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Shapiro-Wilk test for normality

A Pearson correlation coefficient is computed between the order statistics in the sample and scores which represent what the order statistics should be if the population is normal. The test statistic (W) is the square of the correlation coefficient.

The null hypothesis is that the population has a normal distribution; the alternative hypothesis is that the population does not have a normal distribution.

There are various formulations to obtain the correlation coefficient. This one given by Conover (1999) avoids estimation of the rankits by multiplying the difference between specified order statistics by tabulated coefficients.

Then order the sample from smallest to largest, and compute the test statistic (W) as:

Algebraically speaking -

W   =   [Σai(Yn-i+1 − Yi) ]2
Σ(Yi)2
where:
  • ai are tabulated coefficients (see for example Table A16 in Conover (1999)),
  • n is the number of observations,
  • Yi is the ith order statistic,
  • Yi' are the original observations and
  • is the sample mean