Arrays - Building Java Programs: A Back to Basics Approach - page 478

Table 7.3 Expected Distribution
Under Benford's Law

First Digit Frequency

1 30.1%

2 17.6%

3 12.5%

4 9.7%

5 7.9%

6 6.7%

7 5.8%

8 5.1%

9 4.6%

To explore this phenomenon, let's look at two different sequences of numbers: one
that grows linearly and one that grows exponentially. If you start with the number 1
and add 0.2 to it over and over, you get the following linear sequence:

1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6, 2.8, 3, 3.2, 3.4, 3.6, 3.8, 4, 4.2, 4.4, 4.6, 4.8,
5, 5.2, 5.4, 5.6, 5.8, 6, 6.2, 6.4, 6.6, 6.8, 7, 7.2, 7.4, 7.6, 7.8, 8, 8.2, 8.4, 8.6, 8.8,
9, 9.2, 9.4, 9.6, 9.8, 10

In this sequence there are five numbers that begin with 1, five numbers that begin
with 2, five numbers that begin with 3, and so on. For each digit, there are five num-
bers that begin with that digit. That's what we expect to see with data that goes up by
a constant amount each time.
But consider what happens when we make it an exponential sequence instead.
Let's again start with 1 and continue until we get to 10, but this time let's multiply
each successive number by 1.05 (we'll limit ourselves to displaying just two digits
after the decimal, but the actual sequence takes into account all of the digits):

1.00, 1.05, 1.10, 1.16, 1.22, 1.28, 1.34, 1.41, 1.48, 1.55, 1.63, 1.71, 1.80, 1.89,
1.98, 2.08, 2.18, 2.29, 2.41, 2.53, 2.65, 2.79, 2.93, 3.07, 3.23, 3.39, 3.56, 3.73,
3.92, 4.12, 4.32, 4.54, 4.76, 5.00, 5.25, 5.52, 5.79, 6.08, 6.39, 6.70, 7.04, 7.39,
7.76, 8.15, 8.56, 8.99, 9.43, 9.91, 10.40

In this sequence there are 15 numbers that begin with 1 (31.25%), 8 numbers that
begin with 2 (16.7%), and so on. There are only 2 numbers that begin with 9 (4.2%).
In fact, the distribution of digits is almost exactly what you see in the table for
Benford's Law.
There are many real-world phenomena that exhibit an exponential character. For
example, population tends to grow exponentially in most regions. There are many other
data sets that also seem to exhibit the Benford pattern, including sunspots, salaries,
investments, heights of buildings, and so on. Benford's Law has been used to try to
detect accounting fraud under the theory that when someone is making up data, they
are likely to use a more random process that won't yield a Benford style distribution.