Lilliefors Test

Introduction: Lilliefors Test

The Lilliefors Test is a statistical test used to check whether a dataset follows a normal distribution when the population mean and variance are unknown. It is a modified version of the Kolmogorov-Smirnov (K-S) Test, specifically adapted for real-world situations where parameters must be estimated from the data itself.

Background

The test was introduced by Hubert Lilliefors in 1967. It extends the Kolmogorov-Smirnov Test by addressing a key limitation: the K-S test assumes the mean and variance of the normal distribution are known in advance. Since this is rarely the case in practice, the Lilliefors Test provides a more practical approach.

Key Elements / Features

• Estimation of parameters: Unlike the K-S test, the Lilliefors Test estimates the mean and variance from the sample data.
• Test statistic (D): Similar to the K-S test, it measures the largest difference between the sample’s cumulative distribution and the expected normal distribution.
• Critical values: Because parameters are estimated, critical values differ from those in the standard K-S test and are provided by Lilliefors’ tables or modern software.

Applications / Examples

• Quality management: Verifying if product measurements follow a normal distribution before applying process control.
• Education: Testing whether student exam scores are normally distributed when population parameters are not known.
• Research: A practical choice in studies where researchers need a normality check but only have sample data.

For example, a researcher analysing exam results may apply the Lilliefors Test to see if the data align with normality. If results are significant, non-parametric alternatives such as the Mann-Whitney U test may be used.

Relevance / Impact

The Lilliefors Test is widely respected because it solves a practical limitation of the Kolmogorov-Smirnov Test. By allowing parameter estimation, it is more realistic for applied research. Although not as powerful as the Shapiro-Wilk Test for small samples, it remains an important option in statistical analysis.

See also

• Kolmogorov-Smirnov Test
• Shapiro-Wilk Test
• Anderson-Darling Normality Test
• Jarque-Bera Test