Kolmogorov Smirnov Test Tool

Math Probability • Non Parametric and Computational Probability

Written by STEM Calculators Team Published February 15, 2026 Updated February 24, 2026

Compare an empirical CDF to a theoretical CDF (one-sample) or compare two empirical CDFs (two-sample), compute the KS statistic \(D\), and visualize the maximum deviation on a CDF plot.

Tip: Use “Fill example” to load data and run the test immediately.

Test mode

One-sample: \(D=\sup_x\lvert F_n(x)-F(x)\rvert\). Two-sample: \(D=\sup_x\lvert F_{n_1}(x)-F_{n_2}(x)\rvert\).

Theoretical CDF (one-sample)

For custom CDF, write an expression in x using explicit multiplication (e.g., 2*x, not 2x).

Data list (sample 1)

Allowed separators: commas, spaces, semicolons, new lines.

Normal parameters

\(\mu\)

\(\sigma\) (must be \(>0\))

Ready

CDF comparison plot (animated)

Animation ready

Scrub 0% t = 0.00

The badge inside the plot shows computed \(D\) and the approximate p-value. The moving line shows the deviation at the scrubbed position.

Enter data and click “Calculate”.

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Theory

What the Kolmogorov–Smirnov (KS) test measures

The KS test compares cumulative distribution functions (CDFs). It looks at the largest vertical gap between two CDF curves. Because it uses the CDF (not bins), it avoids histogram choices and remains fully non-parametric.

Key idea. The KS statistic is the maximum absolute deviation between CDFs.

\[ D=\sup_{x}\left|F_1(x)-F_2(x)\right| \]

In one-sample form, \(F_1\) is the empirical CDF from data and \(F_2\) is the theoretical CDF. In two-sample form, both are empirical CDFs from two data sets.

Empirical CDF and the KS statistic

Empirical CDF. For sample values \(x_1,\dots,x_n\), the empirical CDF is a step function:

\[ F_n(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}\{x_i\le x\} \]

The empirical CDF jumps by \(1/n\) at each data point (after sorting). This is why the KS maximum can occur either just before or at a jump.

One-sample KS. Compare the step curve to a theoretical CDF \(F(x)\):

\[ \begin{aligned} D &= \sup_x \left|F_n(x)-F(x)\right| \\ &= \max_{i}\left(\left|\frac{i}{n}-F(x_i)\right|,\;\left|\frac{i-1}{n}-F(x_i)\right|\right) \end{aligned} \]

Two-sample KS. Compare two empirical CDFs:

\[ D=\sup_x\left|F_{n_1}(x)-F_{n_2}(x)\right| \]

The plot in the calculator draws both CDFs and marks the maximum vertical segment (the computed \(D\)).

Approximate p-value (Kolmogorov distribution)

For moderate/large samples, an asymptotic approximation gives a p-value using the Kolmogorov distribution. A practical correction improves finite-sample performance:

One-sample scaling.

\[ \begin{aligned} \lambda &= \left(\sqrt{n}+0.12+\frac{0.11}{\sqrt{n}}\right)D \\ p &\approx 2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}\lambda^{2}} \end{aligned} \]

Two-sample scaling. Use an effective sample size:

\[ \begin{aligned} n_{\text{eff}} &= \frac{n_1n_2}{n_1+n_2} \\ \lambda &= \left(\sqrt{n_{\text{eff}}}+0.12+\frac{0.11}{\sqrt{n_{\text{eff}}}}\right)D \\ p &\approx 2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}\lambda^{2}} \end{aligned} \]

Interpretation: small p-values indicate that the observed CDF gap \(D\) is unlikely if the two CDFs were the same. Large p-values mean the data are consistent with the reference distribution (or the two samples are consistent with a common distribution).

Practical note: If you estimate parameters (e.g., \(\mu,\sigma\) for a normal) from the same data, the classical one-sample KS p-value is optimistic. That setting often uses a Lilliefors-type adjustment (not applied here).

Worked example (conceptual)

Suppose we test whether a small dataset looks like a normal distribution with known \(\mu\) and \(\sigma\). After sorting, we evaluate the theoretical CDF at each sample point and compare it to the empirical step values \(i/n\) and \((i-1)/n\).

Process (at a sample point \(x_i\)).

\[ \begin{aligned} F(x_i) &= \Phi\!\left(\frac{x_i-\mu}{\sigma}\right) \\ \Delta_i^{(+)} &= \left|\frac{i}{n}-F(x_i)\right| \\ \Delta_i^{(-)} &= \left|\frac{i-1}{n}-F(x_i)\right| \end{aligned} \]

Then \(D=\max_i\{\Delta_i^{(+)},\Delta_i^{(-)}\}\). The calculator highlights the corresponding vertical segment on the CDF plot.

CDF comparison plot (animated)

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