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One Commonly used data mining tool based on statistics is Benford Analysis. Today we have

Benford Expert WVU Professor Mark Nigrini with us. Professor Nigrini, will you tell

us about Benford Analysis?

Sure, Frank Benford was a physicist in the 1920s.

He noticed that the first few pages of his logarithm tables were more worn than the last

few pages. He concluded that he was looking up the logs of numbers with low first digits

more often than he was looking up the logs of numbers with high first digits.

The first digit of a number is the leftmost digit. We have 32,340 students at WVU and

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the first digit of that number is a 3. There are 9 possible first digits. Zero can never

be a first digit. Minus signs are ignored when we calculate the first digit.

Benford examined 20 lists of numbers with 20,000 records in total. His results showed

that 30.6 percent of the numbers started with a “1” and 18.5 percent of the numbers

started with a “2.” This means that 49 percent of the numbers started with a 1 or

a 2 while the other 51 percent started with 3, 4, 5, 6, 7, 8, or 9.

He made some assumptions about the properties of numbers and using some calculus he calculated

the expected frequencies of the digits in natural numbers. In the first position there

is a large bias towards the low digits. Zero can be a second digit and so from the second

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digit onwards there are ten possible digits. The bias gets less and less as we move on

to the second and third digits and from the fourth digit onwards the ten possible digits

are, for all practical purposes, equally likely.

Why is this so? If we take the Dow Jones index at 1,000 where it has a first digit 1, we

need a 100 percent increase before the 1 becomes a first digit 2. At 5,000 the Dow only needs

a 20 percent increase to change the first digit 5 to a first digit 6. At 9,000 the Dow

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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