The **Shapiro-Wilk test **is a statistical test used to check if a continuous variable follows a normal distribution. The null hypothesis (H_{0}) states that the variable is normally distributed, and the alternative hypothesis (H_{1}) states that the variable is NOT normally distributed. So after running this test:

**If p ≤ 0.05:**then the null hypothesis can be rejected (i.e. the variable is NOT normally distributed).**If p > 0.05:**then the null hypothesis cannot be rejected (i.e. the variable MAY BE normally distributed).

## Information that should be reported

When reporting the Shapiro-Wilk test, the following should be mentioned:

- The
**reason**why the test was used. - The
**results**of the test: the value of the test statistic**W**and the**p-value**associated with it. - The
**consequences/interpretation**of these results.

Here are 2 examples:

### Example 1: Reporting a Shapiro-Wilk test with p **≤** 0.05

Since we had a small sample size, determining the distribution of the variable X was important for choosing an appropriate statistical method. So a Shapiro-Wilk test was performed and showed that the distribution of X departed significantly from normality (W = 0.96, p-value < 0.01). Based on this outcome, a non-parametric test was used, and the median with the interquartile range were used to summarize the variable X.

### Example 2: Reporting a Shapiro-Wilk test with p > 0.05

Since we had a small sample size, determining the distribution of the variable X was important for choosing an appropriate statistical method. So a Shapiro-Wilk test was performed and did not show evidence of non-normality (W = 0.92, p-value = 0.11). Based on this outcome, and after visual examination of the histogram of X and the QQ plot, we decided to use a parametric test. Also, the mean with the standard deviation were used to summarize the variable X.

## Important notes for reporting a Shapiro-Wilk test

### 1. A p > 0.05 does not prove the alternative hypothesis:

A Shapiro-Wilk test with **a p > 0.05 does not mean that the variable is normally distributed**, it only means that you cannot reject the null hypothesis which states that the variable is normally distributed.

This is why in the example above, where we reported a Shapiro-Wilk test with a p > 0.05, we used the words: *“the Shapiro-Wilk test did not show evidence of non-normality*“.

So be aware of incorrect interpretations like the following:

- “
*The Shapiro-Wilk test indicates that the variable has a normal distribution*“. - “
*The normality assumption was verified using the Shapiro-Wilk test*“. - “
*Normality of the data was confirmed by a Shapiro-Wilk test*“. - “
*The distribution of the variable was determined using the Shapiro-Wilk test*“.

### 2. The probability of getting a p < 0.05 depends on the size of your sample

A large enough sample size will make the Shapiro-Wilk test detect the smallest deviation from normality, in this case the p-value will be < 0.05 even if the variable is, in fact, normally distributed. Conversely, a very small sample size will reduce the statistical power of the Shapiro-Wilk test to reject the null hypothesis, in this case the p-value will be ≥ 0.05 even if the data clearly do not come from a normal distribution.

For these reasons many data analysts prefer to assess normality visually and/or using common sense, as sometimes the shape of a distribution can be decided theoretically, especially if the variable has some natural boundaries. For instance, “age” and “bacterial count” cannot have values less than zero, however they may have a very high although improbable upper bound (i.e. a right tail). Therefore, these variables should be ruled as following non-normal distributions.

### 3. Reporting the results of Shapiro-Wilk tests on many variables

For reporting the results of Shapiro-Wilk tests on many variables, use 1 of the following templates:

Example 1:

The distributions were significantly non-normal for the variables X

_{1}(W = 0.93, p < 0.01), X_{2}(W = 0.95, p < 0.01), and X_{3}(W = 0.91, p < 0.01) according to Shapiro-Wilk tests.

Example 2:

Shapiro-Wilk tests showed that neither X

_{1}(W = 0.93, p < 0.01) nor X_{2}(W = 0.95, p < 0.01) were normally distributed.