A remarkable phenomenon in probability theory is that of universality – that many seemingly unrelated probability distributions, which ostensibly involve large numbers of unknown parameters, can end up converging to a universal law that may only depend on a small handful of parameters. One of the most famous examples of the universality phenomenon is the central limit theorem; another rich source of examples comes from random matrix theory, which is one of the areas of my own research.
Analogous universality phenomena also show up in empirical distributions – the distributions of a statistic from a large population of “real-world” objects. Examples include Benford’s law, Zipf’s law, and the Pareto distribution (of which the Pareto principle or 80-20 law is a special case). These laws govern the asymptotic distribution of many statistics which
- (i) take values as positive numbers;
- (ii) range over many different orders of magnitude;
- (iiii) arise from a complicated combination of largely independent factors (with different samples of arising from different independent factors); and
- (iv) have not been artificially rounded, truncated, or otherwise constrained in size.
Examples here include the population of countries or cities, the frequency of occurrence of words in a language, the mass of astronomical objects, or the net worth of individuals or corporations. The laws are then as follows:
- Benford’s law: For , the proportion of whose first digit is is approximately . Thus, for instance, should have a first digit of about of the time, but a first digit of only about of the time.
- Zipf’s law: The largest value of should obey an approximate power law, i.e. it should be approximately for the first few and some parameters . In many cases, is close to .
- Pareto distribution: The proportion of with at least digits (before the decimal point), where is above the median number of digits, should obey an approximate exponential law, i.e. be approximately of the form for some . Again, in many cases is close to .
Benford’s law and Pareto distribution are stated here for base , which is what we are most familiar with, but the laws hold for any base (after replacing all the occurrences of in the above laws with the new base, of course). The laws tend to break down if the hypotheses (i)-(iv) are dropped. For instance, if the statistic concentrates around its mean (as opposed to being spread over many orders of magnitude), then the normal distribution tends to be a much better model (as indicated by such results as the central limit theorem). If instead the various samples of the statistics are highly correlated with each other, then other laws can arise (for instance, the eigenvalues of a random matrix, as well as many empirically observed matrices, are correlated to each other, with the behaviour of the largest eigenvalues being governed by laws such as the Tracy-Widom law rather than Zipf’s law, and the bulk distribution being governed by laws such as the semicircular law rather than the normal or Pareto distributions).
To illustrate these laws, let us take as a data set the populations of 235 countries and regions of the world in 2007 (using the CIA world factbook); I have put the raw data here. This is a relatively small sample (cf. my previous post), but is already enough to discern these laws in action. For instance, here is how the data set tracks with Benford’s law (rounded to three significant figures):
Countries | Number | Benford prediction | |
1 | Angola, Anguilla, Aruba, Bangladesh, Belgium, Botswana, Brazil, Burkina Faso, Cambodia, Cameroon, Chad, Chile, China, Christmas Island, Cook Islands, Cuba, Czech Republic, Ecuador, Estonia, Gabon, (The) Gambia, Greece, Guam, Guatemala, Guinea-Bissau, India, Japan, Kazakhstan, Kiribati, Malawi, Mali, Mauritius, Mexico, (Federated States of) Micronesia, Nauru, Netherlands, Niger, Nigeria, Niue, Pakistan, Portugal, Russia, Rwanda, Saint Lucia, Saint Vincent and the Grenadines, Senegal, Serbia, Swaziland, Syria, Timor-Leste (East-Timor), Tokelau, Tonga, Trinidad and Tobago, Tunisia, Tuvalu, (U.S.) Virgin Islands, Wallis and Futuna, Zambia, Zimbabwe | 59 () | 71 () |
