Kolmogorov-Smirnov Test

Introduction: Kolmogorov-Smirnov Test

The Kolmogorov-Smirnov (K-S) Test is a non-parametric statistical test used to determine whether a dataset follows a specified distribution, often the normal distribution. It can also compare two datasets to see if they come from the same distribution.

Background

Developed by Andrey Kolmogorov and Nikolai Smirnov in the 1930s, the test measures the maximum difference between the cumulative distribution of the sample and the cumulative distribution of the reference model. Unlike many normality tests, the K-S test can be applied to any continuous distribution.

Key Elements / Features

• One-sample K-S test: Compares a sample with a theoretical distribution (e.g., normal, uniform).
• Two-sample K-S test: Compares two samples to check if they follow the same distribution.
• Test statistic (D): Measures the largest vertical distance between the sample distribution and the reference distribution.
• p-Value: A low p-value (e.g., <0.05) indicates the sample does not follow the specified distribution.

Applications / Examples

• Quality control: Testing if manufacturing measurements follow a normal distribution.
• Finance: Checking if stock returns follow a normal or heavy-tailed distribution.
• Research: Comparing distributions of two groups, such as test scores or survey responses.

For example, a company may use the K-S test to compare customer waiting times against a normal model. If the test shows a significant difference, managers may choose non-parametric methods for further analysis.

Relevance / Impact

The Kolmogorov-Smirnov Test is valued for its flexibility, as it does not assume any specific distribution. However, it is less powerful than specialised tests like the Shapiro-Wilk Test for small datasets. It remains a key tool for researchers who need a general-purpose method to assess distribution fit.

See also

• Anderson-Darling Normality Test
• Shapiro-Wilk Test
• Lilliefors Test
• Jarque-Bera Test