Kolmogorov-Smirnov Normality Test
The Kolmogorov-Smirnov (K-S) test is a non-parametric test used to determine whether a sample is drawn from a specified distribution. This calculator tests whether your data follows a normal (Gaussian) distribution.
When to Use
Use the Kolmogorov-Smirnov test to check the assumption of normality before applying parametric statistical tests such as:
- Independent samples t-test
- ANOVA (Analysis of Variance)
- Pearson correlation
- Linear regression
How It Works
The K-S test compares the empirical cumulative distribution function (ECDF) of your sample data with the cumulative distribution function (CDF) of a theoretical normal distribution with the same mean and standard deviation.
The test statistic D is the maximum absolute difference between these two distribution functions. A larger D value indicates greater deviation from normality.
Interpreting Results
- p > 0.05: Fail to reject null hypothesis — data appears normally distributed
- p ≤ 0.05: Reject null hypothesis — data significantly deviates from normal distribution
Assumptions
- Data should be measured on at least an ordinal scale
- Observations should be independent
- The test parameters (mean and SD) are estimated from the sample
Limitations
- More sensitive to deviations in the center of the distribution than the tails
- May have low power with small sample sizes
- Consider using Shapiro-Wilk test as an alternative, especially for small samples
The Test Statistic
D = max|Fn(x) - F(x)|
Where Fn(x) is the empirical cumulative distribution function of the sample, and F(x) is the theoretical cumulative distribution function of the normal distribution.
