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 statisticquantifies 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 –

**H _{0}(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.

*kstest*

function in scipy Python –

*kstest*

**Parameters:**

Statistics:This is the calculated value of D, where`D=|F(x)-Sn(x)|`

.

-> ThisDis compared with D_{alpha}where alpha is the level of significance. Alpha is defined as the probability of rejecting the null hypothesis given the null hypothesis(H_{0}) is true. For most of the practical applications, alpha is chosen as 0.05.

p-value:This is calculated with the help of D.

-> If pvalue> alpha, we fail to reject the null hypothesis. Otherwise, we conclude that the numbers are not uniform. Ideally, the p-value should be as large as possible. For perfect uniform distribution pvalue=1 and Statisitics=0.

`from` `scipy.stats ` `import` `kstest` `import` `random` ` ` `# N = int(input("Enter number of random numbers: "))` `N ` `=` `5` ` ` `actual ` `=` `[]` `print` `(` `"Enter outcomes: "` `)` `for` `i ` `in` `range` `(N):` ` ` `# x = float(input("Outcomes of class "+str(i + 1)+": "))` ` ` `actual.append(random.random())` ` ` `print` `(actual)` `x ` `=` `kstest(actual, ` `"uniform"` `) ` `print` `(x)` |

**Output:**

**KS Test** is a very powerful way to automatically differentiate samples from a different distribution. *kstest *function may also be used to check whether the data given follows Normal Distribution or not. It compares the observed versus the expected cumulative relative frequencies of the Normal Distribution. The Kolmogorov-Smirnov test uses the maximal absolute difference between the observed and expected cumulative distribution.

- The Null hypothesis used here assumes that the numbers follow the normal distribution.
- The functioning of the function remains exactly same. Again it returns statistics and p-value. If the p-value is < alpha, we reject the Null hypothesis.

`from` `scipy.stats ` `import` `kstest` `import` `random` ` ` `# N = int(input("Enter number of random numbers: "))` `N ` `=` `10` ` ` `actual ` `=` `[]` `print` `(` `"Enter outcomes: "` `)` ` ` `for` `i ` `in` `range` `(N):` ` ` `# x = float(input("Outcomes of class "+str(i + 1)+": "))` ` ` `actual.append(random.random())` ` ` `print` `(actual)` `x ` `=` `kstest(actual, ` `"norm"` `) ` `print` `(x)` |

**Output:**

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