My All 2020 Mathematics A to Z: Statistics


I owe Mr Wu, author of the Singapore Maths Tuition blog, thanks for another topic for this A-to-Z. Statistics is a big field of mathematics, and so I won’t try to give you a course’s worth in 1500 words. But I have to start with a question. I seem to have ended at two thousand words.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Statistics.

Is statistics mathematics?

The answer seems obvious at first. Look at a statistics textbook. It’s full of algebra. And graphs of great sloped mounds. There’s tables full of four-digit numbers in back. The first couple chapters are about probability. They’re full of questions about rolling dice and dealing cards and guessing whether the sibling who just entered is the younger.

But then, why does Rutgers University have a Department of Mathematics and also a Department of Statistics? And considered so distinct as to have an interdisciplinary mathematics-and-statistics track? It’s not an idiosyncrasy of Rutgers. Many schools have the same division between mathematics and statistics. Some join them into a Department of Mathematics and Statistics. But the name hints at something just different about the field. Not too different, though. Physics and Chemistry and important threads of Economics and History are full of mathematics. But you never see a Department of Mathematics and History.

Thinking of the field’s history, though, and its use, tell us more. Some of the earliest work we now recognize as statistics was Arab mathematicians deciphering messages. This cryptanalysis is the observation that (in English) a three-letter word is very likely to be ‘the’, mildly likely to be ‘one’, and not likely to be ‘pyx’. A more modern forerunner is the Republic of Venice supposedly calculating that war with Milan would not be worth the winning. Or the gatherings of mortality tables, recording how many people of what age can be expected to die any year, and what from. (Mortality tables are another of Edmond Halley’s claims to fame, though it won’t displace his comet work.) Florence Nightingale’s charts explaining how more soldiers die of disease than in fighting the Crimean War. William Sealy Gosset sharing sample-testing methods developed at the Guinness brewery.

You see a difference in kind to a mathematical question like finding a square with the same area as this trapezoid. It’s not that mathematics is not practical; it’s always been. And it’s not that statistics lacks abstraction and pure mathematics content. But statistics wears practicality in a way that number theory won’t.

Practical about what? History and etymology tip us off. The early uses of things we now see as statistics are about things of interest to the State. Decoding messages. Counting the population. Following — in the study of annuities — the flow of money between peoples. With the industrial revolution, statistics sneaks into the factory. To have an economy of scale you need a reliable product. How do you know whether the product is reliable, without testing every piece? How can you test every beer brewed without drinking it all?

One great leg of statistics — it’s tempting to call it the first leg, but the history is not so neat as to make that work — is descriptive. This gives us things like mean and median and mode and standard deviation and quartiles and quintiles. These try to let us represent more data than we can really understand in a few words. We lose information in doing so. But if we are careful to remember the difference between the descriptive statistics we have and the original population? (nb, a word of the State) We might not do ourselves much harm.

Another great leg is inferential statistics. This uses tools with names like z-score and the Student t distribution. And talk about things like p-values and confidence intervals. Terms like correlation and regression and . This is about looking for causes in complex scenarios. We want to believe there is a cause to, say, a person’s lung cancer. But there is no tracking down what that is; there are too many things that could start a cancer, and too many of them will go unobserved. But we can notice that people who smoke have lung cancer more often than those who don’t. We can’t say why a person recovered from the influenza in five days. But we can say people who were vaccinated got fewer influenzas, and ones that passed quicker, than those who did not. We can get the dire warning that “correlation is not causation”, uttered by people who don’t like what the correlation suggests may be a cause.

Also by people being honest, though. In the 1980s geologists wondered if the sun might have a not-yet-noticed companion star. Its orbit would explain an apparent periodicity in meteor bombardments of the Earth. But completely random bombardments would produce apparent periodicity sometimes. It’s much the same way trees in a forest will sometimes seem to line up. Or imagine finding there is a neighborhood in your city with a high number of arrests. Is this because it has the highest rate of street crime? Or is the rate of street crime the same as any other spot and there are simply more cops here? But then why are there more cops to be found here? Perhaps they’re attracted by the neighborhood’s reputation for high crime. It is difficult to see through randomness, to untangle complex causes, and to root out biases.

The tools of statistics, as we recognize them, largely came together in the 19th and early 20th century. Adolphe Quetelet, a Flemish scientist, set out much early work, including introducing the concept of the “average man”. He studied the crime statistics of Paris for five years and noticed how regular the numbers were. The implication, to Quetelet — who introduced the idea of the “average man”, representative of societal matters — was that crime is a societal problem. It’s something we can control by mindfully organizing society, without infringing anyone’s autonomy. Put like that, the study of statistics seems an obvious and indisputable good, a way for governments to better serve their public.

