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• This law is called Benford's Law and appears in many tables of statistics.Other examples are a table of populations of countries, or lengths of rivers.About one-third of countries have a population size which begins with the digit "1" and veryfew have a population size beginning with "9".Here is a table of the initial digits as produced by theFibonacci Calculator:Initial digit frequencies of fib(i) for i from 1 to 100: Digit: 1 2 3 4 5 6 7 8 9Frequency: 30 18 13 9 8 6 5 7 4 100 values Percent: 30 18 13 9 8 6 5 7 4What are the frequencies for the first 1000 Fibonacci numbers or the first 10,000? Are they settlingdown to fixed values (percentages)? Use the Fibonacci Calculatorto collect the statistics. According to Benford's Law, large numbers of items lead to the followingstatistics for starting figures for the Fibonacci numbers as well as some natural phenomenaDigit: 1 2 3 4 5 6 7 8 9Percentage:30181310 8 7 6 5 5 You do the maths... Look at a table of sizes of countries. How many countries areas begin with "1"? "2"? etc.. Use a table of population sizes (perhaps of cities in your country or of countries in the world). It doesn't matter if the figures are not the latest ones. Does Benford's Law apply to their initial digits? Look at a table of sizes of lakes and find the frequencies of their initial digits. Using the Fibonacci Calculator make a table of the first digits of powers of 2. Do they follow Benford's Law? What about powers of other numbers? Some newspapers give lists of the prices of various stocks and shares, called "quotations". Select a hundred or so of the quotations (or try the first hundred on the page) and make a table of the distribution of the leading digits of the prices. Does it follow Benford's Law? What other sets of statistics can you find which do show Benford's Law? What about the number of the house where the people in your class live? What about the initial digit of their home telephone number? Generate some random numbers of your own and look at the leading digits. You can buy 10-sided dice (bi-pyramids) or else you can cut out a decagon (a 10-sided polygon with all sides the same length) from card and label the sides from 0 to 9. Put a small stick through the centre (a used matchstick or a cocktail stick or a small pencil or a ball-point pen) so that it can spin easily and falls on one of the sides at random. (See the footnote about dice and spinners on the "The Golden Geometry of the Solid Section or Phi in 3 dimensions" page, for picture and more details.) Are all digits equally likely or does this device show Benford's Law? Use the random number generator on your calculator and make a table of leading-digit frequencies. Such functions will often generate a "random" number between 0 and 1, although some calculators generate a random value from 0 to the maximum size of number on the calculator. Or you can use the random number generator in the Fibonacci Calculator to both generate the values and count the initial digit frequencies, if you like. Do the frequencies of leading digits of random values conform to Benford's Law? Measure the height of everyone in your class to the nearest centimetre. Plot a graph of their heights. Are all heights equally likely? Do their initial digits conform to Benford's Law? Suppose you did this for everyone in your school. Would you expect the same distribution of heights? What about repeatedly tossing five coins all at once and counting the number of heads each time? What if you did this for 10 coins, or 20? What is the name of this distribution (the shape of the frequency graph)? When does Benford's Law apply?Random numbers are equally likely to begin with each of the digits 0 to 9. This applies to randomlychosen real numbers or randomly chosen integers. Randomly chosen real numbers If you stick a pin at random on a ruler which is 10cm long and it will fall in each of the 10 sections 0cm-1cm, 1cm-2cm, etc. with the same probability. Also, if you look at the initial digits of the points chosen (so that the initial digit of 0.02cm is 2 even though the point is in the 0-1cm section) then each of the 9 values from 1 to 9 is as likely as any other value. Randomly chosen integers This also applies if we choose random integers. Take a pack of playing cards and remove the jokers, tens, jacks and queens, leaving in all aces up to 9 and the kings. Each card will represent a different digit, with a king representing zero. Shuffle the pack and put the first 4 cards in a row to represent a 4 digit integer. Suppose