only needs an 11 percent increase before we get a new first digit, 1, at 10,000. And now

we again need a 100 percent increase before the first digit changes to a 2. The Dow will

have a first digit 1 for far longer than any other first digit.

Benford’s Law does not apply to all sets of numbers.

For it to apply the numbers must reflect the size of some phenomenon;

big numbers must refer to big things.

There must be no built-in maximum or minimum values.

All though zero can be a minimum number.

Tax returns for example have minimum or maximum amounts in various places.

The numbers must not be labels such as highway

numbers, social security numbers, or flight numbers.

A data set that conforms very nicely to Benford’s Law is the populations of the 19,000 towns and

cities in the United States.

Every population count has a first digit and the graph shows the expected Benford proportions

as a line, and the actual proportions as the nine bars. Each possible first digit is shown

as a bar and the proportions are shown on the y-axis. The top of the bar is pretty close

to the line in all cases, meaning we have a very nice fit to Benfords law.

Every number also has first-two digits. The first-two digits range from 10 to 99. We use

a line to represent the expected proportions. The first-two digits also conform closely

to Benford’s Law.

This graph shows streamflow statistics for 140 years that conform almost perfectly to

Benford’s Law. And the last graph here, are

the 80,000 ledger balances for a large company also conformed closely to Benford’s Law.

Now for a little fraud data...

A State of Arizona employee processed 23 checks for non-existent services performed by a fictitious

vendor. The numbers that he invented had many more 7s, 8s, and 9s than would be expected

under Benford’s Law, and for that matter, than would be expected if the digits were

equally likely.

This graph shows the credits issued for kilowatt hours by an electric utility company. We investigated

the spike at “99” and it turned out that several employees were fraudulently giving

customers credits for numbers just below 1 million and just below 100,000 KwH. Those

customers would in turn give the employees a nice present in exchange for their credit.

To summarize,

Benford Law works well to detect invented numbers when,

One person invents all the numbers, or, lots of different people each have some incentive

to manipulate numbers in the same way (such as on tax returns)

It is a useful start that gives us a better understanding of our data

We use it together with other more focused drill down tests to detect fraud, errors,

biases, and other anomalies

We should have a winning combination