PRIMITIVE MATH, LIFE TRAJECTORIES, AND CURVE FITTING
In their book Where Mathematics Comes From linguist George Lakoff and psychologist Rafael Nuñez argue that from a minimal set of inborn skills—an ability to distinguish objects, to recognize very small numbers at a glance and, in effect, to add and subtract them—people extend their mathematical powers via an ever-growing collection of metaphors. Some of these shed light on the way we view our life stories.1
Watching my grandsons Theo and Charlie play with their toy trains suggests that our common experiences of pushing and pulling objects and moving about in the world lead us to form more complicated ideas and to internalize the associations among them. The size of a collection of Cheerios, for example, is gradually associated with the size of a number. (If my grandsons’ scattering of Cheerios is typical, it also leads to the realization that numbers are all over the place.) And combining collections into a single pile is gradually associated with adding numbers, and so on. Another metaphor associates the familiar realm of sticks (or even small Legos) with the more abstract one of measuring and geometry. The length of a stick or a Lego train is associated with the size of a number once some specified segment or single Lego is associated with the number one.
Mathematics may indeed be a more visceral enterprise than people think, as we come to understand most abstract concepts by generalizing, associating, and projecting our physical responses to them.
Scores and scores of such metaphors and analogies underlying other more advanced mathematical disciplines such as probability and statistics can then be developed. Consider the notions of central tendency—average, median, mode, etcetera. They most certainly grew out of commonplace activities and workaday English (or other natural languages) words like usual, customary, typical, same, middling, most, standard, stereotypical, expected, nondescript, normal, ordinary, medium, commonplace, so-so, which in turn grew out of universal experiences. It is hard to imagine prehistoric humans, even those lacking the vocabulary above, not possessing some sort of rudimentary idea of the typical. Any situations or entities or traits—storms, animals, rocks, friendliness—that recurred again and again would, it seems, lead naturally to the notion of a typical or average recurrence.
These considerations suggest that both looking back on our past and thinking of our future lead very naturally to the metaphor of a trajectory for our lives, and not just any trajectory, but the most accurate or descriptive trajectory. Thus the proto-mathematics inherent in my grandsons’ toy trains (and my own lovingly cared-for childhood Lionel trains as well) leads in this natural sense to the notion of a life's progression and biography. Let me extend this metaphor a bit further.
A common problem in statistics is finding the curve or surface that best fits or approximates a relatively small number of data points in space. Without intending to do harm to our notion of the richness of human biographies, I note that there is a suggestive analogy to biography here. It is the similarity between the linking of data points with the curve or surface that best fits them and the telling of the story of someone's life by invoking a relatively small number of remembered events. We infer from the events what must have linked them. If he did this, that, and the other thing, then he must have had such-and-such a period in his life or have been such-and-such a type of person. The events suggest a life's trajectory, but the narrative constructed from them is usually just an inference structured by social norms, conventional biases, passing fads, and personal attitudes. Joe Smith is a person who did X1, X2, X3,…and Xn, so based on these data points we construct his biography.
We can develop the natural analogy between the mathematical notion of a trajectory and a biography even further. When trying to find the curve or surface of best fit through a set of points in space (physical, psychological, social, cultural, organizational), there are statistical techniques that are used. (There are also ideas about exponential and trigonometric functions and about general Fourier series that are germane if one wants to get more technical.)
We don't want to have too many points far away from the curve or surface, just as in telling a life story one doesn't want too many events that are incongruous with the basic arc of the story, such as the great humanitarian as a teenager spitting in customers’ food or Benjamin Franklin's essay on flatulence, titled “Fart Proudly.”2 And just as the statistical techniques used tend to erase the individual data points in favor of the summarizing equation of the best curve or surface, so our biographies often tend to sacrifice the out-of-character events, incidents, and minor intrigues in favor of the basic outline. That's probably true to an extent in the autobiographical segments included herein.
The length of a trajectory or life is usually considered a crucial fact about it, which is probably why we first check the age of people when reading obituaries and why it's usually given in the headline or first sentence. It's always “Waldo Jenkins, 89,” or “Amy Winehouse, 27,” and not “Waldo Jenkins, 5’7”, 176 pounds.”
