How this calculation works
Comparing two samples by their means can miss differences in spread or shape. The two-sample Kolmogorov–Smirnov distance compares their empirical cumulative distributions across all observed values. This page reports the largest absolute gap and its two signed directional components. It intentionally reports a descriptive distance rather than attaching a continuous-null p value to any dataset supplied.
D = maximum over x |FA(x)−FB(x)|A worked example
For A equal to one and two and B equal to two and three, FA is one half at value one while FB is zero. At value two the cumulative fractions are one and one half. The maximum absolute gap is therefore 0.5. At value three both cumulative fractions reach one. The displayed table exposes each step used to find the maximum.
Good to know
No hypothesis-test p value or rejection cutoff is provided. Classical continuous-distribution null formulas do not automatically apply to ties, discrete outcomes or paired measurements. Dependencies between observations and sampling design are not corrected. The distance is useful for describing distributions but cannot establish causal change or equivalence by itself.
Reference: R stats — Kolmogorov Smirnov test and empirical distributions (opens in a new tab) ↗
What this calculation answers
Comparing two samples by their means can miss differences in spread or shape. The two-sample Kolmogorov–Smirnov distance compares their empirical cumulative distributions across all observed values.
This page reports the largest absolute gap and its two signed directional components. It intentionally reports a descriptive distance rather than attaching a continuous-null p value to any dataset supplied.
The Sample percentile calculator examines selected empirical quantiles. The KS distance instead searches the entire empirical cumulative difference.
Use the right input conventions
Enter each numeric sample as a list. The samples may have different sizes. Their values must describe comparable quantities in the same units; row order does not create pairing in this calculation.
Pooled distinct values are sorted and all observations tied at a value advance together. That convention makes the empirical cumulative fractions explicit and avoids arbitrary within-tie ordering.
Interpret the result
The distance ranges from zero for identical empirical distributions to one for completely separated supports. Its value does not indicate which physical cause produced the difference, or whether that difference is substantively important.
A signed gap FA minus FB means a larger fraction of A is at or below that threshold. Read it as a cumulative-distribution comparison, not as the difference between sample means.
Work through the example
For A equal to one and two and B equal to two and three, FA is one half at value one while FB is zero. At value two the cumulative fractions are one and one half.
The maximum absolute gap is therefore 0.5. At value three both cumulative fractions reach one. The displayed table exposes each step used to find the maximum.
Where the model stops
No hypothesis-test p value or rejection cutoff is provided. Classical continuous-distribution null formulas do not automatically apply to ties, discrete outcomes or paired measurements.
Dependencies between observations and sampling design are not corrected. The distance is useful for describing distributions but cannot establish causal change or equivalence by itself.
A few questions, answered
Does a zero empirical distance prove identical population distributions?
No. It means the entered finite samples have identical empirical cumulative distributions.
What changes in the comparison?
Use identical empirical samples and inspect the zero-distance limit.