Kolmogorov-Smirnov Test

Why K-S handles continuous data better than binned tests

Mathematical Audit Benchmark

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Pull floats from a crypto casino and you’ll see values like 0.485293 and 0.129481 — decimals produced by HMAC-SHA256 seed chains. A lazy audit stops at the mean: if the average lands near 0.50, the game gets a pass. That reasoning has a hole in it. A generator that emits only 0.10 and 0.90 also averages 0.50, and it is catastrophically broken.

The fix is to test the shape of the whole distribution, not one summary number. The Kolmogorov-Smirnov test is non-parametric and works on continuous data directly: it compares the cumulative distribution of your sample against a theoretical uniform CDF. No bins, no grouping decisions — and therefore nowhere for a subtle bias to hide.

The math: Calculating the D statistic and p-value

The test overlays two cumulative curves — your sample’s and the theoretical uniform’s — then finds the widest gap between them:

1. The Empirical CDF (F_n(x))

Sort the $N$ floats in your sample from smallest to largest. The empirical step function ($F_n(x)$) then climbs by $1/N$ at every data point:

F_n(x) = (Number of elements in sample ≤ x) / N

2. The Kolmogorov D Statistic

Next, measure the absolute vertical distance between that step function and the theoretical uniform line ($F_0(x) = x$ on the interval $[0,1]$). The largest such distance anywhere along the axis is the test statistic $D$:

D = supremum | F_n(x) - x |

3. Calculating the p-value

A large $D$ alone doesn’t tell you how unlikely it is. The tool converts it using a Marsaglia-Tsang series calculation, which yields the p-value:

• p-value < 0.05: Highly anomalous. The probability of a fair RNG generating this distribution by chance is under 5%. The casino’s RNG is statistically proven to be biased.
• p-value > 0.95: Too perfect to be true. This suggests the data has been artificially smoothed or manipulated to look fair, which is a common indicator of fraud in manufactured audits.

Step-by-step audit: Auditing 1,000 floats

Say your session history gives you 1,000 round floats. Here is the working procedure:

1. Copy the raw list of continuous decimal floats (one per line).
2. Paste the values into the auditor input box.
3. Click “Verify.” The tool will sort the floats, calculate the step function, and compute the $D$ statistic.
4. Read the p-value verdict: a fair, unmanipulated RNG will output a uniform p-value that typically falls between 0.10 and 0.90.

Frequently asked questions