Statistical Tests

This page explains the statistical tests used to validate die value randomness quality. These tests run continuously on rolling windows of beacon output.

Note: For tests comparing rng.dev against drand and NIST beacons (hash byte comparison), see Benchmark Tests.

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Understanding the Results

P-Values

Every test produces a p-value between 0 and 1. This represents the probability of seeing results this extreme (or more extreme) if the data were truly random.

P-Value	Status	Meaning
> 0.05	PASS	No evidence against randomness
0.01 - 0.05	WATCH	Borderline — monitor for patterns
< 0.01	INVESTIGATE	Statistically significant deviation

Expected Failures

Random data will sometimes fail random tests. This is not a bug — it's mathematics.

At significance level α = 0.05:

• 5% of tests will fail even for perfect randomness
• Running 6 tests means ~26% chance at least one fails per window
• This is why we apply Bonferroni correction (divide α by number of tests)

The dashboard shows occasional failures as expected behavior, not system problems. Only persistent, repeated failures indicate potential bias.

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Test Descriptions

1. Chi-Squared Distribution Test

What it tests: Are all die faces appearing with equal frequency?

How it works:

1. Count occurrences of each face (1-6)
2. Compare observed counts to expected counts (n/6 each)
3. Calculate chi-squared statistic: χ² = Σ (observed - expected)² / expected

What it detects:

• Biased die (one face appears more often)
• Manufacturing defects in physical dice
• Software bugs favoring certain values

Interpretation:

• High p-value (>0.05): Distribution looks uniform
• Low p-value (<0.01): Some faces appear too often or too rarely

Example:

1000 rolls, expected 166.7 per face

Face | Observed | Expected | Contribution
-----|----------|----------|-------------
  1  |   158    |  166.7   |    0.45
  2  |   172    |  166.7   |    0.17
  3  |   165    |  166.7   |    0.02
  4  |   170    |  166.7   |    0.07
  5  |   168    |  166.7   |    0.01
  6  |   167    |  166.7   |    0.00
                             ─────────
                     χ² =      0.72

p-value = 0.98 → PASS (no bias detected)

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2. Runs Test (Odd/Even Parity)

What it tests: Do odd and even values alternate appropriately?

How it works:

1. Convert each die value to parity: odd (1,3,5) → 1, even (2,4,6) → 0
2. Count "runs" — consecutive sequences of same parity
3. Compare run count to expected value for random sequence

What it detects:

• Too much alternation (odd-even-odd-even pattern)
• Too much clustering (odd-odd-odd-odd pattern)
• Predictable sequencing

Why odd/even instead of above/below median:

• Discrete die values split cleanly: exactly 3 odd, 3 even
• Avoids ambiguity at median (3.5)
• Better statistical properties for 6-sided dice

Interpretation:

• High p-value: Runs count is normal for random data
• Low p-value: Sequence is too clustered or too alternating

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3. Streak Distribution

What it tests: Do consecutive same-value sequences follow expected lengths?

How it works:

1. Find all "streaks" — consecutive identical values
2. Count streaks of each length (1, 2, 3, 4+)
3. Compare to theoretical distribution

Expected distribution for fair die:

P(streak length k) = (1/6)^(k-1) × (5/6)

Length | Probability | Per 1000 streaks
-------|-------------|------------------
   1   |   83.3%     |      833
   2   |   13.9%     |      139
   3   |    2.3%     |       23
   4   |    0.4%     |        4
  5+   |    0.1%     |        1

What it detects:

• Sticky behavior (too many long streaks)
• Anti-sticky behavior (too few repeats)
• Memory in the random source

Interpretation:

• High p-value: Streak lengths are normal
• Low p-value: Streaks are suspiciously long or short

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4. Transition Matrix Test

What it tests: Does the next value depend on the current value?

How it works:

1. Build 6×6 matrix counting transitions (e.g., "1 followed by 4")
2. For each row, run chi-squared test against uniform distribution
3. Report 6 separate p-values (one per starting value)

Expected behavior:

P(next = j | current = i) = 1/6 for all i, j

Transition matrix should look roughly like:
       → 1    2    3    4    5    6
    1   16.7% 16.7% 16.7% 16.7% 16.7% 16.7%
    2   16.7% 16.7% 16.7% 16.7% 16.7% 16.7%
    ...

