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How to Fake Data if you must Department of Statistics Rachel Fewster

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Who wants to fake data? Electoral finance returns… Toxic emissions reports… Business tax returns…

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Land areas of world countries: real or fake?

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123456789123456789 IIIII III I I II I

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Land areas of world countries: real or fake? 123456789123456789 I I III I IIII I II III 123456789123456789 IIIII III I II I

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Land areas of world countries: real or fake? 123456789123456789 I III I IIII I II III 123456789123456789 IIIII III I II I This one seems more even… This one has as many 1s as 5-9s put together! This one is right!

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Real land areas of world countries 123456789123456789 IIIII III I II I 11 of them begin with digits 1 – 4… Only 5 begin with digits 5 – 9…

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Fridays Newspaper: 123456789123456789 IIII IIII III IIII II IIII II III 10 out of 34 numbers began with a 1… None out of 34 began with a 9!

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The Curious Case of the Grimy Log-books In 1881, American astronomer Simon Newcomb noticed something funny about books of logarithm tables…

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The Curious Case of the Grimy Log-books The books always seemed grubby on the first pages… … but clean on the last pages The first pages are for numbers beginning with digits 1 and 2… The last pages are for numbers beginning with digits 8 and 9…

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The Curious Case of the Grimy Log-books People seemed to look up numbers beginning with 1 and 2 more often than they looked up numbers beginning with 8 and 9. Why? Because numbers beginning with 1 and 2 are MORE COMMON than numbers beginning with 8 and 9!!

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Newcombs Law American Journal of Mathematics, 1881 30% of numbers begin with a 1 !! < 5% of numbers begin with a 9 !!

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The First Digits… Over 30% of numbers begin with a 1 Only 5% of numbers begin with a 9

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The First Digits… Numbers beginning with a 1 Numbers beginning with a 9 There is the same opportunity for numbers to begin with 9 as with 1 … but for some reason they dont!

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0.301 = log 10 (2/1) 0.176 = log 10 (3/2) 0.125 = log 10 (4/3) Chance of a number starting with digit d

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Reactions to Newcombs law Nothing! …for 57 years!

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Enter Frank Benford: 1938 Physicist with the General Electric Company Assembled over 20,000 numbers and counted their first digits! A study as wide as time and energy permitted.

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Populations Numbers from newspapers Drainage rates of rivers Numbers from Readers Digest articles Street addresses of American Men of Science

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About 30% begin with a 1About 5% begin with a 9

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Benford gave the law its name… …but no explanation. Anomalou s numbers !!

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…The logarithmic law applies to outlaw numbers that are without known relationship, rather than to those that follow an orderly course; and so the logarithmic relation is essentially a Law of Anomalous Numbers.

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Explanations for Benfords Law Numbers from a wide range of data sources have about 30% of 1s, down to only 5% of 9s. Benford called these outlaw or anomalous numbers. They include street addresses of American Men of Science, populations, areas, numbers from magazines and newspapers. Benfords orderly numbers dont follow the law – like atomic weights and physical constants What is the explanation?

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Popular Explanations Scale Invariance Base Invariance Complicated Measure Theory Divine choice Mystery of Nature These two say that IF there is a universal law, it must be Benfords. They dont explain why there should be a law to start with!

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In a nutshell … If you grab numbers from all over the place (a random mix of distributions), their digit frequencies ultimately converge to Benfords Law Complicated Measure Theory

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Thats why THIS works well

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It doesnt explain why street addresses of American Men of Science works well! It doesnt really explain WHAT will work well, nor why

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The Key Idea… If a hat is covered evenly in red and white stripes… Photo - Eric Pouhier http ://commons.wikimedia.org/wiki/Napoleon

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The Key Idea… … it will be half red and half white. If a hat is covered evenly in red and white stripes…

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The red stripes and the white stripes even out over the shape of the hat If the red stripes cover half the base, theyll cover about half the hat

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What if the red stripes cover 30% of the base? 0 0.3 1 1.3 2 2.3 3 3.3 4 4.3 5 5.3 6 Then theyll cover about 30% of the hat.

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What if the red stripes cover precisely fraction 0.301 of the base? 0.301 = log 10 (2/1) 0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 Then theyll cover fraction ~0.301 of the hat.

