Kolmogorov-Smirnov Calculator

Calculator

Enter numbers separated by commas, spaces, or new lines. For one-sample tests, choose a reference model. For two-sample tests, provide both datasets.

Example data table

These values resemble normalized detector noise samples from two runs.

Formula used

Empirical CDF: For sorted values x(i), Fn(x(i)) = i/n.

One-sample statistic: D = max_x |Fn(x) - F(x)|, where F(x) is the model CDF.

Two-sample statistic: D = max_x |F(n1)(x) - F(n2)(x)|.

This tool estimates the two-sided p-value using the standard asymptotic Kolmogorov distribution with an effective sample size correction.

How to use this calculator

1. Choose one-sample to compare a dataset to a model.
2. Or choose two-sample to compare two measured datasets.
3. Paste your values in Sample A, and Sample B if needed.
4. For one-sample, select a model and set its parameters.
5. Pick a significance level alpha, then press Submit.
6. Review D, p-value, and the decision statement above.
7. Use CSV or PDF buttons to export results.

Purpose in experimental physics

The Kolmogorov-Smirnov (KS) test quantifies how closely an observed sample follows a reference distribution, or whether two measured samples appear drawn from the same parent process. In physics, it is widely used for validating noise assumptions, checking arrival-time statistics, comparing spectra, and screening simulation outputs against lab data.

One-sample workflow for model checks

Use the one-sample option when you have a theoretical or design expectation, such as Gaussian readout noise or uniform timing jitter across a gate. The calculator builds the empirical cumulative distribution from the sorted sample and compares it to the chosen model CDF. It then reports the maximum deviation D, plus an approximate p-value.

Two-sample workflow for run-to-run stability

Use the two-sample option when comparing two measurement campaigns, two detectors, or a baseline run against a modified configuration. Because it compares two empirical cumulative distributions directly, no parametric model is required. This makes it practical for quick stability checks, drift detection, and identifying subtle distribution shifts beyond mean and variance.

Choosing reference models and parameters

For normal comparisons, set the mean and standard deviation based on your expected operating regime or independent calibration. For exponential behavior, specify the rate lambda for decay-like waiting times. For a uniform check, define the minimum and maximum bounds. Incorrect parameters can inflate D, so keep the model aligned with the physics and instrumentation.

Interpreting the D statistic

D is the largest vertical distance between the two cumulative curves. A small D means the curves nearly overlap across the full value range, including tails. A large D indicates systematic differences, such as heavier tails, offsets, or multimodal behavior. In practice, D captures shape differences that simple summary statistics can miss.

Interpreting p-value and significance level

The p-value estimates how likely it is to observe a deviation at least as large as D under the null hypothesis. If p is below your chosen alpha (commonly 0.05), you have evidence to reject the null. For exploratory screening, alpha 0.10 is sometimes used; for stricter claims, 0.01 is common.

Practical data preparation tips

Use consistent units and remove non-numeric tokens before pasting. For time-series-derived samples, avoid oversampling strongly correlated points; independence improves interpretability. If you suspect outliers are real physics, keep them; if they are artifacts, fix the acquisition pipeline instead of trimming blindly. Document preprocessing to support reproducibility.

Reporting and exporting results

For lab notes, record the test type, sample sizes, D, p-value, and alpha. When comparing conditions, include context such as temperature, bandwidth, integration time, and detector settings. The CSV export supports quick import into analysis notebooks, while the PDF export provides a compact attachment for reports and QA documentation.

FAQs

1) What does the KS test measure?

It measures the maximum difference between cumulative distribution functions, capturing shape and tail differences, not just averages. The statistic D summarizes that maximum deviation.

2) When should I use one-sample versus two-sample?

Use one-sample to compare measured data to an assumed model. Use two-sample to compare two measured datasets directly when you do not want to assume a parametric distribution.

3) Do I need histogram binning?

No. The test works on the sorted raw values through cumulative distributions, so you avoid bin-size choices that can hide or exaggerate differences.

4) How many points do I need?

More is better. With very small samples, the test has limited power and p-values can be unstable. Aim for at least a few dozen points when possible, and interpret borderline results cautiously.

5) What if I estimated model parameters from the same sample?

Then the one-sample p-value can be optimistic because the model was tuned to the data. Prefer parameters from independent calibration, or treat results as descriptive rather than confirmatory.

6) Can I use negative values with the exponential model?

Exponential CDF assumes nonnegative support. Negative values usually indicate a baseline shift or preprocessing issue. Consider re-centering, choosing a different model, or using the two-sample test instead.

7) Why is the p-value labeled approximate?

This calculator uses a standard asymptotic approximation that becomes more accurate as sample sizes grow. For tiny samples or special cases, exact methods may differ slightly.