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The Shapiro Wilk test is effective at validating normality when we have less than 50 data.

Introduction

This very popular test on the part of its simplicity was published in 1965 by Samuel Shapiro and Martin Wilk (Canadian statistician)1. It is particularly effective for samples with less than 50 observations2.

The principle

The test statistic is only the square of the correlation coefficient between the series of quantiles generated from the normal law and the empirical quantiles obtained from the data. Therefore, the more relates is close to 1 and the more our data follow a normal law.

Step 1: Assumptions

We ask the following assumptions:

  • H0: Our data follow a normal law
  • H1: Our data does not follow a normal law

Step 2: Calculate the practical value

The practical value calculation is done in several steps that we describe below.

  1. Classify n observations in order of increasing magnitude
  2. Calculate the differences between x(n-i + 1) -xi
  3. Read in Shapiro Wilk’s specific table the coefficients a relative to each value.
  4. Then calculate the numerator b2 = (Σ (ai * di))2
  5. Then calculate the denominator Z2 = Σ (xI – xcross)2
  6. Finally, calculate the practical value W which represents the ratio between B2 and Z2

Step 3: Calculating the critical value

The critical value of Shapiro Wilk is given in the exact tables of Shapiro Wilk 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 retain H0Our data follow the normal distribution at given level of α risk.
Practical value < Critical valueWe reject H0Our data do not follow the normal law at given level of α risk .

Source

1 – S. Shapiro, M. Wilk (1965) – An Analysis of variance test for noamlity

2 – R. Rafiq (2011) – Test of normality

Standard NF X 06-050 (1995) – Study of the normality of a distribution

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