# Kolmogorov-Smirnov Test (KS Test)

Kolmogorov–Smirnov test a very efficient way to determine if two samples are significantly different from each other. It is usually used to check the uniformity of random numbers. Uniformity is one of the most important properties of any random number generator and Kolmogorov–Smirnov test can be used to test it.

The Kolmogorov–Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ. It is a very efficient way to determine if two samples are significantly different from each other.

The Kolmogorov–Smirnov statistic quantifies a distance between the empirical distribution function of the sample and the cumulative distribution function of the reference distribution, or between the empirical distribution functions of two samples.

To use the test for checking the uniformity of random numbers, we use the CDF (cumulative distribution function) of U[0, 1].

`F(x)= x for 0<=x<=1`

Empirical CDF, `Sn(x)= (number of R1, R2…Rn < x)/N array of random numbers`, the random numbers must be in the range of [0, 1].

#### Hypothesis Used –

H0(Null Hypothesis): Null hypothesis assumes that the numbers are uniformly distributed between 0-1. If we are able to reject the Null Hypothesis, this means that the numbers are not uniformly distributed between 0-1. Failure to reject the Null Hypothesis although does not necessarily mean that the numbers follow the uniform distribution.

Algorithm:

```-> Rank the N random numbers in ascending order.
-> Calculate D+ as max(i/N-Ri) for all i in(1, N)
-> Calculate D- as max(Ri-((i-1)/N)) for all i in(1, N)
-> Calculate D as max(sqrt(N) * D+, sqrt(N) * D-)
-> If D>D(alpha)
Rejects Uniformity
else
It fails to reject the Null Hypothesis.```

Below is the Python implementation of above algorithm :

 `import` `random ` ` `  `N ``=` `int``(``input``(``"Enter the size of random numbers to be produced : "``)) ` `D_plus ``=``[] ` `D_minus ``=``[] ` `_random ``=``[] ` ` `  `# Rank the N random numbers ` `for` `i ``in` `range``(``0``, N): ` `    ``_random.append(random.random()) ` `    ``_random.sort() ` ` `  `# Calculate max(i/N-Ri) ` `for` `i ``in` `range``(``1``, N ``+` `1``): ` `    ``x ``=` `i ``/` `N ``-` `_random[i``-``1``] ` `    ``D_plus.append(x) ` ` `  `# Calculate max(Ri-((i-1)/N)) ` `for` `i ``in` `range``(``1``, N ``+` `1``): ` `    ``y ``=``(i``-``1``)``/``N ` `    ``y ``=``_random[i``-``1``]``-``y ` `    ``D_minus.append(y) ` ` `  `# Calculate max(D+, D-) ` `ans ``=` `max``(sqrt(N) ``*` `D_plus, sqrt(N) ``*` `D_minus) ` `print``(``"Value of D is :"``) ` `print``(ans) `

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