Number Distribution - Statistics

Introduction

Number distribution is a fundamental concept in the statistical analysis of random sequences. When we analyze lottery results or any random process, a natural question arises: do numbers appear in a balanced way, or do some appear more frequently than others? This study examines how numbers are distributed in different scenarios and what this reveals about the nature of randomness.

Imagine rolling a fair die thousands of times. We expect each face (1 to 6) to appear approximately the same number of times. If the die were biased, we would see one face appear much more than the others. Distribution analysis applies this same principle to understand whether numbers in draws follow a uniform pattern or if there are interesting deviations.

What is Uniform Distribution?

Uniform distribution is one where each number has equal probability of appearing. In a truly random generator, all numbers should appear with approximately equal frequency over time. This does not mean that in a small set of draws all numbers will appear equally - in fact, variations are normal. What we expect is that, with many observations, the frequencies balance out.

Practical Example: Mega-Sena

In Mega-Sena, we have 60 possible numbers and 6 numbers are drawn at a time. If we analyze many draws, we expect each number to appear approximately with the same frequency.

Expected frequency = (Number of draws × Numbers per draw) / Total numbers

Concrete example: If we analyze 1000 draws:

• Total numbers drawn: 1000 × 6 = 6,000
• Expected frequency per number: 6,000 / 60 = 100 appearances
• Each number should appear about 100 times (16.67% of draws)

Important: In 1000 draws, it is normal to see numbers with 90 appearances and others with 110 - small variations are expected. What would not be normal is to see one number with 200 appearances while another has only 20.

Frequency Analysis

To analyze number distribution, we collect data on:

Frequency Metrics

• Absolute frequency
• Relative frequency
• Cumulative frequency
• Percentage of appearance

Complementary Analyses

• Chi-square test
• Standard deviation
• Coefficient of variation
• Randomness tests

Chi-Square Test

The chi-square test (χ²) is a fundamental statistical tool for verifying whether the observed distribution differs significantly from the expected distribution. It compares what we observe in the data with what we would theoretically expect, considering normal random variability.

Chi-Square Formula

χ² = Σ (Observado - Esperado)² / Esperado

Where the sum (Σ) is over all possible numbers.

How it works:

• For each number, we calculate the difference between observed and expected
• We square it (to eliminate negative signs and give more weight to large differences)
• We divide by expected (normalizes the difference, giving more weight to deviations in numbers that should appear less)
• We sum all these normalized differences

Interpretation: The higher the χ² value, the greater the difference between observed and expected. Small values indicate that the data are close to the expected uniform distribution.

Interpreting Results

Correct interpretation of results is crucial:

Low Chi-Square Values

Indicate: Indicate: The distribution is close to the expected uniform. Numbers are appearing with similar frequencies, suggesting good randomness.

High Chi-Square Values

Indicate: Indicate: Significant deviations from uniform distribution. Some numbers may be appearing more or less frequently than expected.

Limitations and Considerations

It is important to understand the limitations of this analysis:

⚠️ Important Limitations

• Sample size: Small samples can give misleading results
• Local randomness: Temporal deviations are normal
• Independence: Each draw is independent
• Confirmation bias: We tend to see patterns where none exist

Practical Applications

Number distribution analysis has several applications:

Generator testing: Generator testing: Verify quality of random algorithms
Lottery auditing: Lottery auditing: Ensure integrity of draws
Simulations: Simulations: Validate computational models
Academic research: Academic research: Studies on randomness

Analysis Example

Let's consider a simplified example:

Simulation: 1000 Draws

Numbers 1-30

• Appearances: 492
• Expected: 500
• Difference: -8

Numbers 31-60

• Appearances: 508
• Expected: 500
• Difference: +8

This small difference is normal and expected in random samples.

Conclusions

Number distribution analysis is a valuable tool for understanding the nature of randomness. However, it is crucial to interpret results carefully and consider statistical limitations.

In truly random games, small deviations from uniform distribution are normal and expected. What matters is that, over time, the distribution approaches uniformity.

💡 Important Insight

Distribution analysis teaches us that randomness does not mean perfect uniformity in small samples, but rather the tendency toward uniformity in large samples.

📊 Number Distribution