Benford's Law - Tests of Benford's Law With Common Distributions

Tests of Benford's Law With Common Distributions

Benford's law was empirically tested against the numbers (up to the 10th digit) generated by a number of important distributions. including the uniform distribution, the exponential distribution, the half-normal distribution, the right-truncated normal, the normal distribution, the chi square distribution and the log normal distribution In addition to these the ratio distribution of two uniform distributions, the ratio distribution of two exponential distributions, the ratio distribution of two half-normal distributions, the ratio distribution of two right-truncated normal distributions, the ratio distribution of two chi-square distributions (the F distribution) and the log normal distribution were tested.

The uniform distribution as might be expected does not obey Benford's law. In contrast, the ratio distribution of two uniform distributions is well described by Benford's law. Benford's law also describes the exponential distribution and the ratio distribution of two exponential distributions well. Although the half-normal distribution does not obey Benford's law, the ratio distribution of two half-normal distributions does. Neither the right-truncated normal distribution nor the ratio distribution of two right-truncated normal distributions are well described by Benford's law. This is not surprising as this distribution is weighted towards larger numbers. Neither the normal distribution nor the ratio distribution of two normal distributions (the Cauchy distribution) obey Benford's law. The fit of chi square distribution depends on the degrees of freedom (df) with good agreement with df = 1 and decreasing agreement as the df increases. The F distribution is fitted well for low degrees of freedom. With increasing dfs the fit decreases but much more slowly than the chi square distribution. The fit of the log-normal distribution depends on the mean and the variance of the distribution. The variance has a much greater effect on the fit than does the mean. Larger values of both parameters result in better agreement with the law. The ratio of two log normal distributions is a log normal so this distribution was not examined.

Other distributions that have been examined include the Muth distribution, Gompertz distribution, Weibull distribution, gamma distribution, log-logistic distribution and the exponential power distribution all of which show reasonable agreement with the law. The Gumbel distribution - a density increases with increasing value of the random variable - does not show agreement with this law.

Read more about this topic:  Benford's Law

Famous quotes containing the words tests of, tests, law and/or common:

    The cinema is going to form the mind of England. The national conscience, the national ideals and tests of conduct, will be those of the film.
    George Bernard Shaw (1856–1950)

    Although she feeds me bread of bitterness,
    And sinks into my throat her tiger’s tooth,
    Stealing my breath of life, I will confess
    I love this cultured hell that tests my youth!
    Claude McKay (1889–1948)

    I was not born to be forced. I will breathe after my own fashion. Let us see who is the strongest. What force has a multitude? They can only force me who obey a higher law than I.... I do not hear of men being forced to live this way or that by masses of men. What sort of life were that to live?
    Henry David Thoreau (1817–1862)

    We therefore commit his body to the ground; earth to earth, ashes to ashes, dust to dust; in sure and certain hope of the Resurrection.
    The Burial Service, Book of Common Prayer (1662)