Shapiro Wilk Test

Posted November 7, 2023 by Rohith and Anusha ‐ 2 min read

In the realm of statistics, assumptions play a crucial role in ensuring the validity of the analyses conducted. One fundamental assumption often encountered is the normality of data distribution. Many statistical tests, like t-tests and ANOVA, rely on this assumption. To assess whether a dataset follows a normal distribution, various methods are employed, and one of the most widely used techniques is the Shapiro-Wilk test.

What is the Shapiro-Wilk Test?

  • The Shapiro-Wilk test is a statistical test used to determine whether a given sample of data comes from a normally distributed population.

  • It is particularly useful for small to moderately sized datasets.

  • The test calculates a W statistic based on the sample data and compares it to the expected distribution under the null hypothesis of normality.

How Does the Shapiro-Wilk Test Work?

Null Hypothesis (H0)

  • The data follows a normal distribution.

Alternative Hypothesis (Ha)

  • The data does not follow a normal distribution.

  • If the p-value obtained from the test is less than a predetermined significance level (often denoted as α, commonly set at 0.05), the null hypothesis is rejected, indicating that the data does not come from a normal distribution.

Interpreting the Results

If p-value > α

  • Fail to reject the null hypothesis.

  • There is no significant evidence to conclude that the data is not normally distributed.

If p-value ≤ α

  • Reject the null hypothesis.

  • There is significant evidence to suggest that the data is not normally distributed.

Benefits of the Shapiro-Wilk Test

Sensitivity to Departures from Normality

  • The Shapiro-Wilk test is sensitive to deviations from normality, making it a robust tool for detecting even small departures from the normal distribution.

Applicability to Small Samples

  • Unlike some other normality tests, the Shapiro-Wilk test performs well even with small sample sizes, making it invaluable in various research scenarios.

Limitations and Considerations

Sample Size

  • While the test is suitable for small to moderately sized datasets, it might not be as reliable with very large samples.

Visual Inspection

  • It is often recommended to complement statistical tests like Shapiro-Wilk with graphical methods (e.g., Q-Q plots) for a comprehensive assessment of normality.


  • In conclusion, the Shapiro-Wilk test stands as a robust tool in the hands of statisticians and researchers, providing a reliable method to assess the normality of data distributions.

  • Its ability to handle small sample sizes and detect subtle deviations from normality makes it a valuable asset in various fields, ensuring the soundness of statistical analyses and the reliability of research findings.

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