Shapiro and Wilk's W-test is a test for normality. The Shapiro Wilk test tests the null hypothesis that a sample follows a normal distribution.

The formula of the test statistic is:

where $X$ is the studied sample, $X\left(i\right)$ is the $i$th smallest data in X, $X_i$ is the $i$th data in X, $a_i$ are the coefficients to estimate straightness of the quantile-quantile plot.

The definitions of these coefficients are beyond the scope of this guide.

The null hypothesis that the sample follows a normal distribution is rejected if W is too small.

Null hypothesis: The data is normally distributed

$X≔\left[355.0\,359.5\,379.3\,366.5\,325.1\,334.4\,308.4\,355.6\,381.2\,316.9\,379.0\,338.7\,380.3\,366.4\,368.1\,333.3\,390.7\,337.4\,373.3\,370.0\right]\:$

$\mathrm{Student}:-\mathrm{Statistics}:-\mathrm{ShapiroWilkWTest}\left(X\right)\:$

Shapiro and Wilk's W-Test for Normality
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Null Hypothesis:
Sample drawn from a population that follows a normal distribution
Alt. Hypothesis:
Sample drawn from population that does not follow a normal distribution
 
Sample Size:             20
Computed Statistic:      .935508635130523
Computed p-value:        .207505438819378
 
Result: [Accepted]
This statistical test does not provide enough evidence to conclude that the null hypothesis is false.