IPBWiki/BasicStatistics

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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.

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