Half a century ago the British scientist and novelist C. P. Snow bemoaned the estrangement of what he termed the “two cultures” in modern society — the literary and the scientific. These days, there is some reason to celebrate better communication between these domains, if only because of the increasingly visible salience of scientific ideas. Still a gap remains, and so I’d like here to take an oblique look at a few lesser-known contrasts and divisions between subdomains of the two cultures, specifically those between stories and statistics.
I’ll begin by noting that the notions of probability and statistics are not alien to storytelling. From the earliest of recorded histories there were glimmerings of these concepts, which were reflected in everyday words and stories. Consider the notions of central tendency — average, median, mode, to name a few. They most certainly grew out of workaday activities and led to words such as (in English) “usual,” “typical.” “customary,” “most,” “standard,” “expected,” “normal,” “ordinary,” “medium,” “commonplace,” “so-so,” and so on. The same is true about the notions of statistical variation — standard deviation, variance, and the like. Words such as “unusual,” “peculiar,” “strange,” “original,” “extreme,” “special,” “unlike,” “deviant,” “dissimilar” and “different” come to mind. It is hard to imagine even prehistoric humans not possessing some sort of rudimentary idea of the typical or of the unusual. Any situation or entity — storms, animals, rocks — that recurred again and again would, it seems, lead naturally to these notions. These and other fundamentally scientific concepts have in one way or another been embedded in the very idea of what a story is — an event distinctive enough to merit retelling — from cave paintings to “Gilgamesh” to “The Canterbury Tales,” onward.
The idea of probability itself is present in such words as “chance,” “likelihood,” “fate,” “odds,” “gods,” “fortune,” “luck,” “happenstance,” “random,” and many others. A mere acceptance of the idea of alternative possibilities almost entails some notion of probability, since some alternatives will be come to be judged more likely than others. Likewise, the idea of sampling is implicit in words like “instance,” “case,” “example,” “cross-section,” “specimen” and “swatch,” and that of correlation is reflected in “connection,” “relation,” “linkage,” “conjunction,” “dependence” and the ever too ready “cause.” Even hypothesis testing and Bayesian analysis possess linguistic echoes in common phrases and ideas that are an integral part of human cognition and storytelling. With regard to informal statistics we’re a bit like Moliere’s character who was shocked to find that he’d been speaking prose his whole life.
Despite the naturalness of these notions, however, there is a tension between stories and statistics, and one under-appreciated contrast between them is simply the mindset with which we approach them. In listening to stories we tend to suspend disbelief in order to be entertained, whereas in evaluating statistics we generally have an opposite inclination to suspend belief in order not to be beguiled. A drily named distinction from formal statistics is relevant: we’re said to commit a Type I error when we observe something that is not really there and a Type II error when we fail to observe something that is there. There is no way to always avoid both types, and we have different error thresholds in different endeavors, but the type of error people feel more comfortable may be telling. It gives some indication of their intellectual personality type, on which side of the two cultures (or maybe two coutures) divide they’re most comfortable.
People who love to be entertained and beguiled or who particularly wish to avoid making a Type II error might be more apt to prefer stories to statistics. Those who don’t particularly like being entertained or beguiled or who fear the prospect of making a Type I error might be more apt to prefer statistics to stories. The distinction is not unrelated to that between those (61.389% of us) who view numbers in a story as providing rhetorical decoration and those who view them as providing clarifying information.
The so-called “conjunction fallacy” suggests another difference between stories and statistics. After reading a novel, it can sometimes seem odd to say that the characters in it don’t exist. The more details there are about them in a story, the more plausible the account often seems. More plausible, but less probable. In fact, the more details there are in a story, the less likely it is that the conjunction of all of them is true. Congressman Smith is known to be cash-strapped and lecherous. Which is more likely? Smith took a bribe from a lobbyist or Smith took a bribe from a lobbyist, has taken money before, and spends it on luxurious “fact-finding” trips with various pretty young interns. Despite the coherent story the second alternative begins to flesh out, the first alternative is more likely. For any statements, A, B, and C, the probability of A is always greater than the probability of A, B, and C together since whenever A, B, and C all occur, A occurs, but not vice versa.
This is one of many cognitive foibles that reside in the nebulous area bordering mathematics, psychology and storytelling. In the classic illustration of the fallacy put forward by Amos Tversky and Daniel Kahneman, a woman named Linda is described. She is single, in her early 30s, outspoken, and exceedingly smart. A philosophy major in college, she has devoted herself to issues such as nuclear non-proliferation. So which of the following is more likely?
a.) Linda is a bank teller.
b.) Linda is a bank teller and is active in the feminist movement.
