When Does Randomness End?

Overview

Objectives/Goals The objective of this experiment is to determine whether multiplication, exponentiation, or addition will force a set of random numbers [0,1] to conform to Benford's Law (a most-significant-digit law for natural, social, or mathematically-manipulated numbers) with a statistical significance level of 0.05 by Chi-square testing. Methods/Materials Java was used to create random number sets [0,1] (proven not to conform to Benford's Law) then perform operations on this set with additional random number sets. The results from these programs were analyzed in Microsoft Excel and placed into charts and graphs. Error bars on these graphs were calculated using a poisson distribution. Chi-square values of each calculation were calculated and compared to the statistical significance level of 0.05. Results When multiplying with sets of random numbers, their conformity to Benford's Law becomes 0.05 significant after one multiplication and becomes much more significant with more multiplications. When random sets are added, the conformity is not significant. As the number of additions increases, the set becomes less significant. When random sets are used in exponentials their conformity is much better than 0.05 significance. Conclusions/Discussion Some mathematical operations on random number sets converge rapidly to a Benford's Law distribution. The number of operations can range from one to a few depending on the operation, but a general trend can often be detected after those first few operations. This rapid convergence can explain why numbers in calculations such as income tax returns can conform to Benford's Law.

Competition history

• CSEF 2005 Mathematics & Software · Entry S1223

Resources

• CSEF project summary ↗

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