**ERIC Number:**EJ1093383

**Record Type:**Journal

**Publication Date:**2013

**Pages:**18

**Abstractor:**ERIC

**ISBN:**N/A

**ISSN:**ISSN-0819-4564

**EISSN:**N/A

Benford's Law and Why the Integers Are Not What We Think They Are: A Critical Numeracy of Benford's Law

Stoessiger, Rex

Australian Senior Mathematics Journal, v27 n1 p29-46 2013

A critical numeracy examination of Benford's Law suggests that our understanding of the integers is faulty. We think of them as equally likely to turn up as the first digit of a random real world number. For many real world data sets this is not true. In many cases, ranging from eBay auction prices to six digit numbers in Google to the distribution of numbers in newspapers, the smaller digits are much more likely than the larger ones. Yet most of us are surprised when we first encounter this result. Benford himself described the real world numbers which fit his law as anomalous. Many others have echoed his surprise. How can our understanding of numbers be such that the way we actually use numbers in our world, that is, the authentic use of numbers, is regarded as strange? The distribution of numbers in this way has been explained in the past by Benford's Law. However it seems that Zipf's Law may be just as useful as an explanation of some of the observed distributions. Both laws are likely to apply when numbers describe growth situations with Benford's Law describing compound interest type growth while Zipf's Law represents a slower growth with constant growth (simple interest) compounded at repeated stages. From a critical numeracy perspective we need to understand how the first digit distribution of real world numbers is both ubiquitous but seen as anomalous. Perhaps this is best explained using the work of Kafri (2009) showing that the random distribution of balls and boxes results in a Benford's Law distribution. In this model the digit 1 is represented by a single ball, digit 2 by two balls and so on. In this model the first digits are actually quantities of the single unit digit. Kafri used a thermodynamic randomness to distribute the balls (rather than a lottery style) and Benford's Law is the predicted distribution. Hence our understanding of random as meaning equally likely is too simplistic for real world numbers. The digits in numbers are not distributed as if by lottery. We need to move to a thermodynamic understanding of randomness in which we recognise that digits are a quantity of the unit digit and are distributed amongst all the different positions which make up a number. It is all the different microstates of digit as quantity in all the various positions which are "equally likely"'. A better understanding of the use of numbers in our world seems to be that: (1) Some numbers such as lottery results are random and their first digits are uniformly distributed; (2) Some numbers represent quantities (such as amounts of money) and it is harder to accumulate say, $700 than $100; (3) The initial digit of numbers from random samples taken from a random variety of distributions will fit Benford's Law; (4) The super sample of numbers held in Google is not particularly Benford like; and (5) Growth data resulting from regularly compounding growth follows Benford's Law. However this may be too strong for many real world situations where Zipf's Law may better represent less regular growth

Descriptors: Numbers, Numeracy, Mathematics, Mathematics Instruction, Mathematical Formulas, Probability, Predictor Variables, Fractions, Statistical Distributions

Australian Association of Mathematics Teachers (AAMT). GPO Box 1729, Adelaide 5001, South Australia. Tel: +61-8-8363-0288; Fax: +61-8-8362-9288; e-mail: office@aamt.edu.au; Web site: http://www.aamt.edu.au

**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Grant or Contract Numbers:**N/A