Not done with bluebonnet colonies yet

On April 9th we visited a new place, Turkey Bend Recreation Area in western Travis County. The bluebonnets were still thriving there, despite some unsightly trampled spots where people had obviously plunked themselves or more likely their kids down for pictures among the best-known Texas wildflowers.

In the upper part of the second picture you see Lake Travis, which was created in the 1930s by damming the Colorado River. Given central Texas’s propensity for both droughts and tremendous downpours that cause flash flooding, the water level in Lake Travis has fluctuated a lot. In some years the land on which these bluebonnets are now flowering was under water.

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Three posts back I noted that it’s common to hear politicians and activists bandy about the phrase “common sense.” I said 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. In that post and the next and yesterday’s I gave examples of “common sense” leading to incorrect conclusions. Here’s another, this time from baseball.

Let’s compare two players on a baseball team, Casey and Roger. When the team finished the first half of the season, Casey’s batting average was a whopping .387 (meaning he got a hit 38.7% of the times he was officially at bat). Roger’s batting average during the same period was almost as good at .375.

During the second half of the season (in its own right, not cumulatively from the beginning of the year), both players declined. In that second half of the season Casey batted .246 and Roger batted only .216.

Summarizing: in the first half of the season Casey out-hit Roger, and again in the second half of the season Casey out-hit Roger. Who had the better batting average for the season as a whole?

Almost everyone will say that because Casey outperformed Roger in the first half and also outperformed him in the second half, there’s no doubt that Casey ended up with the higher batting average of the two for the season as a whole.

But it’s time once again for me to say hold your horses, not so fast. In fact it’s easy to show how Roger could still have ended up with the better batting average, despite trailing in each individual half of the season. Here are three charts that do the trick (I’m sorry WordPress doesn’t seem to let me control the formatting the way I’d like).

First Half of the Season
– – – –	At-bats	Hits	Average = Hits ÷ At bats
Casey	31	12	12 ÷ 31 = .387
Roger	152	57	57 ÷ 152 = .375

Second Half of the Season
– – – –	At-bats	Hits	Average = Hits ÷ At bats
Casey	61	15	15 ÷ 61 = .246
Roger	51	11	11 ÷ 51 = .216

Season as a Whole
– – – –	Total At-bats	Total Hits	Average = Total Hits ÷ Total At bats
Casey	31 + 61 = 92	12 + 15 =27	27 ÷ 92 = .293
Roger	152 + 51 = 203	57 + 11 = 68	68 ÷ 203 = .334

So you see Roger did significantly better than Casey for the season as a whole even though Roger had a lower average in each individual half! This is an example of the very interesting phenomenon known as Simpson’s Paradox. What throws people’s “common sense” off here is that Roger had a lot more at-bats than Casey, especially in the first half of the season, when Roger was batting extremely well. You could say that the players were weighted differently. This is akin to the example a few posts back about average rates of speed while driving, where more time was spent at a slow speed than at a fast one. This baseball example is another one that shows you can’t average averages.

© 2021 Steven Schwartzman