Better Than Average

So, there’s this type of statement a lot of us numbers people like to have fun with, because it makes us feel smart:

“Fifty percent of doctors graduated in the lower half of their medical school classes.”  (Hahaha of course they did!)

“Only 10 percent of students managed to achieve above the 90th percentile.”  (Because that’s what percentile means!  Ah ha ha!)

“Eighty percent people think they’re above-average drivers.”  (Hahaha wrong!)

Except that last one could be correct.

Huh?

Here’s the thing: there are several different types of “average.”  Say you have a population of numbers.  Here are, for example, two different ways you can take the average:[1]

The median is the number at which half the population is below that number and half the population is above it.  So if eighty percent of people claim their driving is above the median on some objective scale, that’s patently untrue.

But “average” can also be the mean, which is what you get when you add up all the numbers and divide by the number of data points.  And if the data have a few extreme outliers, this can skew the mean in one direction or the other even though most of the population is around the middle.

For instance, let’s rate drivers on a scale of 1 to 10.  If we have ten drivers rated 1, 1, 1, 1, 5, 5, 6, 6, 7, 9, the mean is 4.2, but the median is 5.  Since 6 data points are above 4.2, 60 percent of our drivers are above the mean—which means we could make the wholly accurate statement that 60 percent of the drivers are above average.

Wheeeee!

In any population of data this is a possibility, but in populations with extreme outliers it will be a certainty.  Multi-billionaires  mean most people in the United States make below-average incomes.  Infant mortality rates in some populations might mean most people have above-average lifespans.  And yes, a few people who get into ridiculous numbers of accidents might mean most people are above-average drivers . . . as long as when we say “average,” we mean “mean.”

UPDATE: Actually, it turns out the studies about driving ability were asking about being above or below the median (and 93 percent of U.S. drivers thought they were in the top 50 percent of the population in terms of driving skill!), but it’s still possible for the respondents to be correct in another dimension: if they were all using different scales for driving ability.  So many ways to twist around statistics!

  1. And there are so many more.

3 thoughts on “Better Than Average

  1. Pingback: It Was the Best of Times, It Was the Worst of Times

  2. InMyBook

    Isn’t the question about driving ability something subjective? People were asked about their driving ability, and of course most people curse out other drivers and overrate their own driving skills. It wasn’t like anything was being measured by any type of objective scale. But you are right in that statistics can be purposefully manipulated or misinterpreted in all types of ways.

  3. slhuang Post author

    Well, yes, you’re absolutely right. :) I was assuming at first some type of objective measure, like “number of accidents / tickets / etc.” — stats like the insurance companies use. But the idea of a subjective measure is sort of what I was trying to get at in the update at the bottom about “different scales,” because you’re right — depending on how they asked the question, it becomes meaningless. “Do you think you’re a better driver than most people” is far less specific than, “do you think you’ve had less accidents than most people.”

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