Shapiro–Wilk test

Shapiro–Wilk test is a statistical test used to determine whether a given sample comes from a normally distributed population. Developed by Samuel Sanford Shapiro and Martin Wilk in 1965, the test is widely used in the field of statistics for testing the normality of data, which is a common assumption in many statistical tests.

Overview

The Shapiro–Wilk test calculates a statistic, W, which measures the closeness of the sample to a normal distribution. The test compares the ordered sample values with the corresponding expected values from a normal distribution. A significant result indicates that the hypothesis of normality should be rejected. The Shapiro–Wilk test is considered to be one of the most powerful tests for assessing normality, especially for small sample sizes.

Calculation

The calculation of the Shapiro–Wilk statistic involves several steps. First, the sample data are ordered in ascending order. Then, each observation is compared to the expected values from a normal distribution. The test statistic W is calculated as the ratio of the squared correlation between the ordered sample values and the normal scores, to the total squared deviation of the sample values from their mean.

Application

The Shapiro–Wilk test is commonly used in situations where the normality of the data is in question. This includes, but is not limited to, before conducting parametric statistical tests that assume normality, such as the t-test or ANOVA. It is also used in quality control and other fields where the normal distribution plays a key role.

Limitations

While the Shapiro–Wilk test is powerful, it has limitations. The test may lose power with very large sample sizes, leading to the rejection of the normality assumption for slight deviations from normality. Additionally, the test is sensitive to outliers, which can significantly affect the test result.

Comparison with Other Tests

The Shapiro–Wilk test is often compared to other normality tests, such as the Kolmogorov-Smirnov test, Anderson-Darling test, and Lilliefors test. Each test has its own advantages and disadvantages, and the choice of test can depend on the specific characteristics of the data and the objectives of the analysis.

Conclusion

The Shapiro–Wilk test remains a popular choice for testing the assumption of normality in statistical analyses. Its ease of use and power for small sample sizes make it a valuable tool in the statistician's toolkit. However, like all statistical tests, it should be used with an understanding of its limitations and in conjunction with other methods of assessing the distribution of data.