When a Center Does Not Hold

Many books and stats “modules” for school misuse measures of center. At least that’s my conclusion right now; I’d be interested in comments.

I was struck by a lesson that presented an oft-used data set: the mean heart rates of a bunch of animals. The lesson presented the data, had students calculate the mean and median, and eventually make a box plot. An example dot plot appears at right.

The intention here is a good one: use real data about interesting things. But we have to stop and ask, what does the mean (or median) of this data set actually mean? The answer: not much. Finding meaning requires an absurd reach: If you randomly choose an animal from a sample that has one of each species (or two if you have an ark), the mean—in this case, 143 beats per minute—is the expected value of that animal’s heart rate. But we’re not going to do that. So why calculate the mean? (Savvy stats teachers will recognize that the unit of investigation—or whatever you want to call it—is a species here rather than an animal.)

[Note: the dot plot above is actually useful if you label the points; then you can see graphically where each species stands.]

I don’t mean that looking at the data is bad—just that we ought to be careful that when we ask students to calculate something, that thing makes sense. And that it’s really easy to screw up.

Heart rate (per second, logged) against animal mass (grams, logged). The red dot is us.

In the case of animal heartbeats, the point of the data set is often to compare heart rate to the size of the animal. The graph at right shows the endpoint of doing some data-munging in which we decide that it’s a good idea to take the logs and all that. The key here is that this graph tells a story that the mean does not: big animals have slow hearts.

I think the problem with computing the mean is that our data were already means (mean heart rates) or, more broadly, the data were aggregate measures. And we run a risk of being absurd when we compute an aggregate measure of aggregate measures.

But if you’re looking at relationships—something multivariate, like in the figure—the fact that the data are pre-aggregated is not a problem.

So when should you calculate means? When the data are not aggregate: when they’re data about individuals. So if we have the resting heart rates of a bunch of teenagers, we use the mean (or median) to get a heart rate for a “typical” teenager. To compare that mean heart rate to people in their 40s, we might compare the means (or simply look at all the data, as in the graph below, from the NHANES data tool in the eeps data zoo). But we would not then average the teen-mean with the 40s-mean to get a grand mean of the two groups. It wouldn’t make sense.

Heart rates for a sample of teenagers and people in their forties, with means.

Where You Stand

One sensible purpose to calculating the mean heart rate of species is simply to know where an animal is in the panoply of animals. Is its heart rate greater than the mean or less? For that, I still have trouble with using the mean. I’d rather just see the distribution and know where the animal is, like in the dot plot above; or take the time to make some direct comparisons. For example, a typical monkey is about the size of a small dog (5 kg) but has about twice the heart rate.

Weighting and States

I have not talked about weighted averages here. Another common egregious averaging is with State-by-State data, for example, mean income. Suppose we calculate the mean of the mean incomes of the States. We might rightly say that some State has a below-average income, but that can be deceptive: since some states have lots more people, the number we calculate—the mean of the means—may not represent a typical person, which is what the reader naturally assumes.

Instead of calculating the mean of the means the simple way, though, you could calculate a weighted average. This will yield the average income of all people, not the average-average State income.