Incidentally, an idea from probability helps clarify a common misunderstanding about life spans. If the average life span is, for example, 80, and someone is 72, it's not the case that that person can expect only 8 more years of life. His conditional life span, given that he's lived 72 years, might well be 93. Likewise, if you were to roll a pair of dice, the probability of getting a sum of 12—two 6s—is 1 in 36, just 1 of the 36 possible outcomes for the dice. But the conditional probability of getting a sum of 12, given that you know you got at least an 11, rises to 1 in 3, the three possible outcomes on the two dice being 6,5; 5,6; and 6,6.
For most people obituaries are a particularly conventional sort of biography, reducing even the most complex individuals to a stylized litany of birth, early schooling, college, career choice, marital history, noteworthy achievements, and survivors. Almost as conventional are most autobiographies of sports and political figures (in contrast to those of autobiographies of scientists and writers, which tend to take place largely in their heads). Whoever the subject might be, the addition of a couple of vivid incidents would, if nothing else, make the biography more interesting. There are too few that produce the reaction “It's truly amazing that somebody who produced the wonderful X could also have done the execrable Y (or vice versa).” Nobel Prize–winning author V. S. Naipaul comes to mind because of his reported serial physical abuse of women.
Viewing life stories in this metaphorical way also sheds some light on a common phenomenon: people often drastically reevaluate their lives after a few emotionally significant events. How can we model the general trajectory of our life, which is to say the curve or surface that best fits its salient events, being significantly changed by only a few such events? One way is by introducing or just emphasizing another dimension with respect to which our life's trajectory looks quite different. That is, assume that for a while the importance of some dimension of our life—wealth, for example—increases. But then we introduce some other dimension—a relationship or a child, for example—and we note that as more time passes, the importance of the wealth dimension decreases and the importance of the other dimension increases. Some such reevaluations are quite common as one's life horizons recede.
If the discovery of the new dimension or criterion with which to evaluate our lives occurs late in life and if we give it great weight, then our evaluation of our life will shift dramatically as will the curve or surface that best fits it. This can be stated in terms of the rates of change with respect to time of these various dimensions or criteria, but the idea is, I hope, clear as a metaphor. “Conversions” of one sort or another provide examples, a couple of which I'll discuss later.
Conceiving of our lives as the curves or surfaces of best fit passing through an appropriate space provides us with another bit of mathstuff that describes and, to an extent, constitutes us. Our trajectory captures our narrative, the narrative of ourself. Moreover, it gives a metaphorical meaning to a number of phrases such as “over the hill,” which feeds into our visceral understanding of the parabolic shape of trajectories followed by rocks and balls and, as suggested, by us. In fact, if we don't push the analogy too hard, basic notions from calculus, somewhat simplistically applied, can help us demarcate the rough divisions of most lives. Youth can be grossly defined as that part of our lives where the first and second derivatives of our development (indicating the rate of change and the rate of change of the rate of change, respectively) are positive, corresponding to increasingly rapid growth. Middle age is more problematic. It is that part of our lives where the second derivative of development is negative while the first derivative remains slightly positive or slightly negative, corresponding to slow growth or slow decline, respectively. Old age begins either when the first derivative first becomes quite negative, corresponding to rapid decline or, as I'd prefer, when the first derivative remains negative, but the second derivative becomes positive again, corresponding to a very slow decline. Would that we approach death only asymptotically and never arrive there.
The notion of a curve or surface of best fit also suggests that short lives in which the latter part is truncated make better stories because it's easier to impose a coherent narrative upon them. Someone who has moved around a lot, has been buffeted by the vagaries of the world, has had several disparate careers and obsessions, and has interacted with sequentially distinct sets of friends and colleagues will have an interesting life, but not an especially coherent story line (or an easily comprehensible arc). Yet another reason to find the curve or surface of best fit in statistical applications is prediction and retrodiction of future and past events. This is less important in biography, but even here we often want to predict what a still-living biographical protagonist will do in the future or retrodict what he or she might have done in the past. Pitfalls such as statistical overfitting (models that are too complex and may only be describing random noise rather than anything significant) have biographical analogues as well.
