On one hand, there is the joke on Garson Keillor's Prairie Home Companion that, in Lake Wobegone, "all the men are strong, all the women are beautiful, and all the children are above average." On the other hand, in defense of an admittedly mediocre nominee for the U.S. Supreme Court a few years ago, one senator argued that there ought to be at least one justice that was just average, because that's what most Americans are. Fortunately (or, more often, unfortunately), "averages" are one of the most manipulated concepts in statistics and, as we will see, it may be quite possible for most (if not all) children to be "above average," while at the same time even more likely that almost no one is, in fact, "average."
As we saw in the section on statistics, confusion sometimes arises over whether a particular figure represents a cumulative total for the whole group, or the average for each individual in the group. Just knowing the figure is an average, however, does not necessarily eliminate confusion, because there are several different kinds of "averages":
But let's get back to the number of children in the average American family. That figure has recently been given as 2.1. Comedians like to tell jokes about the one-tenth of a child, but the decimal makes it clear that this figure is a mean-and, to express it accurately, we should say "the average number of children in the American family." Notice that this does not mean that half of the families have more than 2.1 children and half have less, but that half of all children are in families above and below that mark. In fact, the vast majority of families have less than 2.1 children.
How is that possible? Because there can be no less than 0 children per family, but as many as 15 or more, we say that the population under study is skewed toward the upper end. A graph made of the number of children per family would not produce a nice, symmetrical bell curve, but one that starts high on the left (at 0), goes up a little (at 1 and 2), and then tails off gradually as it moves to the right (and as the number of children increases). So, while this mean average gives us an accurate picture of the distribution of children according to family size, it does not help much if we are interested in distribution of families. In other words, if we want to find the size of the "average American family," that is, the number of children that half of U.S. families have more than, and half less, we might use a median average.
And while we are on American family life, we can see the same sort of problem with another often-cited statistic, the one about half of all marriages ending in divorce. Yet, most people that get married never get divorced--even in California! This statistic is based on another skewed population: its graph would be at the highest point on the left (at 0), and taper off as the number of divorces increases. To see how this works, let's say that 60% of all people who get married never divorce, 10% get married twice, 10% get married three times, 10% get married four times, and 10% get married five times. For every one hundred married people, then, the once-married would have been involved in 60 marriages, the twice-married in 20 marriages, the thrice-married in 30 marriages, those married four times in 40, and the last group in 50 marriages. That's a total of 200 marriages, 100 of which ended in divorce, and 100 of which did not. So you can see that the same information can be used to support what seem to be conflicting claims: that half of all marriages end in divorce, and that most married people (60%) never get divorced.
(Please note that the example above is purely hypothetical. To simplify the discussion, I have not taken into account those that divorce and never remarry, nor have I bothered to adjust the figures to reflect the fact that each marriage involves two people, nor have I figured in those individuals who get married more than five times. The good news is that the popular statistic about "half of all marriages end in divorce" is wrong, anyway: according to recent figures, in the U.S. about one-third of all marriages end in divorce.)
It should be clear that, like statistics generally, averages can be manipulated and misinterpreted. So, when faced with a statistical average, what is a critical thinker to do?
So, having considered the statistic offered, the authority of its source (if given), the appropriateness of the (likely) averaging method, and the validity and soundness of the argument it is being used to support, if you find no reason to reject that figure, or to suspend judgment while investigating further, then accept the evidence.