## Archive for **April 8th, 2021**

## Winecup flower center

In this closeup of a winecup (*Callirhoe* sp.) at the Lady Bird Johnson Wildflower Center on March 25th the shadow struck me as appropriate for the profile of a gnome or ogre or some such creature.

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It’s common to hear politicians and activists bandy about the phrase “common sense.” That’s a loaded and misleading term because some or even many things that a majority of people believe to be common sense are easily shown to be untrue. Over the next week I’ll give some examples, starting now.

Suppose you live in City A. One morning you get on an Interstate highway, drive to a place in City B, and with light traffic you end up averaging 70 mph for the trip. Three days later you return along the same route, but this time traffic is heavy, and in addition rain pours down for much of the time. As a result, you end up averaging a pitiful 30 mph for your return trip from City B to City A. Now here’s my question: what was your average speed for the round trip? Most people who are given these facts and asked that question will say the average speed for the round trip was 50 mph, which they got by averaging 70 and 30: 70 + 30 = 100, and 100 ÷ 2 = 50. It’s common sense, right?

So simple, so easy—and so wrong! People who come up with an answer of 50 mph don’t understand what an average is. An average is the total of one kind of thing divided by the total of another kind of thing. The very label “miles per hour” tells you what to do: take the total mileage traveled on the round trip and divide by the total number of hours spent doing it.

Let’s suppose City A and City B are 210 miles apart. Driving that 210 miles on the way from A to B at an average of 70 mph took you 3 hours. Returning another 210 miles from B to A at an average 30 mph hour took you a whopping 7 hours. The total distance you drove was 210 miles out plus 210 miles back, or 420 miles. The total time you spent was 3 hours out plus 7 hours back, for a total of 10 hours. As a result, 420 miles ÷ 10 hours gives an average speed of 42 miles per hour for the round trip.

Now, most people’s “common sense” would probably have them objecting: Wait a minute, not so fast (which is a convenient play on words in an example about speeds). These people would assume the average speed depends on how far apart City A and City B are. Well, in fact it makes no difference at all how far apart City A and City B are. Pick any distance you like, do the same kinds of calculations I did (which may mean you’ll need to pull out a calculator because the numbers probably won’t come out so pretty), and you’ll still end up with an average of 42 mph for the round trip.

The reason the true round-trip average speed ends up below the “common sense” but wrong average of 50 mph is that you spent more *time* driving at a slow speed of 30 mph than at a fast speed of 70 mph, and that pulls the average speed down. In summary, the truth is that despite “common sense” you can’t generally average averages.

© 2021 Steven Schwartzman