Lilliefors Normality Test

The Lilliefors test is a Kolmogorov-Smirnov normality test that uses the sample mean and standard deviation and corrects the null distribution for having estimated them.

Lilliefors 검정은 표본에서 추정한 평균과 표준편차를 쓰는 정규분포와의 Kolmogorov-Smirnov 거리를 모수 추정을 반영한 귀무분포로 판정하는 정규성 검정이다.

Is it valid to use the Kolmogorov-Smirnov test for normality when I estimate the mean and SD from my data?

평균과 표준편차를 자료에서 추정했을 때 Kolmogorov-Smirnov 검정으로 정규성을 봐도 되나요?

Not with the standard Kolmogorov-Smirnov table, because estimating the parameters makes the fit look better and the test too lenient. The Lilliefors test uses the same maximum distance D but compares it with a null distribution derived for estimated parameters.

When to use Lilliefors Normality Test

• You need a formal normality check with mean and SD estimated from the sample.
• You want a distance-based test that is easy to explain.
• Sample sizes are moderate to large.

When to use something else

• You want the most powerful general normality test; use the Shapiro-Wilk test instead.
• The hypothesized distribution is fully specified in advance; use the one-sample Kolmogorov-Smirnov test instead.

Assumptions and requirements

• Observations are independent and identically distributed.
• The data are continuous.

What the analysis reports

• Maximum distance D
• p-value from the Lilliefors distribution
• Sample mean and SD
• Skewness and excess kurtosis

How to report it

A Lilliefors test [did/did not] indicate a significant departure from normality, D([n]) = [value], p = [value].

Related methods

• Shapiro-Wilk Test
• Kolmogorov-Smirnov Test
• Anderson-Darling Test

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