Archive for December, 2009

GO. More Coin Fraud

In this segment, we give some explanation of how Benford’s Law actually arises in so many settings: why are so many kinds of data logarithmically distributed? And we give a surprising fact about runs of coin tosses, and a new puzzle. 

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GN. Benford’s Law

Benford’s Law is really quite amazing, at least at first glance: for a wide variety of kinds of data, about 30% of the numbers will begin with a 1, 17% with a 2, on down to just 5% beginning with a 9. Can you spot the fake list of populations of European countries?

  List #1 List #2
Russia 142,008,838 148,368,653
Germany 82,217,800 83,265,593
Turkey 71,517,100 72,032,581
France 60,765,983 61,821,960
United Kingdom 60,587,000 60,118,298
Italy 59,715,625 59,727,785
Ukraine 46,396,470 48,207,555
Spain 45,061,270 45,425,798
Poland 38,625,478 41,209,072
Romania 22,303,552 25,621,748
Netherlands 16,499,085 17,259,211
Greece 10,645,343 11,653,317
Belarus 10,335,382 8,926,908
Belgium 10,274,595 8,316,762
Czech Republic 10,256,760 8,118,486
Portugal 10,084,245 7,738,977
Hungary 10,075,034 7,039,372
Sweden 9,076,744 6,949,578
Austria 8,169,929 6,908,329
Azerbaijan 7,798,497 6,023,385
Serbia 7,780,000 6,000,794
Bulgaria 7,621,337 5,821,480
Switzerland 7,301,994 5,504,737
Slovakia 5,422,366 5,246,778
Denmark 5,368,854 5,242,466
Finland 5,302,545 5,109,544
Georgia 4,960,951 4,932,349
Norway 4,743,193 4,630,651
Croatia 4,490,751 4,523,622
Moldova 4,434,547 4,424,558
Ireland 4,234,925 3,370,947
Bosnia and Herzegovina 3,964,388 3,014,202
Lithuania 3,601,138 2,942,418
Albania 3,544,841 2,051,329
Latvia 2,366,515 1,891,019
Macedonia 2,054,800 1,774,451
Slovenia 2,048,847 1,065,952
Kosovo 1,453,000 984,193
Estonia 1,415,681 841,113
Cyprus 767,314 605,767
Montenegro 626,000 588,802
Luxembourg 448,569 469,288
Malta 397,499 464,183
Iceland 312,384 402,554
Jersey (UK) 89,775 94,679
Isle of Man (UK) 73,873 43,345
Andorra 68,403 41,086
Guernsey (UK) 64,587 34,184
Faroe Islands (Denmark) 46,011 32,668
Liechtenstein 32,842 29,905
Monaco 31,987 22,384
San Marino 27,730 9,743
Gibraltar (UK) 27,714 7,209
Svalbard (Norway) 2,868 3,105
Vatican City 900 656

 Looking at these lists we have a clue as to when and how Benford’s Law works. [spoiler]

In one of the lists, the populations are distributed more or less evenly in a linear scale; that is, there are about as many populations from 1 million to 2 million, as there are from 2 million to 3 million, 3 million to 4 million etc. (Well, actually the distribution isn’t quite linear,  because the fake data was made to look similar to the real data, and so has a few of its characteristics.)

The real list, like many other kinds of data, is distributed in a more exponential manner; that is, the populations grow exponentially (very slowly though) with about as many populations from 100,000 to 1,000,000; then 1,000,000 to 10,000,000; and 10,000,000 to 100,000,000. This is all pretty approximate, so you can’t take this precisely at face value, but you’ll see in the list of real data that, very roughly speaking, in any order of magnitude there are about as many populations as in any other– at least for a while. 

Data like this has a kind of “scale invariance”, especially if this kind of pattern holds over many orders of magnitude. What this means is that if we scale the data up or down, throwing out the outliers, it will look about the same as before. 

The key to Benford’s Law is this scale invariance. Data that has this property will automatically satisfy his rule. Why is this? If we plot such data on a linear scale it won’t be distributed uniformly but will be all stretched out, becoming sparser and sparser. But if we plot it on a logarithmic scale, (which you can think of as approximated by the number of digits in the data), then such data is smoothed out and evenly distributed. 

But presto! Look at how the leading digits are distributed on such a logarithmic scale!

log

That’s mostly 1’s, a bit fewer 2’s, etc. on down to a much smaller proportion of 9’s.

[/spoiler]

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Morris: Follow Up: Triel/Truel/Whatever

 

This is the solution to Morris: Trial/Trual/Whatever.  Please look there before reading the solution.

It turns out the right word is truel, first coined in 1954 by Martin Shubik.

Read the rest of this entry »

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Yoak: Miles, Kilometers and Fibonacci Numbers

I’m overdue to post a puzzle, but I’m momentarily tapped out. Here’s a curiosity in the meantime: You can provide a very good estimate of a conversion from miles to kilometers by choosing sequential Fibonacci numbers.  The conversion rate is 1.609344 kilometers to a mile. So this gives us:

1 2 1.609
2 3 3.219
3 5 4.828
5 8 8.047
8 13 12.875
13 21 20.921
21 34 33.796
34 55 54.718
55 89 88.514
89 144 143.232
144 233 231.746
233 377 374.977
377 610 606.723
610 987 981.700
987 1597 1588.423

This leaves you in pretty good shape if you need to get from Cincinnati, OH to Destin, FL at 610 Miles, but what if you need to convert some distance that doesn’t happen to be a Fibonacci number?  Just build it up from parts!

100 miles is 89+8+3.  So in kilometers, that’s 144 + 13 + 5 or 162 kilometers.  (160.9344 by conversion…)

OK.  Here’s a puzzle, sort of.  I found this interesting set of numbers recently:

{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 53, 371, 5141, 99481 }

The series doesn’t continue.  That’s all of them.  What’s special about those numbers?

Comments (6)

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