IPBWiki/BasicStatistics

From MediaWiki

(Difference between revisions)
Jump to: navigation, search
(→Gaussian distribution)
(→Gaussian distribution)
Line 1: Line 1:
== Gaussian distribution ==
== Gaussian distribution ==
-
; Mean: <math>\bar{x}=\frac{1}{N} \sum x_i</math>
+
; Mean: <math>\bar{x}=\frac{1}{N} \sum x_i</math>
-
; Median: ''The individual value from the collection such that 1/2 the observations are less and 1/2 are greater''
+
; Median: ''The value chosen such that half of the observations are smaller and half are greater.''
-
; Mode: ''The most frequently occurring value.''
+
; Mode: ''The most frequently occurring value.''
-
; Variance: <math>\sigma_x^2 = \frac{1}{N-1} \sum (x_i - \bar{x})^2</math>
+
; Variance: <math>\sigma_x^2 = \frac{1}{N-1} \sum (x_i - \bar{x})^2</math>
-
; Standard Deviation (rms/sigma): <math>\sigma_x = \sqrt{\frac{1}{N-1} \sum (x_i - \bar{x})^2}</math>
+
; Standard Deviation (rms/sigma): <math>\sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N-1}} </math> 
 +
 
 +
    * 68.2% of the points are within +/-1<math>\sigma</math>
 +
    * 95.4% of the points are within +/-2<math>\sigma</math>
 +
    * 99.7% of the points are within +/-3<math>\sigma</math>
 +
 
 +
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):  <math>\sigma_{\bar{x}} = \sqrt{\frac{1}{N}} \sigma_x = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N(N-1)}}</math>
 +
 
 +
=== Deviations from the Gaussianity ===
 +
 
 +
'''Skewness''' =  <math>\frac{1}{N} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^3 </math>
 +
 
 +
'''Kurtosis''' =  <math>\frac{1}{N} \sum \left[ \frac{x_i - \bar{x}}{\sigma} \right]^4 </math>
 +
 
 +
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.

Revision as of 12:51, 3 February 2011

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.

Retrieved from "http://wiki.ipb.ac.rs/index.php/IPBWiki/BasicStatistics"
Personal tools