Averages

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":

  • Mean. Generally, to find the average of a list of figures, we total the list and then divide that total by the number of figures on the list. So, to find the average of 6, 10, 6, 2, and 6, we first total the list (30), then divide that total by the number of figures on the list (5). 30 divided by 5 equals 6, which is the average. Or, more accurately, that is the mean average of that list. The mean is the most common kind of average; but just because no information to the contrary is offered, you should not assume that every average is a mean average.

  • Median. The second most common sort of average is a median, which you identify by putting the figures in a list in ascending (or descending) order. The middle number on the list is then the median. Using the list above, for example, we would first put the numbers in ascending order (2, 6, 6, 6, 10), and then count up or down three to find the median, which would be 6.

  • Mode. The third way to figure an average looks at frequency of occurrence. Once again, the figures on the list are put in ascending (or descending) order. The figure that has recurred the most often is the mode. In the list above (2, 6, 6, 6, 10), the mode is 6.

    In the example used above, the average for this particular list was the same, whether mean, median, or mode, but that is unusual. For example, for a list of 1, 2, 3, 7, 7, the mean is (20 divided by 5) is 4, the median is 3, and the mode is 7. But the point in having three very different ways to measure averages is that, depending on the situation, one may be much more informative than the other two. Most often, the most informative average is the mean, but not always.
Let's take the case of the number of children in the average American family. Of course, the definition of "family" has undergone considerable revision in the last fifty years. We don't have time to get into that issue right here, but it's a good reminder that problems in using statistics often have nothing to do with the actual process of gathering or interpreting those figures. For example, until quite recently, the State of California enumerated as "out-of-wedlock" any birth in which the mother's surname was not the same as the father's - despite the fact that for decades a large percentage of women have preferred not to change their names when married. The result was an inflated total for babies born to unmarried parents-and a statistic that could be used as evidence of extramarital sexual activity, or (according to some) a level of immorality, that did not in fact exist.

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?

  1. Remember that statistics, like other sort of evidence, are only used to support claims being used as premises in an argument. The first issue is always the argument's validity, and only once that has been established should you consider its soundness. Solid support will make the premises seem stronger, and the argument more persuasive, but you should always be a little skeptical if that support comes in the form of a statistic, especially a statistical average.

  2. In the case of any average, you should try to determine whether you are dealing with a mean, median, or mode, and how appropriate each of those would be in expressing the information under consideration.

  3. Unfortunately, we often must make judgments without having total confidence in the evidence on which they are based. If you have good reason to doubt the accuracy of a statistic, then suspend judgment on the argument while pursuing that information. However, if the statistic comes from what you consider a reliable source, and if you have no reason to doubt it, then quibbling about it simply because it is a statistical average is probably counterproductive.

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.