Timers and Variability II

So what happened in class? First, you want to see the data, right?

Timer data. Stripes indicate groups.

The basic story so far is that maybe a week ago, I let the students take the measurements, uploading the data—so we could all get everyone’s measurements—using Fathom Surveys. That worked great, but there was of course not enough time to do the analysis, so that got postponed.

And we still haven’t quite gotten through it—though they have had a couple dollops of homework to make progress—at least partly because I’m not sure the best path to take. Next class—Thursday—I finally have enough time allocated to do more, and get to the bottom of something about variation; the next step in this thread is to do The Case of the Steady Hand.

So what actually happened and why am I a little at sea when the data are so interesting?

Let’s begin with the lesson. The kids took the timers and mostly did what they were supposed to do. Beforehand, we discussed what data we would have to collect in order to share the data well; we had three variables in each case: a group name, the number of minutes, and the number of seconds.

Digression: Why minutes and seconds separately? Fathom won’t accept a string like “1:22.85″ and interpret it as minutes, seconds, and hundredths. One work-around is to have the students mentally add 60 and simply record seconds. In this case, that’s not bad. But as a general practice, I believe (and explained) that it’s best to record what the data actually are, as closely as possible, and let the software make any calculations or corrections. If you’re measuring a distance and because of the baseboard (say) you have to add 3/4”, record your actual measurement and then add the 0.75 in a separate column. This avoids all kinds of screwups when you mis-add or forget whether you made the correction or not. End of digression.

Except to say that it also gives us the chance, when doing the analysis, to learn how to make a new column and write the all-important time = 60 * minutes + seconds formula.

Anyway, they took the data and recorded it, and were able to download the whole class data into Fathom. It was obviously enjoyable; something to do that (a) didn’t require a lot of deep thought and (b) involved the computers. So you could kind of chill out for 25 minutes and still accomplish something. The technology worked great, too.

As they were doing this, I noticed that the “alternation” phenomenon was present in other timers. Some observations:

  • At least one group noticed it, and immediately realized that this gave you a great chance to get an advantage in some game. I thought they would publicize it, but they didn’t. I’m looking forward to getting them (or any other group that noticed) a chance to describe their discovery.
  • Several groups, when they did data entry, only entered the seconds in that column, not the hundredths! I had no idea they would leave it out. Amazing. This will make it hard to analyze the variation in their data (when they look at the sand going only one direction).
  • One group, despite all we had done to prevent it, entered the data in “1:21.85” format. This gave us the chance to clean the data set. (I insisted that they re-enter it correctly; then the others just had to eliminate the bogus values.)
  • One group was (is) convinced that if you kind of slam the timer down when you start, it goes faster. I had thought someone would try tapping to make it speed up, but the slam technique was great. “Really?” I asked. “Slamming makes it go faster?” “Yes!” “How do you know?” “We proved it!” “How?” “We did it both ways, and it was always faster on the slam.”

This last one is especially wonderful, as you will recognize if you’ve been following along, and is at the root of why I’ve been a little paralyzed. It’s so great, so creative, and so problematic! What should I do to get them to discover their luscious problem in experiment design? What a great situation in which to see why we use randomization!

But writing this, and reflecting on what I’ve been telling myself (the reason for writing, after all) I see that really I shouldn’t bring it up now at all. It would just overload the activity if I do; if they do, that’s another story. Besides, experiment design is later. We can revisit the situation—and we will have saved the data—when we address that topic.

What I can do, however (and what I will do Thursday) is get them to reflect briefly on their procedure, and then discuss and come to some decisions and refinements about questions I asked in the homework, which were essentially:

  • Suppose you took your own timer and timed it again. What’s a range of values you’re pretty sure it will be in?
  • Now answer that question if instead of your own timer, you picked one at random out of the class box of timers.

I also want them to look at what happens when you try to make the range narrower. When are we willing to accept some timings outside the range? So, for example, I want them to discuss what would happen if we published the range as the interval [Q1, Q3], namely, the width of the box in a box plot (which we have been studying). This could also lead (especially with the all-class data) to looking at percentiles: what if we wanted our interval to capture the next measurement 90% of the time? (Then we would find the 5th and 95th percentile.)

Seeing the graph (way up at the top of th post) suggests another, perhaps simpler task: find your group’s timer in the class set. (This assumes that the timers do not change much over time. I have some sketchy evidence to suggest that they do…)

In any case, you can see that some of the timers are much closer to being the same in both directions, whereas one timer alternates with a difference of 15 seconds! Then, O stats teachers, we’re faced with the question, what’s the best representation for data like that? A box plot is a little iffy, no? Maybe this is a place to talk about the limitations of summary plots; after all, the dot plot—like the one above—tells the whole story.

I’ll be sure to fess up to whatever happens Thursday.

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Tim Erickson

Math-science ed freelancer and sometime math teacher. In 2014–15, at Mills College in Oakland, California.

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