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The main interest of the Kolmogorov Smirnov test is to process ordinal data.

## Introduction

The test of Kolmogorov-Smirnov, created by Andrei Nikolayevich Kolmogorov in 1933 (Russian statistician creator of the University of Moscow), allows to know the level of adjustment of our data to any law.

It is noted that in order to test normality, it is preferable to Normal probability Plot and to test a match to a diverse law, it will be preferred the test of the Chi2, more robust and reliable especially for samples of small sizes (less than 30 observations).

However, this test is among the most reliable for large size samples.

## The principle

The Kolmogorov Smirnov test is a non-parametric test (therefore very useful for ordered data, for example) which allows the H0 hypothesis to be tested according to which the observed data follow a theoretical law. The probability calculations are based on the maximum difference between the theoretical function and the function actually observed.

It is noted that if the use of this test is to compare our data to the normal law, then the test is called Lilliefors test.

## Step 1: Assumptions

We want to test the following assumptions:

• H0: D = D0: Our data follow the theoretical law
• H1: D ≠ D0: Our data do not follow the theoretical law

## Step 2: Calculate the practical value

The practical value is to calculate the greatest difference between the actual cumulation of our data frequencies and the theoretical cumulation calculated from the law that we have determined.

1. Identify class frequencies: The process of calculating distribution graphs is repeated.
2. Create next to a column displaying the theoretical frequency. This may be only “desired” or calculated according to the binomial law or any other laws.
3. Calculate the actual cumulation and the theoretical cumulation.
4. Deduct the difference between the theoretical accumulation and the actual cumulation for each of the intervals.
5. Finally, we deduce the practical value K which is equal to the maximum of the difference calculated previously (green arrow in the diagram opposite) reported to the number of samples.

## Step 3: Calculating the critical value

The critical value of Kolmogorov Smirnov is given in the exact tables of Kolmogorov Smirnov for a given risk and a number of observations n relating to our situation.

## Step 4: Interpretation

In view of our initial assumptions, the interpretation of the test is as follows:

ResultStatistical conclusionPractical conclusion
Practical value ≥ Critical valueWe reject H0Our data do not follow the theoretical law at the level of risk α given.
Practical value < Critical valueWe retain H0Our data follow the theoretical law at the level of risk α given.

## Source

R. Rafiq (2011) – Probabilities and statistics

Standard NF X 06-050 – Study of the normality of a distribution