I should note that this book, focused as it is largely on me and mathematics, is admittedly somewhat solipsistic. (I originally intended to title it The Book of John 3.14.) Not from lack of regard, but I've ignored almost entirely the rich tapestry and interplay between my trajectory and the trajectories of my friends, family, and colleagues, whether from Milwaukee, Philadelphia, New York, or many elsewheres. (Again the reason is not that they are unimportant to me but merely the narrow focus of this endeavor.) My father-in-law, whose kindest appellation for me was “jerko,” would merit a few chapters all by himself. I'm not a novelist, and capturing with appreciation this intricate woven texture over time is beyond me. Trying to give a minimalist account of just myself is already too difficult.
Having mentioned the word novelist and liberally invoked several mathematical metaphors, I would like to cite Leo Tolstoy as an illustrious codefendant in this particular use of mathematical metaphor. The above metaphors and others to come later may seem a bit much, but, though rare, such figurative speculations are not unheard of. Tolstoy famously proposed in War and Peace that calculus be used to model historical change (biographical change writ large). He wrote, “A modern branch of mathematics having achieved the art of dealing with the infinitely small can now yield solutions in other more complex problems of motion which used to appear insoluble.”3 He further argued, “Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.”4 Even if “modern branch” is understood to encompass the calculus of variations, field theory, and other more recent developments, I'm much less sanguine than the appropriately titled Count Tolstoy about there being any such laws, but I applaud his invocation of mathematics in such a context.
Less of a stretch than Tolstoy's: Playing around with Lego trains leads naturally to the notion of life trajectories.
THE ENVIRONMENT AS A PINBALL MACHINE, THE QUINCUNX OF LIFE
The trajectories of our lives through cultural/psychological spaces also suggest the topics of chaos theory and nonlinear dynamics. These now somewhat familiar notions arise when we examine certain systems—economic, ecological, physical, even personal—whose important variables are linked by nonlinear equations, not straight-line relationships. In the case of economic systems, for example, interest rates have an impact on unemployment rates, which in turn affect revenues. Budget deficits, trade deficits, and the total national debt also influence interest rates as well as consumer confidence and the stock market. These rates and totals as well as economic problems in other countries affect almost all economic indices, each of which feeds back on the others, reinforcing (or weakening) them and being, in turn, affected by them. This convoluted complexity suggests that the answer to many, if not most, economic questions is “Duh, I don't know,” an answer no politician would ever dare give. Even human-made computer code worked on by many programmers over many years is so complex as to be often unpredictable.
Similar connections characterize interactions in nature, a more poetic rendition of which is due to the Buddhist monk and poet Thich Nhat Hanh, who writes, “If you are a poet, you will see clearly that there is a cloud floating in this sheet of paper. Without a cloud there will be no water; without water, the trees cannot grow; and without trees you cannot make paper. So the cloud is in here. The existence of this page is dependent on the existence of a cloud. Paper and cloud are so close. Let us think of other things, like sunshine. Sunshine is very important because…”5 What he doesn't add is the unpredictable nature of this ecological complexity.
Complicated nonlinear interactions such as these characterize any modern economy or any natural ecology and often give rise to the butterfly effect. This is the phenomenon whereby a tiny event, say, a butterfly flapping its wings somewhere, can lead to an unpredictable cascade of events resulting over time in a major event, say, a tsunami in some distant locale. The term derives from the work of meteorologist Edward Lorenz modeling the dynamics of weather; the geometrical shape of his model resembled a butterfly.6
Or think of a pinball machine with an irregularly warped base and numerous randomly placed flappers in which tiny differences in the initial speed, angle, and spin with which two balls start their trajectories soon results in their following quite different paths. Balls bouncing off the pegs and obstacles in directions only minusculely different, one will soon hit a peg or bump the other misses or vice versa, after which the balls’ trajectories diverge radically.
The contingency of people's paths and actions suggests the question of how effective any of us, even BBBs (brilliant billionaire Buddhas), might be in solving a given societal problem. The question is, of course, too vague to answer, but maybe no BBB or no small group of them could be sufficiently smart, rich, and compassionate to effect such change. Or maybe—a depressing thought—the law of unintended consequences would prove more powerful than any bunch of BBBs, and their impact would be generally seen as negative. Or, more likely perhaps, they wouldn't do vastly better, or even better at all, than the same-size bunch of randomly selected ordinary people. After all, no one can herd butterflies. These considerations also undermine a similar story I liked as a kid, that of a completely selfless, thoroughly beneficent, nondescript, almost invisible person who nevertheless was responsible for an inordinate amount of good in the world—not Maxwell's thermodynamic demon but a kind of Maxwell's angel, opening the door to the good and just, closing it to evil. This, like Santa Claus, is an appealing story, but, alas, it also is a fantasy.
