Lilliefors Test for Normality
Lilliefors Test for Normality with Mean and Variance Unknown · Also known as: Lilliefors corrected Kolmogorov-Smirnov test, Lilliefors normality test, Lilliefors Testi
The Lilliefors test is a goodness-of-fit test that checks whether a continuous sample comes from a normal (or exponential) distribution when the mean and variance are unknown and estimated from the data. Introduced by Hubert W. Lilliefors in 1967, it adjusts the critical values of the Kolmogorov-Smirnov test so that they remain valid once the distribution's parameters are estimated rather than known in advance.
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When to use it
Use the Lilliefors test when you need to check the normality of a single continuous variable and you do not know the population mean and variance in advance, so they must be estimated from the sample. This is the typical situation in applied work, which is why Lilliefors is preferred over the plain Kolmogorov-Smirnov test whenever parameters are unknown. It requires continuous data and at least about 5 observations, but its statistical power is very low below roughly 10 observations.
Strengths & limitations
- Corrects the Kolmogorov-Smirnov critical values for the case where the mean and variance are estimated from the data, avoiding the overly conservative KS test.
- Distribution-free reference: works without assuming the parameters are known in advance.
- Simple, fast, and applicable to both normal and exponential goodness-of-fit checks.
- Very low statistical power in small samples (n < 10), so it may fail to detect real departures from normality.
- Below 5 observations the test is essentially meaningless.
- Designed for the normal (or exponential) family with estimated parameters; it is not a general-purpose test for arbitrary distributions.
Frequently asked
How does the Lilliefors test differ from the Kolmogorov-Smirnov test?
Both compare the empirical and a theoretical CDF using the maximum absolute gap. The plain Kolmogorov-Smirnov test assumes the distribution's parameters are known in advance; when you estimate the mean and variance from the same data, those critical values become too conservative. Lilliefors recomputes the critical values for this estimated-parameter case, so it is the correct choice when parameters are unknown.
What sample size do I need?
At least about 5 observations are required, but the test has very low power below roughly 10 observations. With small samples it may fail to flag genuine non-normality, so a permutation test is preferable below 10 and bootstrap inference below 5.
What does a non-significant p-value mean?
A large p-value means there is not enough evidence to reject normality — it does not prove the data are normal. Especially in small samples, a non-significant result may simply reflect low power rather than true normality.
Can it test distributions other than the normal?
The original test targets the normal distribution with estimated mean and variance, and it is also commonly applied to the exponential distribution with an estimated rate. It is not a general goodness-of-fit test for arbitrary distributions.
Sources
- Lilliefors, H. W. (1967). On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown. Journal of the American Statistical Association, 62(318), 399-402. DOI: 10.1080/01621459.1967.10482916 ↗
- Dallal, G. E., & Wilkinson, L. (1986). An Analytic Approximation to the Distribution of Lilliefors's Test Statistic for Normality. The American Statistician, 40(4), 294-296. DOI: 10.1080/00031305.1986.10475419 ↗
How to cite this page
ScholarGate. (2026, June 1). Lilliefors Test for Normality with Mean and Variance Unknown. ScholarGate. https://scholargate.app/en/statistics/lilliefors-test
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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