2 | Armenia, Australia, Barbados, British Virgin Islands, Cote d’Ivoire, French Polynesia, Ghana, Gibraltar, Indonesia, Iraq, Jamaica, (North) Korea, Kosovo, Kuwait, Latvia, Lesotho, Macedonia, Madagascar, Malaysia, Mayotte, Mongolia, Mozambique, Namibia, Nepal, Netherlands Antilles, New Caledonia Norfolk Island, Palau, Peru, Romania, Saint Martin, Samoa, San Marino, Sao Tome and Principe, Saudi Arabia, Slovenia, Sri Lanka, Svalbard, Taiwan, Turks and Caicos Islands, Uzbekistan, Vanuatu, Venezuela, Yemen | 44 () | 41 () |
3 | Afghanistan, Albania, Algeria, (The) Bahamas, Belize, Brunei, Canada, (Rep. of the) Congo, Falkland Islands (Islas Malvinas), Iceland, Kenya, Lebanon, Liberia, Liechtenstein, Lithuania, Maldives, Mauritania, Monaco, Morocco, Oman, (Occupied) Palestinian Territory, Panama, Poland, Puerto Rico, Saint Kitts and Nevis, Uganda, United States of America, Uruguay, Western Sahara | 29 () | 29 () |
4 | Argentina, Bosnia and Herzegovina, Burma (Myanmar), Cape Verde, Cayman Islands, Central African Republic, Colombia, Costa Rica, Croatia, Faroe Islands, Georgia, Ireland, (South) Korea, Luxembourg, Malta, Moldova, New Zealand, Norway, Pitcairn Islands, Singapore, South Africa, Spain, Sudan, Suriname, Tanzania, Ukraine, United Arab Emirates | 27 () | 22 () |
5 | (Macao SAR) China, Cocos Islands, Denmark, Djibouti, Eritrea, Finland, Greenland, Italy, Kyrgyzstan, Montserrat, Nicaragua, Papua New Guinea, Slovakia, Solomon Islands, Togo, Turkmenistan | 16 () | 19 () |
6 | American Samoa, Bermuda, Bhutan, (Dem. Rep. of the) Congo, Equatorial Guinea, France, Guernsey, Iran, Jordan, Laos, Libya, Marshall Islands, Montenegro, Paraguay, Sierra Leone, Thailand, United Kingdom | 17 () | 16 () |
7 | Bahrain, Bulgaria, (Hong Kong SAR) China, Comoros, Cyprus, Dominica, El Salvador, Guyana, Honduras, Israel, (Isle of) Man, Saint Barthelemy, Saint Helena, Saint Pierre and Miquelon, Switzerland, Tajikistan, Turkey | 17 () | 14 () |
8 | Andorra, Antigua and Barbuda, Austria, Azerbaijan, Benin, Burundi, Egypt, Ethiopia, Germany, Haiti, Holy See (Vatican City), Northern Mariana Islands, Qatar, Seychelles, Vietnam | 15 () | 12 () |
9 | Belarus, Bolivia, Dominican Republic, Fiji, Grenada, Guinea, Hungary, Jersey, Philippines, Somalia, Sweden | 11 () | 11 () |
Here is how the same data tracks Zipf’s law for the first twenty values of , with the parameters and (selected by log-linear regression), again rounding to three significant figures:
Country | Population | Zipf prediction | Deviation from prediction | |
1 | China | 1,330,000,000 | 1,280,000,000 | |
2 | India | 1,150,000,000 | 626,000,000 | |
3 | USA | 304,000,000 | 412,000,000 | |
4 | Indonesia | 238,000,000 | 307,000,000 | |
5 | Brazil | 196,000,000 | 244,000,000 | |
6 | Pakistan | 173,000,000 | 202,000,000 | |
7 | Bangladesh | 154,000,000 | 172,000,000 | |
8 | Nigeria | 146,000,000 | 150,000,000 | |
9 | Russia | 141,000,000 | 133,000,000 | |
10 | Japan | 128,000,000 | 120,000,000 | |
11 | Mexico | 110,000,000 | 108,000,000 | |
12 | Philippines | 96,100,000 | 98,900,000 | |
13 | Vietnam | 86,100,000 | 91,100,000 | |
14 | Ethiopia | 82,600,000 | 84,400,000 | |
15 | Germany | 82,400,000 | 78,600,000 | |
16 | Egypt | 81,700,000 | 73,500,000 | |
17 | Turkey | 71,900,000 | 69,100,000 | |
18 | Congo | 66,500,000 | 65,100,000 | |
19 | Iran | 65,900,000 | 61,600,000 | |
20 | Thailand | 65,500,000 | 58,400,000 |
As one sees, Zipf’s law is not particularly precise at the extreme edge of the statistics (when is very small), but becomes reasonably accurate (given the small sample size, and given that we are fitting twenty data points using only two parameters) for moderate sizes of .
