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!


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



  1. Rob Stevenson said,

    December 19, 2009 at 6:44 am

    INteresting that you suggested that data can be faked using a knowledge of Benford’s law – this is almost impossible for medium sized data sets due to the fact that anyfair sized  random sampling of the data should also exhibit the law.
    For example if I were a crooked accountant and wanted to reduce the tax bill for a local store I would need to amke sure that the law held for sales data sliced monthly.   Or sliced by department of the store.  Or by sales person.  Or by payment method.
    Being able to fabricate this level of distributed “semi-randomness” is *really* hard.

  2. strauss said,

    December 19, 2009 at 10:37 am

    I guess the point wasn’t that we COULD fake data using Benford’s Law, but rather we should be careful not to get tripped up by some pesky investigator who is using Benford’s Law.

    It’s not too hard to make up a data set from scratch that will resist any kind of slicing: just make sure your “expenses” are chosen randomly on a log scale. That is, if RAND is a random number from 0 to 1, instead of generating expenses of the form RAND*$10,000,000, generate expenses as 10^(RAND*7). No matter how the data is sliced, you’re home free! (Well, with just a few refinements depending on the particular fraud you’re perpetrating.)

    You’re right, though, hiding or manipulating a specific piece of information within a larger data set is trickier. Now I’m no master criminal, but it doesn’t seem quite as bad as you say: mostly we need to carefully smear out the fraud, blending it in against the background, most especially being careful that the manipulated data is chosen with this log distribution.

    (Mix a few bags of chips, DVDs, ipods, and TVs in with the rolexes, sportscars and beachfront property!)

    My consulting fee can be deposited in an unmarked bank account, number available upon request.

  3. Shawn said,

    November 29, 2011 at 12:48 am

    Has anyone figured out which list is fake yet?

  4. Edwin said,

    January 18, 2012 at 11:20 am

    Ok, is it list two? Countries tat start with digits 4, 5, 6 and 8 appear to surpass the expected digit distribution. 
    Although, the 1st list also offers disparities when compared to Benford. I noted that digits 4, 6 and 7 happen a lot more than expected Benford Law. Nonetheless, he 2nd list appears more manipulated than 1st list.

    Which is erroneous? 

  5. strauss said,

    January 24, 2012 at 5:57 pm

    Er,… unfortunately, I’ve forgotten…

  6. Stephen Morris said,

    February 1, 2012 at 6:37 pm

    You can check them on line here: 

    Hat tip to Ben Goldacre: 

    Do read the article he links to.

  7. Stephen Morris said,

    February 1, 2012 at 6:44 pm

    Ooh, just reading Ben Goldacre’s article again.  He links to this great site which tests lots of real world data.  Very nice. 

     There’s one data set which really stands out, you can probably guess before you try it. And, no, it isn’t the russian elections.

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