Friday, November 2, 2012

Some Words about Numbers

When people present arguments, whether in speech or in writing, they often use numbers as evidence to support a conclusion. However, we need to be wary of arguments that use numbers, especially statistics.

The speaker or writer might support his argument by appealing to "recent studies," or the fact that "seventy-five percent of spinach eaters watched at least one episode of Popeye. . .," or "polls indicate that your candidate has approval ratings among women in the low teens." Statistics, of course, can help make an argument cogent, but we need to be careful of the potential fallacies that can result from an improper use of statistics.

As our good friend the late Robert J. Gula writes in Nonsense, "If we're dealing with a statistic, we should ascertain who gathered the statistic, what process was used, how many people were polled, how these people were selected, [and] what specific questions were asked[. . .]" (65).  

Now most of us do not have the time to perform such a thorough review of a statistical study's raw data. This impracticality is partly resolved by relying on the testimony of reputable sources. It is a fair assumption that a study conducted by the CDC or CBO is trustworthy, whereas a study conducted by one's postman may require a significant amount of corroboration before it can be considered trustworthy. I imagine this is an important reason why professors demand certain kinds of sources: they want the statistics to be generated by organizations that are presumed to be reputable.

Assuming the data is reliable, one of the fallacies that a speaker or writer who makes an appeal to numbers might commit is to use the term "average" in an equivocal manner.

For example, you might hear a person say "I'm just an average guy." In such a construction, the term "average" is probably not being used in a quantitative way. What the speaker is trying to convey is the fact that he is "normal," "unspectacular," "similar to the majority of people who populate a particular geographic area."

"Average" is also used to specifically convey a quantitative relationship. When used to convey a quantitative relationship, the term "average" can connote three different measurements. To illustrate these different kinds of measurements, let's take a look at the following set of numbers:


Now, what is the "average" of these five numbers?

Well it depends on how one interprets the term "average." The mean average for these five figures would be 2,000.

The mode (the figure that appears most frequently) for these five figures would be 300.

The median  (the number that is in the middle of the series when those numbers are listed in order, highest to lowest or lowest to highest) for these figures would be 300.

Let's look at how the following data set be used fallaciously.

Suppose the following figures represent the weekly salaries of five employees at a local company. Suppose further that the owner of the local company claims that his employees make a living wage. "Every employee at my company," the owner tells a local journalist,"is well-compensated. In fact, the average salary at my company is 2,000 dollars per week."

How does this bit of reasoning strike you?

It strikes me as weak.

Because the owner is citing the mean average as if it were the mode, his argument is fallacious. Clarifying the meaning of "average" is an important step in producing cogent arguments that use statistical data.

Another point to consider when evaluating an argument that appeals to numbers is that "percentages" are basically irrelevant without context.

For example, an incumbent local politician who has served a single term might claim the following: "Under my administration, the crime rate has dropped by 50%."  Sounds impressive doesn't it? But what if under the previous administration only two crimes were committed? I mean, I guess one crime is better than two crimes, but in this context the statistic cited by the politician seems a little foolish.

To take another example, suppose a CEO claims during an interview with Maria Bartiromo the following: "Our international sales grew 500 percent last quarter." Sounds great, right? It sounds like the CEO is leading the company into the next century. Well, before we applaud too loudly, we may want to know what the total amount of revenue was the previous quarter. If the company only booked 100 dollars in international revenues the previous quarter, then that 500 percent increase seems a little less awesome.

In a lot of arguments, statistics come from surveys or polls. What we need to keep in mind is that the effectiveness of a survey or poll is contingent on, among other things, the quality of the sample. It may be true that "Eighty percent of the people surveyed claimed that candidate X is doing an excellent job." Sounds great doesn't it?

But one wonders about the the quality of the sample. If the sample of the survey was candidate X's friends and family, then eighty percent no longer seems so great. In fact, it seems somewhat suspicious. In order for a survey to contribute to the strength of an argument that draws a general conclusion about a large population, it must be based on a sample that fairly represents that population in both size and diversity.

Finally, we need to be careful when people use the behavior or beliefs of a large population to support a conclusion. TV commercials use this technique a lot.

For example, a car company might broadcast a commercial that says "In the past ten years, ten million people have bought the XYZ. Ten million people can't be wrong." The inference seems to be that because ten million people have bought the XYZ, I should go and buy the XYZ. But if the most important criterion for me in deciding which vehicle to purchase happens to be a high MPG, and if the XYZ has a low MPG in its class, it wouldn't make sense for me to purchase the XYZ regardless of how many vehicles the company has sold.

Numbers can cogently support an argument, but only if they are used appropriately. How the data is collected, interpreted, and disseminated affects its argumentative value. Although most of us may not have the time to scrutinize reams of raw data or check the wording of a survey question, we should be aware of the fact that statistics can be used for logical purposes and for fallacious purposes. Such an awareness is not foolproof, but it does make our cerebral defenses a little more resilient.

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