So here is the dispute. It’s something mathematicians understate when sharing the stories of important pioneers like Francis Galton or Karl Pearson. They were eugenicists. Part of what drove their interest in studying human populations was to find out which populations were the best. And how to help them overcome their more-populous lessers.

I don’t have the space, or depth of knowledge, to fully recount the 19th century’s racial politics, popular scientific understanding, and international relations. Please accept this as a loose cartoon of the situation. Do not forget the full story is more complex and more ambiguous than I write.

One of the 19th century’s greatest scientific discoveries was evolution. That populations change in time, in size and in characteristics, even budding off new species, is breathtaking. Another of the great discoveries was entropy. This incorporated into science the nostalgic romantic notion that things used to be better. I write that figuratively, but to express the way the notion is felt.

There are implications. If the Sun itself will someday wear out, how long can the Tories last? It was easy for the aristocracy to feel that everything was quite excellent as it was now and dread the inevitable change. This is true for the aristocracy of any country, although the United Kingdom had a special position here. The United Kingdom enjoyed a privileged position among the Great Powers and the Imperial Powers through the 19th century. Note we still call it the Victorian era, when Louis Napoleon or Giuseppe Garibaldi or Otto von Bismarck are more significant European figures. (Granting Victoria had the longer presence on the world stage; “the 19th century” had a longer presence still.) But it could rarely feel secure, always aware that France or Germany or Russia was ready to displace it.

And even internally: if Darwin was right and reproductive success all that matters in the long run, what does it say that so many poor people breed so much? How long could the world hold good things? Would the eternal famines and poverty of the “overpopulated” Irish or Indian colonial populations become all that was left? During the Crimean War, the British military found a shocking number of recruits from the cities were physically unfit for service. In the 1850s this was only an inconvenience; there were plenty of strong young farm workers to recruit. But the British population was already majority-urban, and becoming more so. What would happen by 1880? 1910?

One can follow the reasoning, even if we freeze at the racist conclusions. And we have the advantage of a century-plus hindsight. We can see how the eugenic attitude leads quickly to horrors. And also that it turns out “overpopulated” Ireland and India stopped having famines once they evicted their colonizers.

Does this origin of statistics matter? The utility of a hammer does not depend on the moral standing of its maker. The Central Limit Theorem has an even stronger pretense to objectivity. Why not build as best we can with the crooked timbers of mathematics?

It is in my lifetime that a popular racist book claimed science proved that Black people were intellectual inferiors to White people. This on the basis of supposedly significant differences in the populations’ IQ scores. It proposed that racism wasn’t a thing, or at least nothing to do anything about. It would be mere “realism”. Intelligence Quotients, incidentally, are another idea we can trace to Francis Galton. But an IQ test is not objective. The best we can say is it might be standardized. This says nothing about the biases built into the test, though, or of the people evaluating the results.

So what if some publisher 25 years ago got suckered into publishing a bad book? And racist chumps bought it because they liked its conclusion?

The past is never fully past. In the modern environment of surveillance capitalism we have abundant data on any person. We have abundant computing power. We can find many correlations. This gives people wild ideas for “artificial intelligence”. Something to make predictions. Who will lose a job soon? Who will get sick, and from what? Who will commit a crime? Who will fail their A-levels? At least, who is most likely to?

These seem like answerable questions. One can imagine an algorithm that would answer them fairly. And make for a better world, one which concentrates support around the people most likely to need it. If we were wise, we would ask our friends in the philosophy department about how to do this. Or we might just plunge ahead and trust that since an algorithm runs automatically it must be fair. Our friends in the philosophy department might have some advice there too.

Consider, for example, the body mass index. It was developed by our friend Adolphe Quetelet, as he tried to understand the kinds of bodies in the population. It is now used to judge whether someone is overweight. Weight is treated as though it were a greater threat to health than actual illnesses are. Your diagnosis for the same condition with the same symptoms will be different — and on average worse — if your number says 25.2 rather than 24.8.

We must do better. We can hope that learning how tools were used to injure people will teach us to use them better, to reduce or to avoid harm. We must fight our tendency to latch on to simple ideas as the things we can understand in the world. We must not mistake the greater understanding we have from the statistics for complete understanding. To do this we must have empathy, and we must have humility, and we must understand what we have done badly in the past. We must catch ourselves when we repeat the patterns that brought us to past evils. We must do more than only calculate.


This and the rest of the 2020 A-to-Z essays should be at this link. All the essays from every A-to-Z series should be gathered at this link. And I am looking for V, W, and X topics to write about. Thanks for your thoughts, and thank you for reading.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

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