The delicate dependence of the pinballs’ trajectories on very minor differences is similar to the dependence of one's genetics on which a sperm cell haphazardly zigzagging along makes it to the egg first. Although harder to formally model as part of a nonlinear system, the disproportionate effect of trivial events also comes to mind—the accidentally deleted e-mails, missed flights, serendipitous meetings, and odd mistakes that shape and reshape our lives. Just as there is ample evidence that some economic, ecological, and physical systems are subject to the butterfly effect, so, I think, is there reason to believe that we ourselves as well as our relationships are, to a considerable extent, nonlinear systems sensitively dependent on initial conditions. Our biographies are thus, to an extent, an attempt to string together all our contingent traits and interactions into a coherent, yet quite contingent, narrative.
For this reason among many others we are all instantiations of mathematical notions and theorems; in this case the mathstuff is nonlinear dynamics, but it could be almost any branch of the subject. We're something like the prime-numbered thirteen- and seventeen-year-old cicadas, the golden-spiraled sunflowers and pineapples, and the tessellations of honeycombs, but vastly more complicated. We have instantiated within us the properties of numbers, the distributions of probability theory, the rules of logic, the beauty of calculus, the whole pantheon of mathematical patterns and ideas. We have the geometry of the genome's DNA, the network theory of our brain's connections, and the sinusoidal rhythms of our circadian bodily functions. Mathstuff permeates us and our lives, and, if you're someone who subscribes to a Platonic conception of mathematics, you might even be tempted to call it the “divinity” in all of us, a mathematical idealization that allows us to think of ourselves as gods and goddesses rather than as dying animals.
Let me move from Plato's realm to a couple of less ethereal examples of sensitive dependence and contingency—the first a humdrum biographical illustration of nonlinear dynamics, the second a matter of considerable public importance.
My two brothers Paul and Jim, sister Marilynn, and I grew up in the same home, albeit no doubt in different micro-environments. We're similar in many ways (not all as trivial as our liking The Three Stooges), but we also have very different interests. More generally, there are a number of factors relevant to any set of siblings developing differently. Early labeling such as the wild one or the studious one or the warm one and the repelling effect on siblings of the so-called narcissism of small differences are two such factors, but bouncing off different pegs in the pinball machine that is life shouldn't be underestimated and, as noted, is yet another reason to be very dubious of biographers’ (or even autobiographers’) general pronouncements.
We tend to think we've arrived at our present station largely by dint of determination and hard work, but as my father used to say, we're all just farts in a windstorm. Less graphically put, we're all parts of various systems—familial, professional, societal—and these systems impact on us and direct our paths as if we were pinballs whirling through the quincunx of life. Nevertheless, we should heed the aforementioned title of Benjamin Franklin's essay, “Fart Proudly.” That is, we should embrace our contingency even when it's unpleasant.
