When Should We Trust the Numbers?

Exploring the Beauty of Benford’s Law

A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press).

BENFORD’S LAW: A STATISTICAL ANOMALY

Benford’s Law reveals a surprising pattern in numerical data: lower digits, especially ‘1’, appear more frequently as the leading digit in many real-world datasets than would be expected if all digits were equally likely.

This pattern, also known as the Newcomb–Benford Law, is a rule that goes against what most people would assume to be true about numbers.

Surprisingly, the probability of the leading digit being ‘1’ is approximately 30%, instead of about 11%, which would be the case if all digits from 1 to 9 were equally probable. The number ‘2’ would appear as the first digit about 17.6% of the time, ‘3’ around 12.5%, and so forth, with the number ‘9’ having a mere 4.6% chance of being the first digit.

A fascinating aspect of Benford’s Law is its applicability to numerous datasets that reflect natural and societal phenomena, ranging from the populations of cities and countries to the heights of mountains and even utility bills.

Suppose, for example, you select numbers from everyday data like electricity bills, stock prices, or river lengths. You’ll notice that numbers starting with ‘1’ occur more often than those starting with ‘9’. In fact, you’ll find that roughly 30% of these numbers will start with a 1.

One might wonder why this happens. The answer is rooted in the multiplicative processes that often generate the numbers we see in real life. For a number to grow from 1 to 2, it needs to increase by 100%. But to go from 2 to 3, the increase is only 50%. As a result, numbers spend more time with a leading digit of 1 before moving on to higher first digits, which leads to the distribution observed in Benford’s Law. To look at it another way, if you start with £1 and it grows by 10% each day, the leading digit will stay as ‘1’ for a longer duration than ‘2’, and ‘2’ will hold the top spot longer than ‘3’, and so on.

CONDITIONS FOR BENFORD’S LAW

There are specific conditions that must be satisfied for Benford’s Law to apply. One such condition is that the numbers should be of the same nature—mixing areas of lakes with employee numbers, for instance, won’t adhere to the law.

The numbers also shouldn’t have any artificial caps or boundaries. This excludes data such as house numbers or the price ranges of wines in a supermarket.

Moreover, the numbers shouldn’t be designated by a specific numbering system, like postcodes or telephone numbers, nor should there be any significant spikes around particular numbers.

THE ORIGINS OF BENFORD’S LAW

This unusual phenomenon can be traced back to a note in the American Journal of Mathematics in 1881 by astronomer Simon Newcomb. Newcomb noticed that the pages of logarithms used to perform calculations were more worn for certain leading digits than others. Numbers starting with 1, for example, were used much more frequently than those beginning with 9.

However, it was physicist Frank Benford who provided rigorous empirical support for this distribution. His 1938 paper, ‘The Law of Anomalous Numbers’, examined 20,229 sets of numbers, ranging from baseball statistics to areas of rivers and numbers in magazine articles. He confirmed that approximately 30% of these numbers started with 1. The chances of the leading digit being ‘2’ were found to be 17.6%, and for ‘9’, a mere 4.6%.

IMPLICATIONS FOR FRAUD DETECTION

Interestingly, Benford’s Law has found crucial applications in the detection of fraud, particularly in accounting. If declared returns significantly deviate from the Benford distribution, it serves as a red flag for those investigating fraudulent activities.

SCALE INVARIANCE: A UNIQUE PROPERTY

A truly universal law that governs the digits of numbers describing natural phenomena should operate independently of the units used. This property, known as scale invariance, holds true for Benford’s Law. Regardless of the units of measurement, whether they are inches, yards, centimetres, or metres, the overall distribution of numbers would still maintain the same pattern.

THE MATHEMATICS BEHIND BENFORD’S LAW

Benford discovered that the probability that a number starts with n is equal to log (n + 1) − log (n), to base 10, so that the probability that a number starts with 2, say, is equal to log10 (2 + 1) − log10 2 = 0.1761, or 17.61%.

PREDICTING PROPORTIONS OF DIGITS WITH BENFORD’S LAW

In addition to the first digit, Benford’s Law can also predict the proportions of digits in the second number, the third number, and so forth.

CONNECTING WITH NATURE

Benford’s Law is all around us, even within the Fibonacci numbers. To explain, in the Fibonacci sequence each number is the sum of the two preceding ones, resulting in a series like 1, 1, 2, 3, 5, 8, 13, and so forth. This sequence is famously associated with the Golden Ratio (approximately 1.618), which frequently appears in art, design, and even in natural patterns like the spiral pattern of sunflower seeds. The leading digits of these numbers, when they grow large enough, conform to Benford’s Law. This intersection between Benford’s Law and the Fibonacci sequence further accentuates the pervasive and mysterious nature of mathematical patterns in our universe.

CONCLUSION: THE BEAUTY OF BENFORD

Benford’s Law serves as a testament to the fact that natural and other types of data are governed by underlying patterns, often too subtle for the naked eye but unmistakable in their mathematical signatures. The inherent bias towards smaller leading digits reflects the multiplicatively growing nature of many phenomena in our universe, an echo of the processes that shape everything from economics to geology.

While the conditions for Benford’s Law provide boundaries for its application, they also highlight the importance of context in statistical analysis. This law does not claim universal application; rather, it thrives within the appropriate datasets—those untainted by human constructs or arbitrary limits.

In understanding Benford’s Law, we find more than just an anomaly. We uncover a bridge between the abstract world of numbers and the tangible reality we measure, count, and analyse. Whether it’s in unravelling potential fraud, examining natural occurrences, or exploring the mathematical patterns in art and nature, Benford’s Law remains a powerful ally to the inquisitive mind.