Saturday, October 16, 2010

Jesus' Brother and Benford's Law

Just cruising the interwebs and found the following article at New Scientist:
After five years' deliberation, a trial in an Israeli court has still to reach a verdict over the provenance of an ossuary bearing the Aramaic inscription "James, son of Joseph, brother of Jesus".
The proof, or lack of it, is in the patina - the centuries-old crust that accrues on all antique objects. The Israeli justice ministry is prosecuting the ossuary's owner, antiques dealer Oded Golan, for fraud. Golan is charged with scratching the "brother of Jesus" inscription himself and slathering it with a homemade patina.
Five years of deliberation, when there is scientific evidence refuting the claim? Seems excessive. For those who think this might have an effect on Catholicism I'd like to point out that in that time the names Jesus, Joseph, and James were all very common, so even if it had not been fraudulent, there's nothing to suggest that the Jesus mentioned was Our Lord.

More interesting (to me) is an (unrelated) article about Benford's Law:
A subject of fascination to mathematicians, Benford's law states that for many sets of numbers, the first or "leading" digit of each number is not random. Instead, there is a 30.1 per cent chance that a number's leading digit is a 1. Progressively higher leading digits get increasingly unlikely, and a number has just a 4.6 per cent chance of beginning with a 9 (see diagram).
The law is named after physicist Frank Benford, who in 1938 showed that the trend appears in many number sets, from the surface area of rivers to baseball statistics to figures picked randomly from a newspaper. It later emerged that such distributions are "scale-invariant": if you convert the units of the numbers in the set, from metres to yards, say, the set will still conform to Benford's law.
Not all sets of numbers obey this law, but it crops up surprisingly often.
If I may geek out for a moment. I have never heard of this phenomenon, but I wonder if it's as simple as the "rule of numerous small"? I can't find a link to this law, but I remember reading it in an Asimov science essay years ago, in the collection "Far as Human Eye Could See". The rule is simply this. There are more pebbles than boulder, more grains of sand than pebbles, more alleys than superhighways, etc.

The rule makes sense intuitively, and is also backed up mathematically. If you divide a given mass into randomly sized pieces without regard to the size, you must end up with more pieces of smaller mass. Applying this over any set of numbers, a leading 1 will appear more often.

For instance, let's say I measure the length of all the roads in the US. Very few will be very long (there's just not enough room or need for many 4000 mile long roads). Many more will be moderately long (in the 100s of miles). Even more will be just a few miles long. The vast majority will be local roads a mile or less. If you graph them it will look something like this:

This graph depicts length along the bottom and the height of the line represents the number of roads of that length. As you can see there are more small roads than big ones on this graph (note, it's not real data, just an illustration of the concept). No matter what actual lengths you measure (feet, miles, yards, meters), the "1" is to the left of the "9", and so it appears more often.


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