This data set has too few scales in base to illustrate the Pareto distribution effectively – over half of the country populations are either seven or eight digits in that base. But if we instead work in base , then country populations range in a decent number of scales (the majority of countries have population between and ), and we begin to see the law emerge, where is now the number of digits in binary, the best-fit parameters are and :
Countries with binary digit populations | Number | Pareto prediction | |
31 | China, India | 2 | 1 |
30 | ” | 2 | 2 |
29 | “, United States of America | 3 | 5 |
28 | “, Indonesia, Brazil, Pakistan, Bangladesh, Nigeria, Russia | 9 | 8 |
27 | “, Japan, Mexico, Philippines, Vietnam, Ethiopia, Germany, Egypt, Turkey | 17 | 15 |
26 | “, (Dem. Rep. of the) Congo, Iran, Thailand, France, United Kingdom, Italy, South Africa, (South) Korea, Burma (Myanmar), Ukraine, Colombia, Spain, Argentina, Sudan, Tanzania, Poland, Kenya, Morocco, Algeria | 36 | 27 |
25 | “, Canada, Afghanistan, Uganda, Nepal, Peru, Iraq, Saudi Arabia, Uzbekistan, Venezuela, Malaysia, (North) Korea, Ghana, Yemen, Taiwan, Romania, Mozambique, Sri Lanka, Australia, Cote d’Ivoire, Madagascar, Syria, Cameroon | 58 | 49 |
24 | “, Netherlands, Chile, Kazakhstan, Burkina Faso, Cambodia, Malawi, Ecuador, Niger, Guatemala, Senegal, Angola, Mali, Zambia, Cuba, Zimbabwe, Greece, Portugal, Belgium, Tunisia, Czech Republic, Rwanda, Serbia, Chad, Hungary, Guinea, Belarus, Somalia, Dominican Republic, Bolivia, Sweden, Haiti, Burundi, Benin | 91 | 88 |
23 | “, Austria, Azerbaijan, Honduras, Switzerland, Bulgaria, Tajikistan, Israel, El Salvador, (Hong Kong SAR) China, Paraguay, Laos, Sierra Leone, Jordan, Libya, Papua New Guinea, Togo, Nicaragua, Eritrea, Denmark, Slovakia, Kyrgyzstan, Finland, Turkmenistan, Norway, Georgia, United Arab Emirates, Singapore, Bosnia and Herzegovina, Croatia, Central African Republic, Moldova, Costa Rica | 123 | 159 |
Thus, with each new scale, the number of countries introduced increases by a factor of a little less than , on the average. This approximate doubling of countries with each new scale begins to falter at about the population (i.e. at around million), for the simple reason that one has begun to run out of countries. (Note that the median-population country in this set, Singapore, has a population with binary digits.)
These laws are not merely interesting statistical curiosities; for instance, Benford’s law is often used to help detect fraudulent statistics (such as those arising from accounting fraud), as many such statistics are invented by choosing digits at random, and will therefore deviate significantly from Benford’s law. (This is nicely discussed in Robert Matthews’ New Scientist article “The power of one“; this article can also be found on the web at a number of other places.) In a somewhat analogous spirit, Zipf’s law and the Pareto distribution can be used to mathematically test various models of real-world systems (e.g. formation of astronomical objects, accumulation of wealth, population growth of countries, etc.), without necessarily having to fit all the parameters of that model with the actual data.
Being empirically observed phenomena rather than abstract mathematical facts, Benford’s law, Zipf’s law, and the Pareto distribution cannot be “proved” the same way a mathematical theorem can be proved. However, one can still support these laws mathematically in a number of ways, for instance showing how these laws are compatible with each other, and with other plausible hypotheses on the source of the data. In this post I would like to describe a number of ways (both technical and non-technical) in which one can do this; these arguments do not fully explain these laws (in particular, the empirical fact that the exponent in Zipf’s law or the Pareto distribution is often close to is still quite a mysterious phenomenon), and do not always have the same universal range of applicability as these laws seem to have, but I hope that they do demonstrate that these laws are not completely arbitrary, and ought to have a satisfactory basis of mathematical support.
