Originating author is Christiane Rousseau.
It is very risky to change too many numbers in some financial statements if one does not know some mathematics. Indeed, most often the numbers appearing in financial statements follow some strange mathematical rule, called Benford’s law, or law of the first significant digit. If one forgets to follow the rule, then the numbers will fail some statistical tests and are likely to be scrutinized with care. Benford’s law claims that if you collect numbers at random and calculate the frequencies of their first significant digits, the numbers with first significant digit should appear around % of the time, while the numbers with first significant digit appear only % of the time. This rule is observed in many other sets of numbers, like powers of or Fibonacci numbers.
We now have satisfying explanations. We are going to share them with you.
Benford law’s concerns the distribution of the first significant digits of numbers. The first significant digit of a positive number is the leftmost nonzero digit of its decimal expression. For instance, the first significant digit of is , that of is , and that of is . Another way to define it which will be useful for our mathematical discussion is to write a positive real number as a number times a power of :
Then the first significant digit of is the integer part of , which can be denoted by . The number is called the mantissa of . We now claim that if you collect numbers at random and compute the frequency of the first significant digit , then is approximately given by . This gives us the frequencies:
Table 1: Frequencies of Benford’s law.
Let us now give a very brief historical note. The phenomenon was first discovered by the astronomer Simon Newcombe (1835-1909) who noticed that the first pages of the logarithmic tables corresponding to small first significant digits looked much more worn than the later pages. His discovery was forgotten and the law was rediscovered by Frank Benford (1883-1948) around 1938. Frank Benford collected tenths of thousands of numbers from all origins following his law. The modern data base of Simon Plouffe which contains millions mathematical constants also follows Benford’s law.
Many sets of numbers that are not random also follow Benford’s law. This is the case with the populations of countries, with the areas of countries, with the length of rivers, etc. Maybe you will stop me and start being skeptic… In which units are these lengths or areas collected? Are lengths in miles or in kilometers? This does not matter… If the lengths of rivers in kilometers follow Benford’s law, then the lengths in miles follow Benford’s law! A change of unit corresponds to a change of scale. We will see that Benford’s law is invariant under change of scale. Moreover, it is the only probability law which is invariant under change of scale.
I told you in the introduction that the Fibonacci numbers follow Benford’s law. But, in a sense, Benford’s law is subjective, since it depends on the basis in which we write our numbers. In some basis with , the nonzero digits are the elements of the set , and Benford’s law in basis says that the frequency of the first significant digit is . Well! Fibonacci numbers follow Benford’s law in any basis Benford’s law is invariant under change of basis. And it is the only non trivial probability law which is invariant under change of basis.
Now it is time to come to explanations. They will require that you remember some of your probability course. But you could prefer to experiment by yourself first before starting reading more serious maths.
1. Invariance under change of scale
Let us consider a simple change of scale obtained by multiplying all numbers of a set of numbers by . If we consider the numbers with first significant digit , then they are changed to numbers with first significant digit either or . It is easy to check that . Indeed,
Similarly, you could check that , etc. But how do you manage if you change from miles to kilometers, i.e. multiply numbers by ? As stated, Benford’s law is too restrictive and we need to generalize it. What does it mean that the first significant digit is ? It means that the mantissa belongs to the interval . So, Benford’s law is a partial probability distribution on the mantissa. The generalized Benford law (which, by abuse of language, we will call Benford’s law) on the mantissa is given by a density function on the interval . When we pick up a number at random, we can compute its mantissa. This gives us a random variable taking values in . We say that it follows Benford’s law if its density function is given by
If represents the probability that , then this means that we must have
It is really a generalization of Benford’s law since
What does it mean to say that a random variable on is invariant under change of scale? It means that, if is a positive real number and if we take the random variable , then the mantissa of the random variable has the same density function as . It is not difficult to show that this is the case when follows Benford’s law, but there are several cases to distinguish depending on the size of . We will do one case and will let you do the other cases. We can write , with the mantissa of . Since the mantissa of is the same as that of , it suffices to consider the case .
What is the tool to show that? You may remember from your probability course that the (cumulative) distribution function is sometimes more useful than the density function for a continuous random variable. The distribution function of a random variable is
If follows Benford’s law, then its distribution function is given by
So we must show that if follows Benford’s law and is the mantissa of for , then the distribution function of is given by (1).
For that purpose, we need to calculate for . is the mantissa of which takes values inside . So , when and when The first case occurs when . For the mantissa of to lie inside , the only possibility is that . Then the mantissa of is equal to .
as expected. The other cases are done in the same way.
The reciprocal is more exciting…
2. Benford’s law is the only probability law on the mantissa which is invariant under change of scale
It seems an impressive statement! Yet, you will see that its proof is not much more complicated than the previous argument. Let be a random variable representing the mantissa and taking values inside . We look for its distribution function , under the hypothesis that is invariant under change of scale. So we need to compute
Hence, we must have and .
