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Shapiro-Wilk Test: What it is and How to Run it

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Statistics Definitions > Shapiro-Wilk Test

What is the Shapiro-Wilk Test?

The Shapiro-Wilk test is a way to tell if a random sample comes from a normal distribution. The test gives you a W value; small values indicate your sample is not normally distributed (you can reject the null hypothesis that your population is normally distributed if your values are under a certain threshold). The formula for the W value is:
Shapiro-Wilk test
xi are the ordered random sample values
ai are constants generated from the covariances, variances and means of the sample (size n) from a normally distributed sample.

The test has limitations, most importantly that the test has a bias by sample size. The larger the sample, the more likely you’ll get a statistically significant result.

How to Perform a Shapiro-Wilk Test

It’s rare that you’ll want to calculate the Shapiro-Wilk by hand. Many software packages can make the calculations for you:

  • Minitab:
    2. Choose NORMALITY TEST
    3. Type your data column in the VARIABLE BOX (do not fill in the reference
    4. Choose RYAN JOINER (this is the same as Shapiro-Wilk)
    5. Click OK
  • R: Although not as popular as SPSS or Excel, R does have the ability to perform the test. The argument is very simple:
    > shapiro.test(sample)
    You can find more information about the argument here.
  • SPSS: This article on shows how to perform the test in SPSS (it also incorporates the Kolmogorov-Smirnov test).
  • Excel: This article has a very good outline of how to run the test in Excel for samples up to 5,000. There are also instructions on how to handle larger samples.
  • SAS: The SAS support site has comprehensive instructions for a variety of Goodness of Fit tests. You can find the documentation here.

Tip: Use this test in combination with a normal probability plot.


Beyer, W. H. CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 2002.
Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.
Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.
Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Stephanie Glen. "Shapiro-Wilk Test: What it is and How to Run it" From Elementary Statistics for the rest of us!

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