What it detects:

• Markov dependencies ("3 is often followed by 5")
• Mechanical bias in physical dice
• Pseudo-random generator weaknesses

Why 6 separate tests instead of one:

• Raw matrix chi-squared violates independence assumptions
• Row-by-row testing is statistically valid
• Pinpoints which transitions are problematic

Interpretation:

• All rows p > 0.05: No transition bias detected
• One row p < 0.01: That starting value has biased follow-ups

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5. Serial Pair Test

What it tests: Do all consecutive pairs appear with equal frequency?

How it works:

1. Extract all consecutive pairs: (1,4), (4,2), (2,6), ...
2. Count occurrences of each of 36 possible pairs
3. Chi-squared test against expected frequency (n/36 each)

Expected behavior:

36 unique pairs, each with probability 1/36 ≈ 2.78%

In 10,000 rolls → ~9,999 pairs → ~278 expected per pair

What it detects:

• Subtle sequential bias invisible to single-value tests
• "1 is more likely to follow 3" patterns
• State-dependent behavior

Difference from transition matrix:

• Transition matrix: conditional probabilities (P(next|current))
• Serial pair test: joint probabilities (P(current AND next))
• Both are valuable; they catch different anomalies

Interpretation:

• High p-value: All pairs appear equally often
• Low p-value: Some pairs are over/under-represented

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6. Shannon Entropy

What it tests: How much information content is in the output?

How it works:

1. Calculate frequency of each face: p_i = count_i / total
2. Compute entropy: H = -Σ p_i × log₂(p_i)
3. Compare to theoretical maximum

Theoretical values:

Maximum entropy for 6-sided die:
H_max = log₂(6) ≈ 2.585 bits

This occurs when all faces are equally likely (p = 1/6)

What it detects:

• Low entropy = predictable output
• Concentrated distribution = fewer effective outcomes
• Information loss from bias

Interpretation:

Observed Entropy	Meaning
~2.585 bits	Perfect — all outcomes equally likely
2.4 - 2.58 bits	Good — minor variation
< 2.4 bits	Concerning — some outcomes dominate
< 2.0 bits	Serious — significant bias present

Note: Entropy is always shown as a metric, not a pass/fail test, because it provides intuitive understanding of randomness quality.

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7. Autocorrelation

What it tests: Is there correlation between values at different time lags?

How it works:

1. For each lag k (1, 2, 3, ... 20):
2. Compute correlation between sequence and itself shifted by k
3. Check if correlations fall within expected bounds

Expected behavior:

For truly random data:
- Autocorrelation at all lags ≈ 0
- 95% confidence bounds: ±1.96/√n

For n = 1000:
- Bounds ≈ ±0.062
- Values outside bounds suggest correlation

What it detects:

• Periodic patterns (every 10th value repeats)
• Trending behavior
• Poor PRNG with short cycles

Important caveat: Die values (1-6) are categorical, not continuous. Autocorrelation assumes numeric distance matters, but "1 → 6" isn't meaningfully different from "2 → 3" in randomness terms.

Interpretation:

• Use for visualization and trend detection
• Don't rely on it as primary randomness test
• Transition matrix is more appropriate for sequential independence

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Rolling Windows

Tests run on multiple window sizes to catch both short-term and long-term anomalies:

Window	Purpose	Update Frequency
100	Short-term fluctuations	Every round
1,000	Medium-term patterns	Every round
10,000	Stable statistics	Every 10 rounds
100,000	Long-term validation	Every 100 rounds

Why multiple windows?

• Small windows: Responsive to recent changes, but noisy
• Large windows: Stable statistics, but slow to detect new problems
• Combined view: Best of both worlds

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Bonferroni Correction

When running multiple tests, false positives accumulate.

Problem:

• 6 tests at α = 0.05
• Probability of at least one false positive: 1 - (0.95)⁶ ≈ 26%

Solution: Bonferroni correction

• Adjusted α = 0.05 / 6 ≈ 0.0083
• Each test must achieve p < 0.0083 to be "significant"
• Family-wise error rate stays at 5%

Dashboard shows:

• Individual test p-values
• Bonferroni-corrected overall status
• Whether any test is "significant" after correction

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What These Tests Cannot Tell You

Statistical tests have limitations:

1. Cannot prove randomness — only detect specific types of non-randomness
2. Cannot detect all manipulation — adversary might pass all tests
3. Cannot predict future values — that's the point
4. Will occasionally fail — 5% false positive rate is expected

The dashboard is evidence of quality, not proof of perfection.

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Further Reading

• Benchmark Tests — Comparing rng.dev against drand and NIST beacons
• NIST SP 800-22 — Statistical test suite for random number generators
• Diehard Tests — Classic battery of randomness tests
• How It Works — Beacon generation process
• Threat Model — Security assumptions and limitations