— 1. Scale invariance —
One consistency check that is enjoyed by all of these laws is that of scale invariance – they are invariant under rescalings of the data (for instance, by changing the units).
For example, suppose for sake of argument that the country populations of the world in 2007 obey Benford’s law, thus for instance about of the countries have population with first digit , have population with first digit , and so forth. Now, imagine that several decades in the future, say in 2067, all of the countries in the world double their population, from to a new population . (This makes the somewhat implausible assumption that growth rates are uniform across all countries; I will talk about what happens when one omits this hypothesis later.) To further simplify the experiment, suppose that no countries are created or dissolved during this time period. What happens to Benford’s law when passing from to ?
The key observation here, of course, is that the first digit of is linked to the first digit of . If, for instance, the first digit of is , then the first digit of is either or ; conversely, if the first digit of is or , then the first digit of is . As a consequence, the proportion of ‘s with first digit is equal to the proportion of ‘s with first digit , plus the proportion of ‘s with first digit . This is consistent with Benford’s law holding for both and , since
(or numerically, after rounding). Indeed one can check the other digit ranges also and that conclude that Benford’s law for is compatible with Benford’s law for ; to pick a contrasting example, a uniformly distributed model in which each digit from to is the first digit of occurs with probability totally fails to be preserved under doubling.
One can be even more precise. Observe (through telescoping series) that Benford’s law implies that
for all integers , where the left-hand side denotes the proportion of data for which lies between and for some integer . Suppose now that we generalise Benford’s law to the continuous Benford’s law, which asserts that (1) is true for all real numbers . Then it is not hard to show that a statistic obeys the continuous Benford’s law if and only if its dilate does, and similarly with replaced by any other constant growth factor. (This is easiest seen by observing that (1) is equivalent to asserting that the fractional part of is uniformly distributed.) In fact, the continuous Benford law is the only distribution for the quantities on the left-hand side of (1) with this scale-invariance property; this fact is a special case of the general fact that Haar measures are unique (see e.g. these lecture notes).
It is also easy to see that Zipf’s law and the Pareto distribution also enjoy this sort of scale-invariance property, as long as one generalises the Pareto distribution
from integer to real , just as with Benford’s law. Once one does that, one can phrase the Pareto distribution law independently of any base as
for any much larger than the median value of , at which point the scale-invariance is easily seen.
One may object that the above thought-experiment was too idealised, because it assumed uniform growth rates for all the statistics at once. What happens if there are non-uniform growth rates? To keep the computations simple, let us consider the following toy model, where we take the same 2007 population statistics as before, and assume that half of the countries (the “high-growth” countries) will experience a population doubling by 2067, while the other half (the “zero-growth” countries) will keep their population constant, thus the 2067 population statistic is equal to half the time and half the time. (We will assume that our sample sizes are large enough that the law of large numbers kicks in, and we will therefore ignore issues such as what happens to this “half the time” if the number of samples is odd.) Furthermore, we make the plausible but crucial assumption that the event that a country is a high-growth or a zero-growth country is independent of the first digit of the 2007 population; thus, for instance, a country whose population begins with is assumed to be just as likely to be high-growth as one whose population begins with .
Now let’s have a look again at the proportion of countries whose 2067 population begins with either or . There are exactly two ways in which a country can fall into this category: either it is a zero-growth country whose 2007 population also began with either or , or it was a high-growth country whose population in 2007 began with . Since all countries have a probability of being high-growth regardless of the first digit of their population, we conclude the identity
which is once again compatible with Benford’s law for since
More generally, it is not hard to show that if obeys the continuous Benford’s law (1), and one multiplies by some positive multiplier which is independent of the first digit of (and, a fortiori, is independent of the fractional part of ), one obtains another quantity which also obeys the continuous Benford’s law. (Indeed, we have already seen this to be the case when is a deterministic constant, and the case when is random then follows simply by conditioning to be fixed.)
