Also, anyone forced to give a number between 1 and 10 often gives 7, with 3 being the 2nd most likely statistic.

In other news, I had 1,234 scoops of ice cream for dinner last night.

I think the problem with your logic here is that you data set is too constrained, which will, as Alan pointed out at the beginning of the article, cause it to fail Benford’s Law. It is not to constrained because it is too small a data set, but because the range is too small. That is, the range between smallest number and largest number is too small. According to Wikipedia, the shortest fully grown man is 2’11″ while the tallest man in recorded history was 8’11″. Thus there is only an approximately 300% increase from shortest to tallest, regardless of what unit system you use. Compare this to the range in river lengths or yearly incomes, to go back to the two examples so heavily cited above. To run the full range of starting-digit-1 to starting-digit-9
requires a 900% increase in value, again regardless of what unit system is used. Bedford’s Law requires that the numbers involved traverse this range at least once , otherwise at least 1 number will always have a 0% occurrence in the opening position and the number set will thus immediately fail the test.

So, I think the bell curve distributions will still follow Bedford’s Law, but only when the distribution covers a sufficiently wide enough range of numbers, not just a sufficiently large enough set. That, it appears, is a key distinction to “too constrained” which it does not seem was made above.

Actually, this Law does NOT work for bell curve distributions. In a perfect bell curve, the median and the mode are the same value. This means high numbers and low numbers are equally rare, with numbers somewhere in between being most common. For instance, look at average height (in feet) for 20-year-old males. It’s a bell curve distribution (or close to one) with 5 as the most common first digit. Sure, you could convert this to meters, and the most common digit would be 1. But that’s just a matter of shifting the curve. If you shifted it again into inches, the most common first digit is 6.

As long as the curve is bell-shaped, the most common first digit will be whatever the first digit of the median is.

But river length is not a bell curve distribution. Neither is amounts on a tax form. In both of those distributions, the median and the mode are not the same. If the distribution were unimodal, which it may not be, the mode would not be the same as the median. There would be a large number of rivers a little bit shorter than the median, and a few rivers extremely longer than the median. The distribution is much more likely to look like this: http://www.gatsby.ucl.ac.uk/~turner/Benford's%20law/Benford%20Tea%20Talk_files/Benfords%20law3.png
This is because short rivers are much more common than long ones. And, similarly, small dollar amounts are much more likely than large ones.

Take a look at that distribution and adjust the scale however you like. The most common data points will begin with 1 in every scale.

0.00000164 lightyears = 9,640,739.7 miles… that’s one LONG river.”[/quote]

which is 1/10 or (.1) the approximate distance from the earth to the sun.

Hmm. . notice how the coverage between 1 and 2 is a lot bigger than between 8 and 9? In fact, it’s about 30% of the area from 1 to 10.
You see, this is just a matter of things GROWING, natural or artificial. When a river is forming, it might grow 10% longer in a century. So if it’s a 100 miles long, it could easily grow by 10miles. But a ONE mile long river is not likely to grow by ten miles. So we get a bunch of short rivers and a few long ones, which looks wierd on a plot, but nice and uniform on a LOG plot! The log plot has wider 1′s, so the (now uniform) data is more likely to be from 1 to 2 than from 8 to 9.

Am I right here??”

No!

Feel free to comment me if I’m mistaken.

1) The law does not just apply rivers. The length of rivers was used as an example, I believe the author started with tax returns…

However, on the subject of rivers…

2) Although I agree that it takes longer to move from 1 to 2 exponentially (or on a log plot) than 8 to 9, you must remember that our system of measurement is essentially arbitrary. If a river is 1.1 miles long, it will take a long time to reach 2 miles exponentially. But 1.1 miles is also 1.76km. it will take a much shorter time to reach “2″ by exponential growth. It doesnt mean that the river is any longer or growing any faster.

In answer to the other person’s comment above, I suspect that if measurements were converted from miles to km or vice versa, the law would still hold.

Hmm. . notice how the coverage between 1 and 2 is a lot bigger than between 8 and 9? In fact, it’s about 30% of the area from 1 to 10.
You see, this is just a matter of things GROWING, natural or artificial. When a river is forming, it might grow 10% longer in a century. So if it’s a 100 miles long, it could easily grow by 10miles. But a ONE mile long river is not likely to grow by ten miles. So we get a bunch of short rivers and a few long ones, which looks wierd on a plot, but nice and uniform on a LOG plot! The log plot has wider 1′s, so the (now uniform) data is more likely to be from 1 to 2 than from 8 to 9.

Am I right here??

When you shift the numeric base, the percentages change, but 1 is still the most common. Of course, if you mix things up, say represent hex using binary (0000,0001,etc.), you get 50% 0′s and 50% 1′s.