Chi-Square + Runs Test

Why one statistical test is not enough

An online casino can cheat you without making the game look obviously biased. They can ensure that the overall game pays out the exact advertised RTP (e.g. 97%), passing basic regulatory audits, but manipulate the *sequence* of outcomes. They might cluster winning rounds during low wagers and cluster losing rounds the second you scale up your stakes.

To verify real randomness, a stream must pass two distinct hurdles: statistical uniformity (equal distribution of values) and serial independence (no predictable patterns or memory).

The two pillars of RNG auditing

Our auditing verifier processes your pasted sequence of floating-point numbers or dice rolls through two rigorous mathematical tests:

1. Chi-Square Goodness-of-Fit Test (Uniformity)

The Chi-Square test checks if the outcomes are spread evenly across all possible bins. For example, if you roll a 6-sided die 6,000 times, each face should appear approximately 1,000 times:

χ² = Σ ((Observed_i - Expected_i)^2 / Expected_i)

Where Observed is the actual count in bin i, and Expected is the theoretical count. The calculator converts this χ² statistic into a precise p-value using the Wilson-Hilferty approximation.

2. Wald-Wolfowitz Runs Test (Independence) (best paired with the Kolmogorov-Smirnov Test)

A stream can be perfectly uniform but completely predictable. For example, the alternating sequence [1, 6, 1, 6, 1, 6...] is perfectly uniform but is obviously not random. The Runs Test tracks the number of continuous sequences of values above and below the median (known as “runs”) to find serial patterns:

Expected_Runs (μ) = ((2 * N1 * N2) / N) + 1

Standard_Deviation (σ) = √((2*N1*N2 * (2*N1*N2 - N)) / (N^2 * (N - 1)))

Where N1 is the number of values above the median, N2 is the number of values below the median, and N is the total count. The calculator derives the standard normal Z-score:

Z = (Observed_Runs - μ) / σ

A high absolute Z-score indicates that the stream has too few runs (indicating clustering) or too many runs (indicating artificial alternation).

Data Sandwich: Auditing 500 crash rounds

Let’s walk through an actual audit. You extract a history of 500 consecutive crash multipliers from an online casino. You convert them to their raw floating-point seeds (between 0 and 1) and paste them into the auditor:

• Chi-Square P-Value: 0.42 (Uniformity check: **PASSED**)
• Runs Test Z-Score: -3.85 (P-value: 0.0001, Independence check: **FAILED**)

The Chi-Square test confirms that all decimals between 0 and 1 are evenly represented. However, the negative Z-score of -3.85 reveals a major problem: there are far fewer runs than expected. The winning and losing rounds are heavily clustered together. This is a clear mathematical indicator of a manipulated, non-random stream.

Frequently asked questions