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Wednesday, 02 December, 2009
Benford’s Law
(with comments)
Kind of interesting: Benford's Law.
Benford's law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 almost one third of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than one time in twenty.
This counter-intuitive result has been found to apply to a wide variety of data sets, including electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers, physical and mathematical constants, and processes described by power laws (which are very common in nature).
I put it to a test. I found an Excel file with birth and death statistics, by state. It had six columns of data: Births, Deaths, Natural Increase, Net International Migration, Net Internal Migration, and Net Migration. I put all of those numbers into a single column of 306 numbers, and created a frequency distribution for the first digit of each number:
Benford's Law was not violated.
- By Dave (TDC). Comment posted 02-Dec-2009 @09:21am:If you plotted that graph to a logarithmic scale, it would be roughly straight. Anyone who understands logs will find nothing unexpected here.
- By LogLady. Comment posted 02-Dec-2009 @09:44am:

Anyone who understands logs will find nothing unexpected here.
Actually, I would have predicted twin peaks. - By L.. Comment posted 02-Dec-2009 @10:02am:Why not use a cherry pie chart then?
- By Bisbonian. Comment posted 02-Dec-2009 @10:05am:Hi Log Lady. I'm livin' in your house.
- By . Comment posted 02-Dec-2009 @11:03am:I guess I don't understand how the choice of how the data is represented on a chart, diminishes the impact of the data itself and how it supports Benford's law.
- By . Comment posted 02-Dec-2009 @12:59pm:a wide variety of data sets, including electricity bills
I want to know why my utility company account numbers are always larger than the total population of the planet. - By Doug. Comment posted 02-Dec-2009 @01:19pm:Amazing. It even applies to historic Bush poll numbers. 1 is the most common leading digit.
- By banjo brad. Comment posted 02-Dec-2009 @02:04pm:I sure hope that my electric bill has 1 as the leading digit more often than a 9! ($134.76 vx $905.23)
- By wally the duck. Comment posted 02-Dec-2009 @02:05pm:They call it a law, but I think of it more as a recommendation.
- By . Comment posted 02-Dec-2009 @02:52pm:Is the lawmaker Gregory Benford of SF fame?
- By exp_err. Comment posted 02-Dec-2009 @03:04pm:This is one of the things used to detect fraud. With made up numbers (in account books, for instance), people try to be "random", and fail to take into account real patterns.
- By . Comment posted 02-Dec-2009 @03:18pm:The IRS uses this. A lot.
- By Volt. Comment posted 02-Dec-2009 @03:21pm:With a transposition to interval sets instead of conventional numeric representation, it can be used to rapidly locate prime numbers, which in turn can be used to crack encryption via methods faster and more elegant than brute force approaches.
It also hints at how our minds are trammeled by how we represent concepts, in this case representing numbers with numerals. - By . Comment posted 02-Dec-2009 @03:52pm:If it were a law it would be opposite
9 would be the most popular
that would be called Tax Law - By . Comment posted 02-Dec-2009 @04:05pm:By Banjo Brad:
I sure hope that my electric bill has 1 as the leading digit more often than a 9! ($134.76 vs $905.23)
I was delighted when my electric bills last summer had nines as leading digits instead of ones. The cooler weather created a bill of $95.00 a month instead of the $130+ a month I had to pay in 2008. - By Ross. Comment posted 02-Dec-2009 @05:01pm:>>I want to know why my utility company account numbers are always larger than the total population of the planet.
Grouping?, each part of the number will mean something, 100 might mean you in the east part of wherever for example, 101, in the west etc. etc - By wally the duck. Comment posted 02-Dec-2009 @05:11pm:It also hints at how our minds are trammeled by how we represent concepts, in this case representing numbers with numerals.
What would be the analogous situation in a set of binary numbers? Would you just look at frequency of 0 and 1 or would it need to be something different than that? - By Blue. Comment posted 02-Dec-2009 @05:42pm:account numbers are always larger than the total population of the planet
Because they can. And even worse are the IT idjuts who insist on issuing you a different account number for every bill, or even different numbers for different payment option on each bill as well.
There are 20 million people in Australia, and our transport authority insists on using a transaction number to track on line payments. They even take out a different provider number each year.
Why don't they just use the rego number of the vehicle and a single provider number for registration renewal?
So instead of QDOT, BF7268 (or numerical equivalent), I have
48298, 1011 4069 6361 13 for six month renewal, or
48298, 1211 4069 6361 86 for 12 month renewal.
Sheer stupidity. - By Volt. Comment posted 02-Dec-2009 @05:58pm:Wally,
Notice there is no zero on J-Walk's chart. Binary numbers, base 2, reduce to a triviality. Other bases work fine.
Regardless of the base, however, it's still a variation of conventional numeral representation. - By . Comment posted 02-Dec-2009 @06:15pm:You folks are a lot smarter than I am. I had to read the article twice before I fully understood...
- By wally the duck. Comment posted 02-Dec-2009 @08:33pm:I see that. So in all systems #1 is #1, but in binary it is the ONLY one.
Thanks. - By . Comment posted 03-Dec-2009 @07:20am:Lots of reasons why 1's rule:
- We all like cheap as opposed to expensive
- Who wants to finish 2nd or 3rd...
- It was a lot quicker on an old Rotary Dial
This post is probably more suited to a Pyschology class than the ... J-Walk Blog - By Mr. Pointer Outer. Comment posted 03-Dec-2009 @11:18am:Does it work for the second and subsequent digits of numbers?
- By Miss Cellania. Comment posted 03-Dec-2009 @12:19pm:How is this counterintuitive? That's the way I'd expect the graph to look.


