IPBWiki/BasicStatistics

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== Gaussian distribution ==
== Gaussian distribution ==
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; Mean: <math>\bar{x}=\frac{1}{N} \sum x_i</math>
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; Mean: <math>\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}}</math>  
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; Median: ''The individual value from the collection such that 1/2 the observations are less and 1/2 are greater''
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; 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.''
; Mode: ''The most frequently occurring value.''
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; Variance: <math>\sigma_x^2 = \frac{1}{N-1} \sum (x_i - \bar{x})^2</math>
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; Variance: <math>\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{N-1} </math>
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; Standard Deviation (rms/sigma): <math>\sigma_x = \sqrt{\frac{1}{N-1} \sum (x_i - \bar{x})^2}</math>   
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; Standard Deviation (rms/sigma) and standard error (error bars): <math>\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)}}</math>   
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The standard deviation tells us something about the expected value of a single observation.
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    * 68.2% of the points are within +/-1<math>\sigma</math>
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    * 95.4% of the points are within +/-2<math>\sigma</math>
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    * 99.7% of the points are within +/-3<math>\sigma</math>
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If the data are normally distributed
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=== Deviations from the Gaussianity ===
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    * 68% of the points will lie within &plusmn 1 sigma
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; Skewness:
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    * 95% of the points will lie within &plusmn 2 sigma
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    * 99.7% of the points will lie within &plusmn 3 sigma
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Usually we accept a variation as statistically significant only if it is more than 3 sigma from the mean.
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<math>s_3 = \frac{1}{N-1} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^3 \ \ \ \ \ \ \ \ \ \ \ \ \ \Delta s_3 = \sqrt{\frac{6}{N}}</math>
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; Error bars (standard error/standard deviation of the mean):  <math>\sigma_{\bar{x}} = \sqrt{\frac{1}{N}} \sigma_x = \sqrt{\frac{1}{N(N-1)} \sum (x_i - \bar{x})^2}</math> 
 
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* Moments of the Gaussian distribution
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;Kurtosis: 
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  '''variance''' =  <math>\frac{1}{N-1} \sum \left[(x_i - \bar{x})\right]^2
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<math>s_4 = \frac{1}{N-1} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^4 - 3 \ \ \ \ \ \ \ \ \Delta s_4 = \sqrt{\frac{24}{N}}</math>
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  '''skewness''' =  <math>\frac{1}{N} \sum \left[\frac{(x_i - \bar{x})}{\sigma}\right]^3  
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  '''kurtosis''' =  <math>\frac{1}{N} \sum \left[\frac{(x_i - \bar{x})}{\sigma}\right]^4 - 3
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Kurtosis excess = Kurtosis - 3   # To assign the value zero to a normal distribution.
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Gaussian being symmetric with respect to the mean, has a skewness of zero.

Latest revision as of 12:20, 8 February 2011

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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