Although most people choose b.), this option is less likely since two conditions must be met in order for it to be satisfied, whereas only one of them is required for option a.) to be satisfied.
(Incidentally, the conjunction fallacy is especially relevant to religious texts. Imbedding the God character in a holy book’s very detailed narrative and building an entire culture around this narrative seems by itself to confer a kind of existence on Him.)
Yet another contrast between informal stories and formal statistics stems from the extensional/intensional distinction. Standard scientific and mathematical logic is termed extensional since objects and sets are determined by their extensions, which is to say by their member(s). Mathematical entities having the same members are the same even if they are referred to differently. Thus, in formal mathematical contexts, the number 3 can always be substituted for, or interchanged with, the square root of 9 or the largest whole number smaller than pi without affecting the truth of the statement in which it appears.
In everyday intensional (with an s) logic, things aren’t so simple since such substitution isn’t always possible. Lois Lane knows that Superman can fly, but even though Superman and Clark Kent are the same person, she doesn’t know that Clark Kent can fly. Likewise, someone may believe that Oslo is in Sweden, but even though Oslo is the capital of Norway, that person will likely not believe that the capital of Norway is in Sweden. Locutions such as “believes that” or “thinks that” are generally intensional and do not allow substitution of equals for equals.
The relevance of this to probability and statistics? Since they’re disciplines of pure mathematics, their appropriate logic is the standard extensional logic of proof and computation. But for applications of probability and statistics, which are what most people mean when they refer to them, the appropriate logic is informal and intensional. The reason is that an event’s probability, or rather our judgment of its probability, is almost always affected by its intensional context.
Consider the two boys problem in probability. Given that a family has two children and that at least one of them is a boy, what is the probability that both children are boys? The most common solution notes that there are four equally likely possibilities — BB, BG, GB, GG, the order of the letters indicating birth order. Since we’re told that the family has at least one boy, the GG possibility is eliminated and only one of the remaining three equally likely possibilities is a family with two boys. Thus the probability of two boys in the family is 1/3. But how do we come to think that, learn that, believe that the family has at least one boy? What if instead of being told that the family has at least one boy, we meet the parents who introduce us to their son? Then there are only two equally like possibilities — the other child is a girl or the other child is a boy, and so the probability of two boys is 1/2.
Many probability problems and statistical surveys are sensitive to their intensional contexts (the phrasing and ordering of questions, for example). Consider this relatively new variant of the two boys problem. A couple has two children and we’re told that at least one of them is a boy born on a Tuesday. What is the probability the couple has two boys? Believe it or not, the Tuesday is important, and the answer is 13/27. If we discover the Tuesday birth in slightly different intensional contexts, however, the answer could be 1/3 or 1/2.
Of course, the contrasts between stories and statistics don’t end here. Another example is the role of coincidences, which loom large in narratives, where they too frequently are invested with a significance that they don’t warrant probabilistically. The birthday paradox, small world links between people, psychics’ vaguely correct pronouncements, the sports pundit Paul the Octopus, and the various bible codes are all examples. In fact, if one considers any sufficiently large data set, such meaningless coincidences will naturally arise: the best predictor of the value of the S&P 500 stock index in the early 1990s was butter production in Bangladesh. Or examine the first letters of the months or of the planets: JFMAMJ-JASON-D or MVEMJ-SUN-P. Are JASON and SUN significant? Of course not. As I’ve written often, the most amazing coincidence of all would be the complete absence of all coincidences.
I’ll close with perhaps the most fundamental tension between stories and statistics. The focus of stories is on individual people rather than averages, on motives rather than movements, on point of view rather than the view from nowhere, context rather than raw data. Moreover, stories are open-ended and metaphorical rather than determinate and literal.
In the end, whether we resonate viscerally to King Lear’s predicament in dividing his realm among his three daughters or can’t help thinking of various mathematical apportionment ideas that may have helped him clarify his situation is probably beyond calculation. At different times and places most of us can, should, and do respond in both ways.
John Allen Paulos is Professor of Mathematics at Temple University and the author of several books, including “Innumeracy,” “Once Upon a Number,” and, most recently, “Irreligion.”
Full article and photo: http://opinionator.blogs.nytimes.com/2010/10/24/stories-vs-statistics/