The main difficulty of the proof lies in interpreting what it means that is invariant under change of scale. Since and are the same events, then we have that
As before, let us consider the case of some so that ( depends on ). Then, for , is equal to its mantissa. Since is invariant under change of scale, then the mantissa of has the same distribution function as . Hence,
Combining with (2) we see that satisfies
provided that is not too large. We must find from the functional equation (3). Let’s see how to do this. If we let , this yields
which can be written
since . Let us take the limit when . We must recognize on each side a quotient whose limit is a derivative. On the left side it is , the limit of which is , and on the right side it is which tends to . Hence, we get the
differential equation with separable variables
the solution of which is . Since we have , and since , then . Hence, and we are done!
3. Why numbers collected from all origins follow Benford’s law?
An answer was provided by Theodore Hill in 1995, and we will discuss briefly his idea. Of course, not all sets of numbers follow Benford’s law. For instance, if you consider the height of humans in meters then, up to a few exceptions, only the first significant digits and will occur, and if you convert the size in feet (a foot is approximately cm) you will change the distribution law of the first significant digit. So this set of numbers is not invariant under change of scale. But, suppose we consider a large set of numbers coming from all origins and we change of scale. There are different subsets of numbers with their particular scale. Because the set is large and the numbers come from all origins then, most likely all different scales are present. Multiplying all numbers present in the set by a positive constant induces a permutation of the scales present in the new set. So, as a whole, we could expect the set of numbers to behave as if it had no special scale. Hence, it will follow Benford’s law.
This explanation is good for sets of numbers coming from all origins. But it does not explain why areas of countries, populations of countries, or lengths of rivers should follow Benford’s law. We will discuss very recent explanations (2008!) for this case given by Gauvrit, Delahaye and Fewster. Their explanation is also valid for large sets of numbers coming from all origins.
4. Sets of numbers spreading over several orders of magnitude are likely to follow Benford’s law!
We are working in base and we have seen that positive numbers can be written as , where and . We could consider as the order of magnitude and we say that there are several orders of magnitude if there are several values of for our set of numbers. (Note that such a property is invariant under change of scale!) To simplify the explanation, suppose that the numbers lie within . Then the numbers with signicant digit are the ones in the set
and similar sets with the other digits. It is better to move to the logarithm in base of these numbers: . Then . Let us show that if a random variable on obeys Benford’s law, then the random variable is simply uniform on . For that, it suffices to show that the distribution function of is that of the uniform random variable on , namely
Indeed, when ,
If belongs to the set , then belongs to the set :
and similarly for the other digits. Suppose that taking a random number in our set is a random variable taking values inside . Then takes values inside . Recall that the probability that some random variable belongs to some set is equal to the area under the graph of the density function over the set. If the density function of over was uniform as in Figure 3 (a) we would be done. Most often however, it will not be the case as in Figure 3 (b). But this is why it is so important that the original set of numbers is spread over several orders of magnitude. The different portions corresponding to a given first significant digit are spread horizontally over several segments, whose sum of lengths is of the order of of the total width. So, even if the height of is not the same from one segment to the other, one could hope that the mean height be of the same order of magnitude for the different digits. When this occurs, then the data follow Benford’s law.
5. How to test if a set of numbers follows Benford’s law?
If you have followed a statistics course, then you probably studied the goodness-of-fit test. This test allows you to check if some data follow some probability distribution. Suppose you want to make the test with a set of numbers. You just need to construct a table, in which represents the number of numbers in your set which have first
significant digit . Of course, . The represent the numbers of numbers that should have first significant digit if your set was following Benford’s law, namely .
Table 2: The table for the goodness-of-fit test.
You then calculate
and you look in a table of the at the line corresponding to degrees of freedom. If you are to make a test with % of error, then you accept that the data fit Benford’s law if and you reject otherwise. This is a fast recipe, but if you are to make such tests with your students, then just take some time to get more familiar with the details of the test and its meaning.
6. Invariance of Benford’s law under change of basis
This could be modeled in a similar way as the invariance under change of scale. It is more tricky though, since we cannot limit our work to the mantissa. Indeed, if , then the part also need to be converted to the new basis. And, in fact, the main difficulty is to express in mathematical terms what it means for a random variable to be independent under change of basis. We skip the details.
Benford’s law is fascinating: it defies the intuition, and is it something that you can test yourself and also adapt for a classroom activity. It used to be a curiosity, but it is now a standard tool to detect frauds. Of course, more and more tax evaders learn of it. But pay attention: the first significant digit is not the only thing to take care of. The generalized Benford’s law allows to derive a law for the second significant digit, third significant digit, etc. You can try to find it by yourself: just think in which unions of intervals the mantissa of a number should be so that its second significant digit is .