The second example of an extremely significant, decidedly unintended result of a relatively tiny event is nightmarish, at least for me. It concerns the role I played in getting George W. Bush elected president in 2000. That I was the butterfly whose fluttering cascaded into Bush's election still pains me. I had written an op-ed for the New York Times titled “We're Measuring Bacteria with a Yardstick” in which I argued that the vote in Florida had been so close that the gross apparatus of the state's electoral system was incapable of discerning the difference between the candidates’ vote totals. Given the problems with the hanging chads, the misleading ballots (in retrospect, aptly termed butterfly ballots), the missing and military ballots, a variety of other serious flaws, and the six million votes cast, there really was no objective reality of the matter.7
Later when the Florida Supreme Court weighed in, its chief justice Charles T. Wells cited me in his dissent from the majority decision of the rest of his court to allow for a manual recount of the undervote in Florida. Summarizing the legal maneuverings, I simply note that in part because of Wells's dissent the ongoing recount was discontinued, the case went to the US Supreme Court, and George Bush was (s)elected president.8
Specifically, Judge Wells wrote, “I agree with a quote by John Allen Paulos, a professor of mathematics at Temple University, when he wrote that, ‘the margin of error in this election is far greater than the margin of victory, no matter who wins.’ Further judicial process will not change this self-evident fact and will only result in confusion and disorder.”9 (Incidentally, the CNN senior political analyst at the time, Jeff Greenfield, cited the quote in his book on the 2000 presidential election, Oh, Waiter! One Order of Crow!, and wrote “the single wisest word about Florida was delivered not by a pundit but by mathematician John Allen Paulos.” I doubt, however, that Greenfield thought it was reason to stop the recount.)10
I was surprised and flattered, I admit, that I was cited by the judge but also very distressed that my words were used to support a position with which I disagreed. Vituperative e-mails I received didn't help. Many were angry that I would support Bush. Some were clearly demented. With all due respect to these correspondents and the esteemed judge, I believed and still believe that the statistical tie in the Florida election supported a conclusion opposite to the one Wells drew. The tie seemed to lend greater weight to the fact that Al Gore received almost half a million more popular votes nationally than did Bush. If anything, the dead heat in Florida could be seen as giving Gore's national plurality the status of a moral tiebreaker. At the very least the decision of the rest of the court to allow for a manual recount should have been honored since Florida's vote was pivotal in the Electoral College. Even flipping a commemorative Gore-Bush coin in the capitol in Tallahassee would have been justified since the vote totals were essentially indistinguishable.
Historical counterfactuals are always dubious undertakings, but I doubt very much that the United States would have gone to war in Iraq had Gore been president. I also think strong environmental legislation would have been pursued and implemented under him. Was I responsible for Bush's presidency? No, of course not; butterflies can't be held responsible for the unpredictable tsunamis that in retrospect can be traced to their fluttering and to a myriad of other intermediate events. Still, every once in a while, the guiltifying thought that the unwarranted Iraq War was my fault does occur to me.
BIOGRAPHIES AND THE TEXAS SHARPSHOOTER
One unavoidable problem associated with the idea of a life's trajectory is illustrated by the old story about a Texas rifleman who was thought to be an expert sharpshooter. The shooter's secret was that he would fire shots at a barn or a sign and then when many of the bullet holes happened to cluster together or form some sort of pattern, he would draw a target or other shape around them and then strut about claiming to be a sharpshooter.
Given that any life has countless incidents, situations, and characters, it's clear that biographers might be especially prone to the Texas sharpshooter fallacy. (The fallacy reminds me of the observation that meteors always seem to land in craters.) More generally, in this era of Big Data, the National Security Agency (NSA), and state and corporate surveillance, we must always be wary of such cherry-picking of data. What makes the fallacy especially common in biographies is that once a reputation takes hold, almost the only events that are recorded are those consistent with it. Bad guys seem to do only bad things; compassionate people perform only considerate acts; mass killers are always said to be loners; parents are usually characterized as family men or devoted mothers; George Washington and Abraham Lincoln were always completely honest; and so on. It's as if the Texas sharpshooter's bullets make no holes when they deviate too far from the target or conventional trajectory.
A possible recent example came to light on the hundredth anniversary of the birth of British mathematician and computer scientist Alan Turing. Turing did seminal work on the eponymous universal Turing machines and theoretical computer science in general and performed invaluable lifesaving service during World War II on cryptography. His achievements can hardly be overstated. Literally. Despite them, he was convicted of indecency because of a homosexual affair in the ’50s and opted for hormone treatments rather than prison. He died of cyanide poisoning a couple of years later. After his death he became something of a symbol of gay oppression, and biographers wrote of his being harassed, despondent, and suicidal. Philosophy professor Jack Copeland has suggested, however, that there was no evidence for the standard story of depression and suicide and that the cyanide poisoning was accidental.11 We'll never know the truth, but there is certainly a tendency to prefer narratives, such as the first, that comport with our expectations.
More generally, biographers can vehemently insist on a biased view of a person that is unsupported by much evidence. The person is assumed to be manipulative, saint-like, laid-back, greedy, or whatever. But such views don't exist in isolation and exert a distorting effect on other matters.