In particular, we see an absorptive property of Benford’s law: if obeys Benford’s law, and is any positive statistic independent of , then the product also obeys Benford’s law – even if did not obey this law. Thus, if a statistic is the product of many independent factors, then it only requires a single factor to obey Benford’s law in order for the whole product to obey the law also. For instance, the population of a country is the product of its area and its population density. Assuming that the population density of a country is independent of the size of that country (which is not a completely reasonable assumption, but let us take it for the sake of argument), then we see that Benford’s law for the population would follow if just one of the area or population density obeyed this law. It is also clear that Benford’s law is the only distribution with this absorptive property (if there was another law with this property, what would happen if one multiplied a statistic with that law with an independent statistic with Benford’s law?). Thus we begin to get a glimpse as to why Benford’s law is universal for quantities which are the product of many separate factors, in a manner that no other law could be.
As an example: for any given number , the uniform distribution from to does not obey Benford’s law; for instance, if one picks a random number from to then each digit from to appears as the first digit with an equal probability of each. However, if is not fixed, but instead obeys Benford’s law, then a random number selected from to also obeys Benford’s law (ignoring for now the distinction between continuous and discrete distributions), as it can be viewed as the product of with an independent random number selected from between and .
Actually, one can say something even stronger than the absorption property. Suppose that the continuous Benford’s law (1) for a statistic did not hold exactly, but instead held with some accuracy , thus
for all . Then it is not hard to see that any dilated statistic, such as , or more generally for any fixed deterministic , also obeys (5) with exactly the same accuracy . But now suppose one uses a variable multiplier; for instance, suppose one uses the model discussed earlier in which is equal to half the time and half the time. Then the relationship between the distribution of the first digit of and the first digit of is given by formulae such as (4). Now, in the right-hand side of (4), each of the two terms and differs from the Benford’s law predictions of and respectively by at most . Since the left-hand side of (4) is the average of these two terms, it also differs from the Benford law prediction by at most . But the averaging opens up an opportunity for cancelling; for instance, an overestimate of for could cancel an underestimate of for to produce a spot-on prediction for . Thus we see that variable multipliers (or variable growth rates) not only preserve Benford’s law, but in fact stabilise it by averaging out the errors. In fact, if one started with a distribution which did not initially obey Benford’s law, and then started applying some variable (and independent) growth rates to the various samples in the distribution, then under reasonable assumptions one can show that the resulting distribution will converge to Benford’s law over time. This helps explain the universality of Benford’s law for statistics such as populations, for which the independent variable growth law is not so unreasonable (at least, until the population hits some maximum capacity threshold).
Note that the independence property is crucial; if for instance population growth always slowed down for some inexplicable reason to a crawl whenever the first digit of the population was , then there would be a noticeable deviation from Benford’s law, particularly in digits and , due to this growth bottleneck. But this is not a particularly plausible scenario (being somewhat analogous to Maxwell’s demon in thermodynamics).
The above analysis can also be carried over to some extent to the Pareto distribution and Zipf’s law; if a statistic obeys these laws approximately, then after multiplying by an independent variable , the product will obey the same laws with equal or higher accuracy, so long as is small compared to the number of scales that typically ranges over. (One needs a restriction such as this because the Pareto distribution and Zipf’s law must break down below the median.) These laws are also stable under other multiplicative processes, for instance if some fraction of the samples in spontaneously split into two smaller pieces, or conversely if two samples in spontaneously merge into one; as before, the key is that the occurrence of these events should be independent of the actual size of the objects being split. If one considers a generalisation of the Pareto or Zipf law in which the exponent is not fixed, but varies with or , then the effect of these sorts of multiplicative changes is to blur and average together the various values of , thus “flattening” the curve over time and making the distribution approach Zipf’s law and/or the Pareto distribution. This helps explain why eventually becomes constant; however, I do not have a good explanation as to why is often close to .
— 2. Compatibility between laws —
Another mathematical line of support for Benford’s law, Zipf’s law, and the Pareto distribution are that the laws are highly compatible with each other. For instance, Zipf’s law and the Pareto distribution are formally equivalent: if there are samples of , then applying (3) with equal to the largest value of gives
which implies Zipf’s law with . Conversely one can deduce the Pareto distribution from Zipf’s law. These deductions are only formal in nature, because the Pareto distribution can only hold exactly for continuous distributions, whereas Zipf’s law only makes sense for discrete distributions, but one can generate more rigorous variants of these deductions without much difficulty.