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Think of X as a random number… We want the probability that X has first digit = 1 Let the hat be a probability density curve for X Then AREAS on the hat give PROBABILITIES for X

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Think of X as a random number… We want the probability that X has first digit = 1 Let the hat be a probability density curve for X Then AREAS on the hat give PROBABILITIES for X Pr(1 < X < 5) = 0.95 Area = 0.95 from 1 to 5 Total area = 1

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In the same way …. 0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 If the red stripes somehow represent the X values with first digit = 1, and the red stripes have area ~ 0.301, then Pr(X has first digit 1) ~ 0.301.

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So X values with first digit=1 somehow lie on a set of evenly spaced stripes? Write X in Scientific Notation:

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So X values with first digit=1 somehow lie on a set of evenly spaced stripes? Write X in Scientific Notation: r is betwee n 1 and 10 n is an integer

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For example… r is betwee n 1 and 10 n is an integer

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For example… For the first digit of X, only r matters!

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For example… For the first digit of X, only r matters! 1 < r < 2 r > 2

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Take logs to base 10… Or in other words…

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r is betwee n 1 and 10 n is an integer

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r is betwee n 1 and 10 n is an integer

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r is betwee n 1 and 10 n is an integer

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X has first digit 1 precisely when log(X) is between n and n + 0.301 for any integer n n = 0 : n = 1 : n = 2 : X from 1 to 2 X from 10 to 20 X from 100 to 200

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n is an integer X has first digit 1 precisely when log(X) is between n and n + 0.301 for any integer n n = 0 : n = 1 : n = 2 : STRIPES!!

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n = 0 : n = 1 : n = 2 : 0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 X values with first digit = 1 satisfy: and so on! The hat is the probability density curve for log(X)

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n = 0 : n = 1 : n = 2 : 0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 X values with first digit = 1 satisfy: The hat is the probability density curve for log(X) X from 1 to 2 X from 10 to 20 X from 100 to 200

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0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 So X values with first digit=1 DO lie on evenly spaced stripes, on the log scale! The PROBABILITY of getting first digit 1 is the AREA of the red stripes, ~ approx the fraction on the base, = 0.301.

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Weve done it! Weve shown that we really should expect the first digit to be 1 about 30% of the time!

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0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 The log scale distorts: small numbers (e.g. 100) are stretched out; larger numbers (e.g. 900) are bunched up. The first digit corresponds to regularly spaced stripes on the log scale. Intuitively… So the smallest numbers (first digit = 1) are stretched out, and get the highest probability!

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0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 We need a lot of stripes to balance out big ones and little ones! We get one stripe every integer… So we need a lot of integers! When is this going to work? The distribution of X needs to be WIDE on the log scale!

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0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 X ranges from 0 to 6 on the log scale… So it ranges from 1 to 10 6 on usual scale! When is this going to work? 1.. 2.. Miss a few... 999,999.. 1,000,000

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0 0.301 1 1.301 2 2.301 3 3.301 4 4.301 5 5.301 6 These are Benfords Outlaw Numbers! All we need is a distribution that is: WIDE (4 – 6 orders of magnitude or more) Reasonably SMOOTH … Then the red stripes will even out to cover about 30% of the total area.

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In Real Life… World Populations: From 50 for the Pitcairn Islands … To 1.3 x 10 9 for China… Wide (9 integers => 9 stripes) First digits very good fit to Benford!

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In Real Life… World Populations: From 50 for the Pitcairn Islands … To 1.3 x 10 9 for China…

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Electorate populations? From 583,000 to 773,000 in California: Of course not! All the first digits are 5, 6, or 7… The hat has less than one stripe! Benford doesnt work here.

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But naturally occurring populations are a different story! Cities in California: - from 94 in the city of Vernon… - to 3.9 million in Los Angeles… Yes! Its Benford! Wide enough (5 integers => 5 stripes)

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Powerball Jackpots? - from $10 million to $365 million… Not bad! Orders of magnitude only 1.5 … … but sometimes you just hit lucky! Data with kind permission from www.lottostrategies.com

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Your tax return….? If you plan to fake data, you should first check whether it ought to be Benford! BUT the IRD has a few other tricks up its sleeve too….

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To find out more: A Simple Explanation of Benfords Law by R. M. Fewster The American Statistician, to appear. PDF from www.stat.auckland.ac.nz/~fewster/benford.html Judy Patersons CMCT course, Term 1 2009: Centre for Mathematical Content in Teaching Centre for Mathematical Content in Teaching Thanks for listening!

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