An analogy from geometry is helpful. Were a physicist so inclined, he could assert without inconsistency, but contrary to Einstein and others, that space was Euclidean and flat rather than non-Euclidean and curved. If he did so, however, he would have to account for astronomical phenomena that can be explained quite simply and naturally in a non-Euclidean framework. The benighted physicist would be compelled to introduce fictitious forces and accelerations to save his assumption that space was Euclidean and flat. Which geometry/physics combination to use is to some extent a matter of convention, but some conventions are better than others. In the same way, to insist that a person is, for example, always manipulative when by most accounts this is not the case requires that one weirdly interpret the actions of some people, question the motivations of others, or assert the naïveté of still others. Insisting, for example, that a politician, say, President Barack Obama, is an evil socialist, as some Tea Party members do, requires that they undertake wholesale distortions of their standards of judgment.
The so-called conjunction fallacy, or Linda problem, suggests a related pitfall of just-so stories with little evidentiary value. The more details in a biographical anecdote or even in an everyday conversation, the more plausible and engaging the account becomes. Alas, it also grows less probable. The reason is simple: the more details there are, the less likely it is that the conjunction of all (or most) of them is true.
If, for example, Senator Jones seems to be an extremely devoted and happily married family man who lives modestly in a small house, which is more likely: (a) Jones accepted an illegal campaign contribution from a supporter or (b) Jones accepted an illegal campaign contribution from a supporter and used it to pay for his daughter's expensive medical treatments? Despite the more 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.
As with the Texas sharpshooter foible, approaches to biography or even everyday storytelling that depend on the conjunction fallacy are quite common. It's interesting watching how some people effortlessly embroider, exaggerate, gerrymander, and invent details to concoct a compelling little anecdote out of the sparsest and most ordinary of incidents. Munchausen syndrome, whereby healthcare providers and/or patients exaggerate reports and add false details to obtain sympathy, attract attention, or portray themselves as heroes, is an extreme example.
My predilection has usually been just the opposite. I find excessive enthusiasm suspect and often feel compelled to report neutral facts that undermine the tendentious slant of any story I read and thereby drain it of much of its drama. I can be an irritating killjoy. This deflating habit is one reason why I would make a very bad novelist or biographer. My wife, who has also taught at Temple University for many years with more vivacity than I can muster, is quite different. Before her teaching career, she wrote romance novels and occasionally exploited my debunking psychology in doing so. She would ask what I would do or say in some situation, and she then made sure her male protagonist did or said something radically different. Given any conflict, I generally look for possible misunderstandings that might contribute to it and then search for common ground, not an approach conducive to much bodacity.
This is not always the best strategy, as the unlikely story of the nonfunctioning guillotine and the condemned engineer shows. Other potential guillotine victims were released when the blade didn't fall. The engineer, however, once his head was placed on the block, noted, “Oh I see the problem. The rope has a kink and has come off the pulley on the left side and can't lift the blade.” The problem was duly corrected and the engineer beheaded.
Clearly misunderstandings should sometimes not be dispelled, but my professorial bias leads me to think that on the whole dispelling them is good cognitive hygiene. More specifically, biographies would be better with more tentative analyses and fewer apocryphal events. Biographies would also be better off, albeit occasionally less entertaining, if they were less extreme, whether in a hagiographic or a demonizing way.
Like accounts of fictional characters, biographies, too, often depict people who are in so many ways more extraordinary than the people we meet in everyday life. Biographical subjects seem to react more, emote more, and have more well-defined opinions, motives, and goals than the people we know. They're focused, and their opinions, motives, and goals lead more frequently to decisions and actions. Because of this, they're often more predictable than people in real life, more likely to struggle with others for what they want, less likely to dither and vacillate. Most everything about biographical subjects is more clear-cut and logical. Wishy-washy, conflicted, and uncertain they are not. They also appear to be much more self-governed and less buffeted by the vagaries of chance.
People are, of course, quite different from each other in countless subtle and not-so-subtle ways, but it's hard for me, a confirmed ditherer and vacillator, to credit these stark unidirectional differences between the proverbial people on the street and the subjects of (auto)biographies. Often only a few lucky breaks are sufficient to make an average shmo biography-worthy, and then his or her otherwise complex but superficially nondescript story will be retroactively recast in a more heroic mold. (Incidentally, this use of “average shmo” is more consistent with the median than it is with the mean or mathematical average of a group of people.)