we have King, Five, King, Nine. This will represent "0509" or the integer 509 whose first digit is 5. The integer is as likely to begin with 0 (a king) as 1 (an ace) or 2 or any other digit up to 9. But if our "integer" began with a king (0), then we look at the next "digit". These have the same distribution as if we had chosen to put down just 3 cards in a row instead of 4. The first digits all have the same probability again. If our first two cards had been 0, then we look at the third digit, and the same applies again. So if we ignore the integer 0, any randomly chosen (4 digit) integer begins with 1 to 9 with equal probability. (This is not quite true of a row of 5 or more cards if we use an ordinary pack of cards - why?)So the question is, why does this all-digits-equally-likely propertynot apply to the first digits of each of the following: the Fibonacci numbers, the Lucas numbers, populations of countries or towns sizes of lakes prices of shares on the Stock Exchange Whether we measure the size of a country ora lake in square kilometres orsquare miles (or square anything), does not matter - Benford's Law will still apply. So when is a number random? We often meant that we cannot predict the next value. If we toss a coin, we can never predict if it will be Heads or Tails if we give it a reasonably high flip in the air. Similarly, with throwing a dice - "1" is as likely as "6".Physical methods such as tossing coins or throwing dice or picking numbered balls from a rotating drum as in Lottery games are always unpredictable.The answer is that the Fibonacci and Lucas Numbers are governed by a Power Law.We have seen that Fib(i) is round(Phii/5) and Lucas(i) isround(Phii). Dividing by sqrt(5) will merely adjust the scale - which does not matter.Similarly, rounding will not affect the overall distribution of the digits in a large sample.Basically, Fibonacci and Lucas numbers are powers of Phi. Many natural statistics are also governedby a power law - the values are related to Bi for some base value B. Such data would seem toinclude the sizes of lakes and populations of towns as well as non-natural data such as the collection ofprices of stocks and shares at any one time. In terms of natural phenomena (like lake sizesor heights of mountains) the larger values are rare and smaller sizes are more common. So there arevery few large lakes, quite a few medium sized lakes and very many little lakes. We can see this with theFibonacci numbers too: there are 11 Fibonacci numbers in the range 1-100, but only one in the next 3 ranges of 100 (101-200, 201-300,301-400) and they get increasingly rarer for large ranges of size 100. The same is true for any othersize of range (1000 or 1000000 or whatever). You do the maths... Type a power expression in the Eval(i)= box, such as pow(1.2,i) and give a range of i values from i=1 to i=100. Clicking the Initial digits button will print the leading digit distribution. Change 1.2 to any other value. Does Benford's Law apply here? Using Eval(i)=randint(1,100000) with an i range from 1 to 1000 (so that 1000 separate random integers are generated in the range 1 to 100000) shows that the leading digits are all equally likely. Benford's Law for Fibonacci and Lucas Numbers, L. C. Washington, The Fibonacci Quarterly vol. 19, 1981, pages 175-177.

The original reference: The Law of Anomalous Numbers F Benford, (1938) Proceedings of the American Philosophical Society vol 78, pages 551-572. The Math Forum's archives of the History of Mathematics discussion group have an email from Ralph A. Raimi (July 2000) about his research into Benford's Law. It seems that Simon Newcomb had written about it much earlier, in 1881, in American Journal of Mathematics volume 4, pages 39-40. The name Benford is, however, the one that is commonly used today for this law. MathTrek by Ivars Peterson (author of The Mathematical Tourist and Islands of Truth) the editor of Science News Online has produced this very good, short and readable introduction to Benford's Law. M Schroeder Fractals, Chaos and Power Laws, Freeman, 1991, ISBN 0-7167-2357-3. This is an interesting book but some of the mathematics is at first year university level (mathematics or physics degrees), unfortunately, and the rest will need sixth form or college level mathematics beyond age 16. However, it is still good to browse through. It has only a passing reference to Benford's Law: The Peculiar Distribution of the Leading Digit on page 116. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ..More..

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