IPBWiki/BasicStatistics

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Gaussian distribution

Mean
\bar{x}=\frac{1}{N} \sum x_i
Median
The value chosen such that half of the observations are smaller and half are greater.
Mode
The most frequently occurring value.
Variance
\sigma_x^2 = \frac{1}{N-1} \sum (x_i - \bar{x})^2
Standard Deviation (rms/sigma)
\sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{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

A variation is statistically significant if it is more than 3 sigma from the mean (i.e. encompasses 99.7% points).

Standard error (error bars)
\sigma_{\bar{x}} = \sqrt{\frac{1}{N}} \sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N(N-1)}}

Deviations from the Gaussianity

Skewness = \frac{1}{N} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^3

Kurtosis = \frac{1}{N} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^4

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.

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