[Math Talk #6] Benford's Law (Is it Always True?) [Part 1]

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Benford's Law (Part 1, Is it True?)

1. What is Benford's Law?

Benford's law, also called Newcomb-Benford's law, law of anomalous numbers, and first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. While Newcomb was inspecting numerous data's including logarithms, he noticed that people were looking up more numbers whose logarithms started with the digit 1.

Inspired by this inspection, in 1938, American Mathematician Frank Benford [2] stated in his paper <The Law of Anomalous numbers> [3] what is called Benford sequence.

Benford Sequence.

A sequence of positive numbers is said to be Benford of base if the probability of observing the first digit of in base is is

More precisely, we would have

where

The limit implies that the ratio between total number and the number of s having first digit where runs over through should converge to as .

2. Where does it Come From?

2-1. Intuitive Approach

Many people would think that the probability that any positive number in base having first digit is just

Let's just examine what it means without any reasoning. It means that we expect equal probability of having each digit , so that the uniform probability mass function is

Is this true...? Well, first let's do some computer experiment.

2-2. A Simple Computer Experiment

Take the sequence to be just enumeration of positive integers, . Define a function

such that

Here is the plot of when base (decimal expression) up to .

where

Blue Line corresponds to n = 1
Orange Line corresponds to n = 4
Green Line corresponds to n = 7
Red Line corresponds to n = 9

2-3. Analyzing the plot - [4]

The flucutation for each specific , is due to the numbers of the form

all such numbers are counted, which skyrockets the graph. For example, if , then numbers of the form

will be added for numerator of .

The local maxima and minima of specific graph are special. By direct calculation, -th local minima occurs on the number of the form

and maxima occurs on the number of the form

. By direct calculation, minimum and maximum would be

and

Letting implies , so that

and

You can see that since the limit of maxima and minima does not match, the limit

does not exist. From the definition of Benford Sequence in Section 1, we can deduce that

Set of Natural numbers viewed as a trivial sequence is NOT Benford of base 10!

Also, any sequence drawn from a single set can NOT be Benford of base 10.

3. So what is the Point?

The easiest way to understand the conclusion is to think a a random set of number as not one number but two: you have the number itself and the highest possible number that it could be. Therefore, you have a potential range of possible numbers for every random number. Also, for you to have a good data set the potential range should be random for each random number. If the potential range is the same for each random number, then Benford's law will not correctly predict the leading number.

For example, suppose you have a test score of 50 students. If the test score ranges 1 through 100 for all students, you can not expect the test score statistic to follow Benford.

However, suppose you have an accounting data. Numbers in accounting are highly random, each has different potential range, so you can expect that the numbers appear would roughly follow Benford.