In some literature, Zipf’s law is applied primarily near the extreme edge of the distribution (e.g. the top of the sample space), whereas the Pareto distribution in regions closer to the bulk (e.g. between the top and and top ). But this is mostly a difference of degree rather than of kind, though in some cases (such as with the example of the 2007 country populations data set) the exponent for the Pareto distribtion in the bulk can differ slightly from the exponent for Zipf’s law at the extreme edge.
The relationship between Zipf’s law or the Pareto distribution and Benford’s law is more subtle. For instance Benford’s law predicts that the proportion of with initial digit should equal the proportion of with initial digit or . But if one formally uses the Pareto distribution (3) to compare those between and , and those between and , it seems that the former is larger by a factor of , which upon summing by appears inconsistent with Benford’s law (unless is extremely large). A similar inconsistency is revealed if one uses Zipf’s law instead.
However, the fallacy here is that the Pareto distribution (or Zipf’s law) does not apply on the entire range of , but only on the upper tail region when is significantly higher than the median; it is a law for the outliers of only. In contrast, Benford’s law concerns the behaviour of typical values of ; the behaviour of the top is of negligible significance to Benford’s law, though it is of prime importance for Zipf’s law and the Pareto distribution. Thus the two laws describe different components of the distribution and thus complement each other. Roughly speaking, Benford’s law asserts that the bulk distribution of is locally uniform at unit scales, while the Pareto distribution (or Zipf’s law) asserts that the tail distribution of decays exponentially. Note that Benford’s law only describes the fine-scale behaviour of the bulk distribution; the coarse-scale distribution can be a variety of distributions (e.g. log-gaussian).
69 comments
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11 February, 2019 at 2:07 am
lawyer
i think still a lot of research is yet to be done on Benford’s Law
13 April, 2019 at 2:30 pm
Robert Fernholz
You can find a mathematical explanation for the universality of Zipf’s law here: https://arxiv.org/pdf/1707.04285.pdf
15 April, 2019 at 2:08 am
Oded Kafri
Benford law is an outcome of simple probability. There is nothing more to explore. Read for example “A Novel Approach to Probability”, Scientific Research Publishing-Advances in Pure Mathematics.2016, 6, 201-211 https://arxiv.org/pdf/1606.03696
8 January, 2021 at 9:32 am
Alex Fakos
This is such a great post! Thank you for taking the time to make math accessible to the general public.
18 March, 2021 at 11:37 am
Cost-effectiveness distributions, power laws and scale invariance | Stijn Bruers, the rational ethicist
[…] of a country. So we can consider the distribution of country population sizes. This distribution is close to a power law (with a few outliers, such as India), having estimated power parameter p = 0,9. Now we can consider […]
16 February, 2022 at 8:40 am
Giovanni Ciriani
Wonderful article, thanks! I’d never heard of Benford’s law before, but it makes sense, as Oded Kafri pointed out in a previous comment. I think that with a sufficiently large data set it could be expanded to two or more digits, for instance the percentage of numbers in the data set starting with 12 is log(13/12) = 3.5%
9 September, 2022 at 11:06 am
Eric Toombs
IDG what the big deal is. It’s just 3 different regimes of the log normal distribution. Over many orders of magnitude, the central limit theorem goes from random independent additions and subtractions to random independent multiplications and divisions. Very obvious.
3 October, 2022 at 11:01 pm
What are the odds? | What's new
[…] If these corrupt officials are indeed choosing their predetermined winning numbers randomly, then the probability would in fact be just the same probability as , and in this case the seemingly unusual event would in fact have no effect on the odds of the alternative hypothesis, because it was just as unlikely for the alternative hypothesis to generate this multiples-of-nine pattern as for the null hypothesis to. In fact, one would imagine that these corrupt officials would avoid “suspicious” numbers, such as the multiples of , and only choose numbers that look random, in which case would in fact be less than and so the event would actually lower the odds of the alternative hypothesis in this case. (In fact, one can sometimes use this tendency of fraudsters to not generate truly random data as a statistical tool to detect such fraud; violations of Benford’s law for instance can be used in this fashion, though only in situations where the null hypothesis is expected to obey Benford’s law, as discussed in this previous blog post.) […]