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BY OUR STUDENT CONTRIBUTORS
Fresh Pisuttisarun Ever since the COVID-19 pandemic began, people have been skeptical about the numbers of cases being reported around the world. Many have claimed, although with little supporting evidence, that health officials are tampering with the numbers, in order to avert public fearmongering, economic decline, and international shame. As an ordinary person without access to global healthcare databases or CIA-level intelligence, it seems impossible to investigate this. However, an extraordinary law of mathematics gives us some fascinating insights into the situation! Benford’s Law states that real-world data follows a special pattern in which the leading digits of individual pieces of data appear in descending frequency from 1 to 9. Let’s take the population of countries as an example. The number of countries whose population starts with a 1 — China (1.4 billion), Mexico (130 million), Greece (10 million), Tonga (110,000), etc. — is higher than the number of countries with a population that starts with a 2. The pattern continues: 2 appears more frequently than 3, 3 appears more frequently than 4, and so on. This law can be extrapolated to all sorts of real-world data. Areas of countries, heights of skyscrapers, genome data, and macroeconomic spending are all known to obey Benford’s Law. In fact, the law is so powerful that it is even used to detect fraud in election results and credit card transactions! So how can this be applied to the COVID-19 numbers? Well, the number of national COVID-19 cases, being real-world data, is expected to obey Benford’s Law. If it doesn’t, that could suggest widespread fraudulent health reports. The blue bars in the graph shows how frequently each digit appears as the leading digit of COVID-19 cases in a country. The red bars show the frequency that would be expected if this data set perfectly obeyed Benford’s Law. Visibly, this seems like a good fit; the differences seem small enough to be accounted for by natural randomness. To mathematically confirm this impression, a chi-squared test can be used to test for goodness of fit. This statistical test takes into account the differences between the observed and expected values (the difference in height between the blue and red bars) to determine whether Benford’s Law is a good enough fit for our observed data set.
Turns out... yes! Benford’s Law is a great fit for the data set at a 1% significance level, which seems to suggest that the number of COVID-19 cases retains real-world veracity. However, this does not mean that the numbers have not been tampered with at all. There could be dishonest numbers in the set, but these do not become noticeable. It only means that the health officials around the globe are generally being honest. We can rest assured that most countries are not reporting randomly made-up numbers! But why does Benford’s Law work? Given that the number of COVID-19 cases in a given country is totally random, shouldn’t that flatten the curve (pun intended) into a uniform distribution? This part of the law is harder to grasp, so hold tight. Imagine you are a country during the COVID-19 pandemic. The number of COVID-19 cases in your country is not randomly generated by a computer — it’s accumulated. You first start off with 0 cases, then as the disease spreads, the number goes up, one-by-one, from 0 to 1 to 2 to 3, etc. When you count, you always count 1 before 2, 2 before 3, 3 before 4, etc. You count the 10s before the 20s, the 20s before the 30s, etc. You count the 100s before the 200s, the 200s before the 300s, etc. Therefore, if I were to ask what number you’re on at any given moment, you are more likely to be on a number that starts with 1 than 2, 2 than 3, 3 than 4, etc. The number of COVID-19 cases is no different: the cases accumulate. Every country with 2 cases must have had 1 case at some point before; a country with 5,000 cases must have had 3,000 cases at some point before; a country with 900,000 cases must have had 600,000 cases at some point before, and so on. Therefore, if we stop the world and ask all the countries for their number of cases, as long as the health officials are honestly counting and not randomly generating numbers, the results should obey Benford’s Law. If you're interested in other applications of Benford's Law, check out this article which outlines how it can be used to help you do well on multiple choice tests!
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