Monday Morning Math: the First Digit Law

You can catch criminals with math!  You might expect that if you were to record a bunch of numbers, the first digit would be equally likely to be 1, 2, 3, 4, 5, 6, 7, 8, or 9, but it turns out that for many numbers (costs for a company, time spent working on something) the first digit is usually small:  1 is the first digit about 30% of the time, while 9 is the first digit less than 5% of the time! Here’s a picture of the distribution:

from Gknor 

This rule is generally known as the First Digit Law, although it is also called Benford’s Law after Frank Benford (who himself called it “The Law of Anomalous Numbers” in a 1938 paper) or the Newcomb-Benford Law in recognition that  Simon Newcomb had noted it more than 50 years earlier, in 1881, in “Note on the Frequency of Use of the Different Digits in Natural Numbers”.

There are also some restrictions on what kind of numbers follow the First Digit Law:  According to Statistics How To:

Benford’s law doesn’t apply to every set of numbers, but it usually applies to large sets of naturally occurring numbers with some connection like:

• Companies’ stock market values,
• Data found in texts — like the Reader’s Digest, or a copy of Newsweek.
• Demographic data, including state and city populations,
• Income tax data,
• Mathematical tables, like logarithms,
• River drainage rates,
• Scientific data.

The law usually doesn’t apply to data sets that have a stated minimum and maximum, like interest rates or hourly wages. If numbers are assigned, rather than naturally occurring, they will also not follow the law. Examples of assigned numbers include: zip codes, telephone numbers and Social Security numbers.

(TwoPi, in a discussion about this, mentioned that books of logarithm tables tend to be dirtier in the beginning than at the end, in a visual application of the law.) According to J. Carlton Collins in the Journal of Accounting the data set should be somewhat large, at least 500 entries ideally.  Still, it’s a pretty impressive rule, and one that doesn’t quite make intuitive sense to me.

So about catching criminals?  Forensic accountants use this rule to catch people who falsify invoices, because falsified data doesn’t usually follow this expected pattern.  Go math!

Wanted for Insurance jobs: A few good mathematicians

Probably more than a few, really.

There have been a few Math Mistakes in the News this summer with regard to insurance companies, though unfortunately I can’t seem to dig up what all of the actual mistakes are.

Back on May 5 the California Department of Insurance released a press statement that Anthem Blue Cross had been making some math mistakes and they were going to be under extra scrutiny.  According to the statement:

The errors identified included:

• Error #1: Double counting of aging in the calculation of underlying medical trend for the projection of total lifetime loss ratio.
• Error #2: Anthem overstated the initial medical trend used to project claims for September 2009 for known risk factors.

Both of these errors are errors of math and not differences in actuarial opinion.

I didn’t see anything about this costing a particular amount of money, though a June 25 article from the Los Angeles Times indicates that they canceled a rate increase of up to 39% for many of their California customers customers as a result.

Then, less than two months later, Aetna Inc. also had some math woes.  According to the same LA Times article,

Connecticut-based Aetna Inc. had sought an average 19% increase in rates for its 65,000 individual customers, but pulled back after multiple math errors in its paperwork were found by its own staff and by an independent consultant working for the state.

I was tempted to write “Bummer” but there’s really nothing bummerish about not having a 19% rate increase.  There’s no direct statement of what the mistakes are, just that “There were multiple errors … in the way [Aetna] annualized premiums and in the compounding of the rate increase,” according to California Insurance Department spokesman Darrel Ng.

Another article that same day on a Consumer Watchdog site, quoted Watchdog president Jamie Court as saying, “It’s amazing how insurers are making mathematical errors when they’re not used to regulators checking their math.”

Incidentally, a similar error was more recently discovered across the pond.  According to citywire,

Yorkshire and Clydesdale Bank today said it is in the process of mailing around 18,000 variable rate mortgage customers to apologise for miscalculating their monthly repayments and to suggest ways customers can repay what they owe.

The bank said the calculating error, which was exacerbated by last year’s unprecedented falls in interest rates, led to the bank collecting less than the contractual minimum monthly payment required for customers to pay their mortgage within their agreed term.

Moral?  Check your math.

Seriously, though this topic is serious enough, several of our math grads in recent years have gone into different aspects of insurance and finding errors can be a pretty important part of their jobs, whether it’s part of the official description or not.

Math Careers: Computer Animation (Games and Movies)

Video gaming.  Creating cool movies.  If you want to do computer animation, you need math.

One of our recent graduates ended up working in just this field.  Actually, he wasn’t even a math major:  he was a theater major who picked up a math minor in his last two years.   He’s now doing graphics programming, and it requires a ton of math:  he mentioned that the most important course turned out to be Linear Algebra.   In all fairness, since he’s officially a software engineer I have to assume he had some computer science as well.  But we don’t even offer a computer science degree, and he started work in this right out of college, so I don’ t know how much formal computer science training he had.

The article Math in the Movies from 2007 gives similar information.  There’s a 90-second video there, which I can’t seem to reproduce here, but in part of it the announcer Cindy Demus(?) says:

Trigonometry helps rotate and move characters, while algebra creates the special effects that make images shine and sparkle.  Calculus helps light up a scene and new math techniques turn images like this [flat and blocky] into this [smooth and more realistic].

Then Tony DeRose, a computer scientist from Pixar Animation Studios, added

I remember as a mathematics student thinking, “Well, where am I ever going to use simultaneous equations?” And I find myself using them every day, all the time now.”

So with some sort of computer background and good math skills, that job could be yours as well.

Math Careers: Astronaut

Additional training required.

Witness Lieutenant Colonel Robert Shane Kimbrough (who goes by Shane). He’s heading up into the sky Friday on the Space Shuttle Endeavor with Colonel Eric Boe and flight engineer Sandra Magnus. Lt. Col Kimbrough didn’t get his undergraduate degree in math (it was in Aerospace Engineering), but he got his MS in Operations Research from Georgia Tech and taught math at West Point. He’s given talks to school kids about how important math and science were in his being able to become an astronaut. And most recently he recognized his high school Calculus teacher (Sandy Sturgeon) in an official NASA letter because she was “instrumental in helping him get where he is today” and made calculus both real and fun.

For more information, see his official NASA biography or this recent article on ajc.com.

This is the premier post in a new series inspired by the common question, “What can I do with a degree in Math?”