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:

where:

x_{i} are the ordered random sample values

a_{i} 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:**- Click BASIC STATISTICS
- Choose NORMALITY TEST
- Type your data column in the VARIABLE BOX (do not fill in the reference

box) - Choose RYAN JOINER (this is the same as Shapiro-Wilk)
- 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 Laerd.com 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.**MATLAB:**Instructions for the test are given on the*MathWorks site*.**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.

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