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The Lilliefors test is a variation of the Kolmogorov Smirnov test to test normality.

Introduction

The Lilliefors test is a variation of the Kolmogorov Smirnov Adapted to test normality. The test statistic is the same simply, the law of calculating the critical value as well as the data of comparisons are calculated via a normal law.

The principle

The principle remains identical to the Kolmogorov Smirnov test.

Step 1: Assumptions

The following assumptions are tested:

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

Step 2: Calculate the practical value

The practical value of the test is calculated according to the same principle as the Kolmogorov Smirnov test.

Step 3: Calculating the critical value

The critical value is found in exact tables for values less than or equal to (50 data).

Beyond that, the critical value is approximated according to the following formula:

  • for α = 1%: 1.035/((0.83 + N)/√ n – 0.01)
  • for α = 5%: 0.895/((0.83 + N)/√ n – 0.01)
  • For α = 10%: 0.819/((0.83 + N)/√ n – 0.01)
  • for α = 15%: 0.775/((0.83 + N)/√ n – 0.01)
  • for α = 20%: 0.741/((0.83 + N)/√ n – 0.01)

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 normal law at the level of risk α given.
Practical value < Critical valueWe retain H0Our data follow the normal distribution at given level of risk.

Source

H. Lilliefors (1967) – On the Kolmogorov Smirnov test for normality with mean and variance unknown

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