Kolmogorov-Smirnov Test

The continuity advantage of the K-S test

When you audit a crypto casino, you get a stream of decimal floats (such as 0.485293, 0.129481, etc.) generated by their HMAC-SHA256 seeds. A basic audit might average these numbers and find that they equal roughly 0.50, concluding that the game is fair. This is a severe statistical oversight. An RNG could output exclusively 0.10 and 0.90, averaging 0.50, while being completely broken.

To verify true fairness, you must check the **uniformity** of the entire distribution. The Kolmogorov-Smirnov test is a non-parametric, continuous test that compares the cumulative distribution of your sample against a theoretical uniform distribution. Because it requires no binning, it is highly sensitive to subtle, hidden biases that other tests miss.

The math: Calculating the D statistic and p-value

The K-S test operates by plotting the Cumulative Distribution Function (CDF) of your sample against the uniform CDF and locating the maximum deviation:

1. The Empirical CDF (F_n(x))

First, the $N$ floats in your sample are sorted in ascending order. The empirical step function ($F_n(x)$) increases by $1/N$ at each data point:

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

2. The Kolmogorov D Statistic

We calculate the absolute vertical distance between your sample’s step function and the theoretical uniform line ($F_0(x) = x$ on the interval $[0,1]$). The test statistic $D$ is the supremum (maximum) of these differences:

D = supremum | F_n(x) - x |

3. Calculating the p-value

To evaluate if $D$ is statistically anomalous, the tool runs a Marsaglia-Tsang series calculation to compute 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

Suppose you have extracted 1,000 round floats from your session history:

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. Check 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