. A histogram is simply a series of rectangles, in this case one for each income interval of $10,000, with the height of each rectangle proportional to the number of people in a particular income range. Notice we also draw a line that approximates the overall shape of the histogram. That line represents something extremely important in the study of statistics. The histogram here is obtained by sorting the incomes in bins of $10K, but the curve gives a mathematical formula to compute the approximate number of people at any income value. For example, while the histogram can only tell how many people make between $120K and $130K, the curve can give an accurate estimate of exactly how many people are making, say, $126K. Curves like these have the intuitive name, distribution functions, as in this case, it happens to represent the distribution of household incomes in that borough. The particular bell-shaped distribution that we see here in the figure is the most common, as well as the most important, one in the field of statistics, and you can tell how much so by its name—normal distribution. One of the most fundamental results in probability and statistics, the central limit theorem, states that if you take the average of a large but fixed number of random variables (like the average of the numbers that appear face-up in tossing a thousand dice simultaneously), and keep on doing so multiple times (keep tossing the dice hundreds of times—the average would vary with each throw, since this is quite random), and then you plot how many times you get a particular average value, you will always end up with a normal distribution like the one shown here. It’s a very powerful statement because it predicts a very precise mathematical distribution for a set of random and arbitrary variables! This gives a glimpse of why normal distributions are so important in statistical modeling, because finding patterns in randomness is the very name of the game—with obvious relevance for a bulk of important things in science and in life!
, you will notice that the average income in this borough seems to be about $150K. Suppose you already happen to know that. Then if you meet someone, you would certainly do a bit of stereotyping that the person is well off in the upper middle class but perhaps not a multimillionaire. And, since most of the people seem to have incomes between $100K and $200K, there is a good chance that you would be right. This is an example of the first kind of stereotyping. Since you know the average, and most of the distribution curve lies close to the average, you know that your guess is more likely than not going to be close to that average. However, you can also see in that there are quite a few families making less than $60K and a fair number making above $250K, which means that your stereotyping of the person that you just met could also very well be way off the mark.
, we see three possible ones with $15K within their ranges. If the actual one is Distribution 1, your guess might have been correct because the average income is $25K, but if it were Distribution 2, or even more so, Distribution 3, then $15K is an outlier very far from the average income in the borough, and you would be completely wrong in your presumption!
Figure 7.2 If the first person we meet from a borough had an annual household income of $15K this year, from that we cannot conclude anything about the income distribution of the borough. Three (of the infinite possible) distributions are shown here, all containing $15K within their ranges, but each one has a very different average—and anyone or none of them could be right!
to see that.
To come full circle back to Asimov’s vision of the future of humanity—aren’t we already partial believers when we say, “History repeats itself?” After all, anything that repeats is predictable. Because in the end, on average we humans are much more alike than we are different, and on the scale of large numbers of people and over timespans of decades and generations, our behavior is more predictable than we would like to admit. On average, we know how a human life plays out in any part of the world: spend early years getting educated and then the all-consuming search for a mate, get married, get a job, build a career, buy a house, have kids, kids grow up, you grow old, develop various levels of health problems, worry about your finances and retirement plans, perhaps move around from place to place, job to job a few times, and then you retire, and so on. There are many variations of these, some drastically different from others, but on average this is what plays out for most people in most parts of the civilized world.
While history does not always repeat itself exactly, the patterns of human behavior en masse certainly remain surprisingly uniform across time as well as space, so that in a broad sense certain aspects of history do seem to follow recurrent patterns. But such recurrences have long enough periodicity—perhaps several generations—so that it is easy to overlook them altogether since our collective memory is short and seems to reset after a generation or two. The recent financial crisis caught most of us by surprise; well, if we read our history, it is just the last one in a long line of similar crises brought on by eerily similar causes—particularly uncontrolled greed. Or take wars, for instance. Nobody wants wars; they can fool us all they want with the propaganda about the glory of war, but nobody wants to get killed for glory, and yet there are wars all the time, because the same dynamics of conflict play out over and over again with groups and nations, large and small: the aggressor who wants to take control of something (material or ideological) that the other has or refuses to give up, and the defender who is compelled to resist. Typically, both believe that God is on their side. Likewise, all the empires ever built were generally convinced at their height of glory that they will last forever, but they all vanished, and they always will. The list goes on. History might never become a predictive science, but patterns of collective human behavior are predictable enough that we ignore history at our own peril.