3
$\begingroup$

Can someone explain to me the mechanics of the Shapiro–Wilk test for normality?

I have that: $$W = \frac{ \sum (x_{(j)}-\bar{x})(m_{(j)}-\bar{m})}{\sum (x_{j}-\bar{x})^{2} \sum m_{j}^2}\,\,,$$ where $m_{j}$ is the expected value of the $j$th order statistic from $N(0,1)$.

So, what is $x_{j}$ here? Is the $m_{j}$ from a theoretical distribution, or from an observed sample? What is the test statistic comparing?

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ Note it's "Shapiro-Wilk test" - there's confusion between the two statisticians Samuel S. Wilks and Martin Wilk, but it was the latter involved in the Shapiro-Wilk test. (No doubt the confusion is increased by the English use of possessive "'s"). $\endgroup$
    – Silverfish
    Oct 2, 2015 at 22:37

1 Answer 1

Reset to default
3
$\begingroup$

It's basically a sample correlation between empirical quantile and quantile from the normal distribution. So, if the true distribution is normal, we can expect that the correlation would be high.

Improve this answer
$\endgroup$
2
  • $\begingroup$ So .... let me check check and see if my interpretation is correct..... The "Shapiro Wilk" test is basically the measure of correlation for the Q-Q plot? $\endgroup$
    – LotsofQuestions
    Oct 5, 2015 at 15:08
  • 1
    $\begingroup$ It's similar to it, but isn't actually that correlation. The Shapiro-Francia test is effectively that sort of correlation. $\endgroup$
    – Glen_b
    Oct 26, 2018 at 22:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.