The law of Benford , or law of the abnormal numbers because it is surprising when it is discovered it, watch that in the life of the every day, figure 1 is more frequent than the 2, itself more frequent than the 3, etc

In a general way, the law gives the theoretical value F of the frequency of appearance of the first decimal D of a result of measurement expressed in a bases B given by means of a unit.

$f\; =\; \backslash \; log\_\; \{B\}\; \backslash \; left\; (1\; +\; \backslash \; frac\; 1\; D\; \backslash \; right)$

Decimal system

In particular, for the Decimal system (bases 10), one thus has:

$f\; =\; \backslash \; log\_\; \{10\}\; \backslash \; left\; (1\; +\; \backslash \; frac\; 1\; D\; \backslash \; right)$

What leads to the table of following results:

There exists also a discrete on-presentation of the first figures with regard to the second significant figure. This on-presentation tends to be cancelled beyond that.

The examples illustrating this law are numerous: take the continuation of the first 100 squares, the frequency of the numbers starting with 1 is definitely higher than the frequencies of the squares starting with 2,3,4 etc etc By drawing up the list of 100 numbers, products of two or three randomly drawn numbers (in a great interval), again the frequency of the numbers starting with 1 is definitely higher than the other frequencies.

The numerical continuations which converge exactly as the law of Benford stipulates it are, in fact, rather rare: among those Ci, one can quote the Suite of Fibonacci, the continuation of N! … In the real life, the decrease of the probabilities according to the first figure is largely noted but convergence towards the values of the law of Benford is only approximate.

On the contrary, this law is not checked if the set of data comprises constraints as for the scale of the probable values: the size of the individuals does not follow, obviously, the law of Benford since it quasi totality of measurements starts with the figure “1”.

Explanation

The precise form of the law of Benford can be explained if it is admitted that the logarithms of the numbers are uniformly distributed. That means that number is likely as many to be between 100 and 1000 (logarithm between 2 and 3) that it has chances to be between 10.000 and 100.000 (logarithm between 4 and 5). For many whole of numbers, and particularly those which grow exponentially, like the sales turnovers of companies and the stock exchange courts, this assumption is reasonable.

Outline demonstration of the law of Benford

Let us choose a real number stictement positive pertaining to an interval.

One seeks the probability of his first quantifies not no one, indépendemment very other characteristic.

That corresponds in the search of a measurement on the unit, presumedly measurable, with:

1. $P\; (\backslash \; mathrm\; \{1^\; \{er\}\; ~chiffre\; =\; 1\})\; =\; \backslash \; frac\; \{m\; (I\; \backslash \; course\; \backslash \; \{\backslash \; ldots\; \backslash \; cup\; \#$ P\; (\backslash \; mathrm\; \{1^\; \{er\}\; ~chiffre\; =\; 2\})\; =\; \backslash \; frac\; \{m\; (I\; \backslash \; course\; \backslash \; \{\backslash \; ldots\; \backslash \; cup\; \#$ P\; (\backslash \; mathrm\; \{1^\; \{er\}\; ~chiffre\; =\; 3\})\; =\; \backslash \; frac\; \{m\; (I\; \backslash \; course\; \backslash \; \{\backslash \; ldots\; \backslash \; cup\; \#\; etc$$$

It is supposed that is built like a union of products of the interval by realities {{maths|a_i > 0}}; i.e.: {{maths|I {{=}} ∪ a_i × [1; 10 [}} for {{maths|a_i > 0}}. Therefore, one works in the multiplicative group of strictly positive realities (because thus the topology of this group is built).

The whole of strictly positive realities provided with the multiplication being a topological Group separable and locally compact, there exists one and only one measurement (with a multiplying coefficient near) which is invariant by the law of group: the Measurement of Haar of the group.

This measurement is $m\; =\; \backslash \; tfrac\; \{\backslash \; mathrm\; dx\}\; \{X\}$.

Let us take I = one a: : $m\; (I)\; =\; \backslash \; int\_1^\; \{10\}\; \backslash \; frac\; \{\backslash \; mathrm\; dx\}\; \{X\}\; =\; \backslash \; ln\; (10)\; -\; \backslash \; ln\; (1)\; =\; \backslash \; ln\; (10)$

And one a:

$P\; (\backslash \; mathrm\; \{1^\; \{er\}\; ~chiffre\}\; =\; K)\; =\; \backslash \; frac\; \{m\; (=\; \backslash \; ldots\; =\; \backslash \; frac\; \{\backslash \; ln\; (k+1)\; -\; \backslash \; ln\; (K)\}\{\backslash \; ln10\}\; =\; \backslash \; frac\; \{\backslash \; ln\; (1+\; \backslash \; frac\; \{1\}\; \{K\})\}\{\backslash ln10\}$

As measurement is invariant by the product, while taking with $a\_i=10^n$, one arrives at the same result (one can also check it by calculation).

Outline made starting from a work exposed on the site of the University Paris 5. It misses in these two exposed a convincing argument on the need for using the measurement of Haar.

History

This distribution would have been discovered first once in 1881 by Simon Newcomb, a American Astronome, after he had realized wear (and thus of the use) preferential of the first pages of the tables of logarithms (then compiled in works). Frank Benford, in the neighborhoods of 1938, noticed in its turn this unequal wear, believed being the first to formulate this law which bears its name unduly today and arrived at same the results after having indexed tens of thousands of data (lengths of rivers, price stock exchange, etc).

Application

The law of Benford is used with the the United States, like in other countries, of which the France, to detect tax evasions, following the ideas presented in 1972 by Hal Varian.

References

Random links:

Lauri Ylönen | Polytetrafluoroethylene | R. (album) | Mow Flora (partido) | If long way