Portraits of Wildflowers

Perspectives on Nature Photography

Not done with bluebonnet colonies yet

with 19 comments

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.

* * * * * * * * *

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-batsHitsAverage = Hits ÷ At bats
Casey3112 12 ÷ 31 = .387
Roger15257 57 ÷ 152 = .375

Second Half of the Season
– – – –At-batsHitsAverage = Hits ÷ At bats
Casey6115 15 ÷ 61 = .246
Roger5111 11 ÷ 51 = .216

Season as a Whole
– – – –Total At-batsTotal HitsAverage = Total Hits ÷ Total At bats
Casey31 + 61 = 9212 + 15 =27 27 ÷ 92 = .293
Roger152 + 51 = 20357 + 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

Written by Steve Schwartzman

April 11, 2021 at 4:45 AM

19 Responses

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  1. Great capture of the bluebonnets showing off their beauty, Steve! I am astounded to learn that they would survive after their colony has been underwater. The water level of the Arrow Lakes (the dammed-up Columbia River) also fluctuates a lot. Only grasses and a few hardy flowers survive.

    Peter Klopp

    April 11, 2021 at 8:28 AM

    • Seven years ago I was amazed to find large colonies of bluebonnets flowering at an adjacent recreation area on land that had clearly been underwater, and which I saw go back underwater in later years after we’d had lots of rain. Somehow the seeds had remained viable, or else new seeds were blown or carried in from elsewhere. This year that adjacent place doesn’t have masses of bluebonnets, while Turkey Bend does, as you see in today’s post. So much in nature is hit and miss.

      Steve Schwartzman

      April 11, 2021 at 9:44 AM

  2. I remember the videos of planes pulling water out of Lake Travis during the Bastrop fire, and I remember how low the water level was. These are far more pleasant views. There’s ongoing grousing in some areas about the LCRA’s policies, but they’ve done a good job of creating spots for recreation — and photo taking — along the river. It’s interesting to see the juxtaposition of bluebonnets and water, although I suppose in your area it’s not at all uncommon.


    April 11, 2021 at 8:30 AM

    • You lost me with this baseball example. Let me go get some peanuts and Cracker Jacks, and make another run at it. Maybe I can improve my average comprehension!


      April 11, 2021 at 8:35 AM

    • I hadn’t thought about those planes since 2011 till you mentioned them here. In dry years Lake Travis has gone below 1/3 full, which begins to get scary, as that’s where Austin gets its water. You’re right that because of the Highland Lakes it’s not at all unusual in this part of the state to see bluebonnets near a body of water. I wasn’t able to turn up any photographs of bluebonnets by the Gulf of Mexico.

      Steve Schwartzman

      April 11, 2021 at 9:53 AM

  3. Dazzling view!

    Eliza Waters

    April 11, 2021 at 9:22 AM

  4. I have caught up again now, and enjoyed your lovely photos and educational posts. Not enough time is spent in school on what was called “word problems” back then.

    Lavinia Ross

    April 11, 2021 at 11:57 AM

    • Thanks, Lavinia. You see that I’ve found plenty to photograph, even thought the sustained freeze in February delayed most wildflowers.

      Many of the old math word problems were contrived, but not all. It’s easy to create more realistic ones.

      Steve Schwartzman

      April 11, 2021 at 4:42 PM

  5. What a beautiful spread of Bluebonnets! I really love the flow of the landscape in both of your images. Bluebonnet seeds have an extremely hard exterior. The only way I could get any to grow here years ago was to soak the seeds a couple of nights and then graze them gently on a rough surface. So it does make sense that they would germinate well, having been soaked a good bit under water.

    While I am no math whiz, I did enjoy what we called “story problems” in school. Somehow, having a scenario to work out, made the problem solving more enjoyable.


    April 13, 2021 at 8:54 AM

    • What made the first bluebonnet colony special for me was the sandstone ridge in back; I don’t remember ever seeing bluebonnets in a place like that. It wasn’t in the main part of the park, where the bluebonnets are at their most expansive, but off to one side of the entrance road shortly after the booth where we paid.

      As you said, the presumed long soaking of the seeds seems to have had a beneficial effect, as it did in an adjacent park on the lake that astounded me with the extent of its bluebonnets in 2015.

      As for math, arithmetic seems to have sprung up originally as a way of dealing with “story problems” like buying and selling of goods, taxation, and the like. Later some very smart people abstracted away from that to create “disembodied” mathematics as purely a pursuit of the mind.

      Steve Schwartzman

      April 13, 2021 at 1:53 PM

  6. I’m really impressed with the huge fields of flowers you’ve been sharing lately. I don’t think I’ve yet found anywhere here with such impressive displays. I really love seeing these.

    Todd Henson

    April 13, 2021 at 6:46 PM

    • You’ve been seeing why Texas is duly famous for its wildflower displays, especially in the spring. Even so, I can’t imagine how much better those displays must have been 200 years ago, before agriculture, ranching, and finally commercial development destroyed so much of nature.

      Steve Schwartzman

      April 13, 2021 at 8:18 PM

  7. Why be done when they are still offering such glorious views?

    Steve Gingold

    April 16, 2021 at 4:03 AM

    • This was a week ago, so I imagine fading has begun now. In compensation, the Indian blankets are beginning to come into their own.

      Steve Schwartzman

      April 16, 2021 at 6:58 AM

  8. WOW … what a beautiful sight!


    April 16, 2021 at 12:14 PM

    • That’s Texas for you. Just about every year there are at least some places where the bluebonnets look this good.

      Steve Schwartzman

      April 16, 2021 at 2:22 PM

  9. […] vast colonies of bluebonnets. This turned out not to be an expansive year for them there (we found broad stands at Turkey Bend on the way home), but the white prickly poppies (Argemone albiflora) along the Willow City Loop […]

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