# IPBWiki/BasicStatistics

## Gaussian distribution

Mean
$\bar{x}=\frac{1}{N} \sum x_i \ \ \ \ \ \ \ \ \Delta \bar{x}=\sqrt{\frac{\sum (x_i - \bar{x})^2}{N\ (N-1)}} = \frac{\sigma}{\sqrt{N}}$
Median
The value chosen such that half of the observations are smaller and half are greater than this value.
Mode
The most frequently occurring value.
Variance
$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N-1}$
Standard Deviation (rms/sigma) and standard error (error bars)
$\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N-1}} \ \ \ \ \ \ \ \ \Delta \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{2\ N\ (N-1)}}$
   * 68.2% of the points are within +/-1σ
* 95.4% of the points are within +/-2σ
* 99.7% of the points are within +/-3σ


### Deviations from the Gaussianity

Skewness

$s_3 = \frac{1}{N-1} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^3 \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta s_3 = \sqrt{\frac{6}{N}}$

Kurtosis

$s_4 = \frac{1}{N-1} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^4 - 3 \ \ \ \ \ \ \ \ \Delta s_4 = \sqrt{\frac{24}{N}}$

Kurtosis excess = Kurtosis - 3 # To assign the value zero to a normal distribution.

Gaussian being symmetric with respect to the mean, has a skewness of zero.