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Chi2 test

When checking the normality of a distribution our first intuition would be to draw a histogram of the distribution of the observed variables to see how closely it resembles a normalized Gaussian curve.
chi21
This is the exact principle behind the Chi2 test that adds to this intuition a small dose of statistical calculations. The principle is as follows:
chi22
For each bar of the histogram we can calculate:
  • Ni: The number of parts actually observed (here 10)
  • Npi: The number of parts theoretically observed if it were a normal distribution (here 9.2)
  • represents the “number of badly arranged parts”
We then calculate ,and find that D follows a distribution law due to the n-2 degrees of freedom (N being the number of classes). Consequently, we can calculate the probability of getting such a value
courbe chi2
For example; for a histogram composed of 7 classes, if we calculated one d at 11.07 then we calculate that there are 5% that obtain this value, or more if the distribution law for the parts is actually normal.
The result of the test would therefore be 5% and we generally conclude in the following way:
  • If X < 5%: the variables distribution law is not considered to follow a normal distribution
  • If X >= 5%: the normality hypothesis is accepted and we consider that this distribution law follows a normal distribution.

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