Kolmogorov-Smirnov test: another goodness of fit test for comparing two distributions. Here the measurements need not be collected into "bins," but are instead re-arranged and placed in order of magnitude:

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N independent samples are collected and sorted in numerical order in array X as x $_{1}\protect$ ... x $_{N}$ , x $_{i+1}\geq x_{i}$
P(x $_{i}$ ) is the cumulative distribution of the sample: the fraction of the n observations which are less than or equal to x $_{i}$ , P(X $\leq x_{i}$ )= i / N
F(X) is the reference cumulative distribution, the probability that a random value will be less than or equal to x. Here we want F(x $_{i}$ ), the fraction of the distribution to the left of x $_{i}$ which is a value from the array.
we define the statistic q= $\sqrt{N}max\vert F(x_{i})-P(x_{i})\vert$
If agreement of the two distributions will be acceptable subject to it being not less likely than f% (e.g. f=5, corresponding to less than 5% probability of the theoretical distribution, randomly sampled, giving rise to the recorded data), for the large value of N (in practice N $\geq$ 30, f can be approximated by equation f=100e $^{-2q^{2}}\{1-2q/3\sqrt{N}+O(N^{-1})\}$

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Amaury LATAILLADE 2002-11-04