim 1/(N-1)$. Thus the difference of the two types will only be significant with small amount of data. To make it better for computation, here we use the definition with $N$.

$\sigma = \sqrt{\frac{\sum_{i=1}(x_i - \mu)^2}{N}}$

Standard error (SE)

Standard error is a very important statistic value, because we usually choose error bar in this form during experimental data analysis.

$SE = \frac{\sigma}{\sqrt{N}} = \sqrt{\frac{\sum_{i=1}(x_i - \mu)^2}{N^2}}$

Reference

[1] http://berkeleysciencereview.com/errorbars-anyway/

[2] https://en.wikipedia.org/wiki/Standard_deviation