1The “first significant digit” of a number is its leftmost non-zero digit. For example, the first significant digit of the number 325 is 3 and the first significant digit of 0.8732 is 8. It might be expected that the first significant digits of any given series of numbers, or of a set of numbers measuring any given phenomenon, are randomly distributed. Nothing of the sort: in most series found in the real world, figure 1 appears more often than figure 2, which in turn appears more often than figure 3, and so on. The purpose of this note is to illustrate this rule, known as Benford’s law, using data for the populations of all world countries, and to show its underlying logic, which in this particular case, relies on the pattern of population growth.

I. Benford’s law

2In 1881, the mathematician and astronomer Simon Newcomb noticed that the first volumes of the tables of logarithms in the library of his institution were more worn than subsequent ones. This meant that there were more frequent consultations for numbers starting with 1 or 2 than for numbers starting with 8 or 9. In 1938, the engineer Benford made the same observation about the same tables of logarithms, independently of Newcomb’s work. In his article, Benford compiled numerous sets of data, from physical constants to baseball results and various number series he found in newspapers. The average number of appearances of the first significant digits followed a logarithmic law already proposed by Newcomb (Bedford, 1938):

3— where F_d is the frequency of appearance of the first significant digit d.

4Benford’s law predicts that first significant digits will be distributed as described in Table 1.

Table 1

Predicted frequency of appearance of the first significant digit according to Benford’s law

First significant digit Predicted frequency 1 0.301 2 0.176 3 0.125 4 0.097 5 0.079 6 0.067 7 0.058 8 0.051 9 0.046

Predicted frequency of appearance of the first significant digit according to Benford’s law

5The sum of probabilities can be shown to be equal to 1:

6In practice, it is difficult to apply this law universally; some numerical series do not conform to it. Hill (1995) demonstrated, however, that for these series, random samples taken from the complete series did follow Benford’s law. Next, we will see how the law works for the current populations of different countries worldwide.

II. The distribution of world populations

7INED regularly publishes a list of country-specific statistics worldwide. Here, we will use the 1997 list, which gives information on 198 countries or geopolitical entities. Figure 1 compares the observed distribution with the distribution predicted by Benford’s law. The difference between the distributions is small, confirming that the distribution of population among countries does in fact follow Benford’s law.

Figure 1

Observed frequency of the first significant digit of the 1997 population size for 198 countries and predicted frequency according to Benford’s law

Observed frequency of the first significant digit of the 1997 population size for 198 countries and predicted frequency according to Benford’s law

8If we perform the same type of calculations for surface areas and population densities, the results are similar (Figures 2 and 3).

Figure 2

Observed frequency of the first significant digit of the 1997 surface area for 198 countries and predicted frequency according to Benford’s law

Observed frequency of the first significant digit of the 1997 surface area for 198 countries and predicted frequency according to Benford’s law

Figure 3

Observed frequency of the first significant digit of the 1997 population density for 198 countries and predicted frequency according to Benford’s law

Observed frequency of the first significant digit of the 1997 population density for 198 countries and predicted frequency according to Benford’s law

III. An explanation for population sizes

9First, it must be made clear that there is no bias resulting from rounding the population size of small countries to 1 as a first significant digit, the smallest number in the table being 0.02 (million).

10The explanation lies elsewhere. To work it out, one needs to look at the distribution of population sizes for a population with constant growth rate, for example 2% annually. Table 2 shows that at a constant annual rate of growth, population sizes with first significant digit 1 are more common than others, and follow Benford’s law. If we extend these calculations over the long term, the observed frequency of first significant digits follows Benford’s law perfectly; this is also true with other constant growth rates.

Table 2

Population size over 172 time periods with a 2% growth rate^*

Periods 1- 36 37-56 57-71 72-82 83-91 92-99 100-106 107-111 112-117 118-152 153-172 100.00 203.99 303.12 407.95 507.24 606.20 710.26 815.86 900.78 1014.43 2028.74 102.00 208.07 309.18 416.11 517.39 618.32 724.46 832.18 918.80 1034.71 2069.31 104.04 212.23 315.36 424.44 527.73 630.69 738.95 848.83 937.17 1055.41 2110.70 106.12 216.47 321.67 432.93 538.29 643.30 753.73 865.80 955.92 1076.52 2152.91 108.24 220.80 328.10 441.58 549.05 656.17 768.81 883.12 975.03 1098.05 2195.97 110.41 225.22 334.67 450.42 560.03 669.29 784.18 994.53 1120.01 2239.89 112.62 229.72 341.36 459.42 571.24 682.68 799.87 1142.41 2284.69 114.87 234.32 348.19 468.61 582.66 696.33 1165.26 2330.38 117.17 239.01 355.15 477.98 594.31 1188.56 2376.99 119.51 243.79 362.25 487.54 1212.33 2424.53 121.90 248.66 369.50 497.29 1236.58 2473.02 124.34 253.63 376.89 1261.31 2522.48 126.82 258.71 384.43 1286.54 2572.93 129.36 263.88 392.11 1312.27 2624.39 131.95 269.16 399.96 1338.51 2676.88 134.59 274.54 1365.28 2730.42 137.28 280.03 1392.59 2785.02 140.02 285.63 1420.44 2840.72 142.82 291.35 1448.85 2897.54 145.68 297.17 1477.83 2955.49 148.59 1507.38 151.57 1537.53 154.60 1568.28 157.69 1599.65 160.84 1631.64 164.06 1664.27 167.34 1697.56 170.69 1731.51 174.10 1766.14 177.58 1801.46 181.14 1837.49 184.76 1874.24 188.45 1911.73 192.22 1949.96 196.07 1988.96 199.99 *The table should be read sequentially by column.

Population size over 172 time periods with a 2% growth rate^*

11Demographers use the shortcut of dividing 70 by the growth rate (in%) of a population to find the population’s doubling time. This result can be derived as follows:

• the pace of the growth is given by ;
• where P(u) = population size at time u;
• r = growth rate.

• then ;
• and where r’ is the constant growth rate in %.

12To move from size 200 to size 300, it only takes 21 units of time:

13It takes 15 units of time to grow from 300 to 400:

14and only 5 units from 800 to 900:

15The number of time periods necessary for a population to grow from size (d)00 to (d+1)00 is given by the formula:

16— with

17— where M is a constant

18— and F_d the frequency of appearance of the first significant digit d.

19Thus, this formula is determined by the distribution of first significant digits, which is precisely the distribution specified by Benford’s law. Over a long period, the first significant digit of the population size of any given country is therefore more often 1 than 2, 2 than 3, and so on, up to 9.

20To look for a cross-sectional version of this longitudinal regularity for countries is equivalent to drawing a random sample among a set of 198 series that conform to Bedford’s law, assuming that the following two hypotheses hold, which can be reasonably assumed.

21Hypothesis 1: the population sizes of different countries are independent;

22Hypothesis 2: the timing of the onset of the demographic transition in any given country is independent of the first significant digit of its population size.

Conclusion

23The explanation given here for the conformity of population sizes to Bedford’s law does not negate the existence of other data sets, for example the areas of countries, that follow Benford’s law but do not fit this longitudinal argument. Certain advances in probability theory have led to a better understanding of the principal aspects of Benford’s law, but a completely satisfactory explanation still remains to be developed (Hill, 1999).