Leo has gotten very interested in interesting data graphics. To be honest, he’s always been pretty interested in complex drawings; Since he was old enough to draw articulately, he’s drawn very complex mazes and maze-like pictures; never (or rarely) “artistic” sorts of things (like paintings). And he’s always been interested in the probabilities of things happening (like the likelihood of various things happening in his Minecraft play, for example, what’s the probability of an Enderman spawning with and ender-crystals … 1/420, FWIW — dunno!) And todayFiveThirtyEight published their annual favorite crazy graphics posting, which I showed him, and he was fascinated by, and went on to read the last 4 year’s worth as well!
So in today’s math session we started into “real” stats, with t-tests. Now, it turns out to be harder than you’d think to find a good example of something to run a hypothesis-testing experiment. I considered loading some dice, or making an unfair coin, or even just programming a load dice/coin. But it’s hard to make something that’s loaded in an interesting way. After a while thinking on this, I came up with what turned out to be a surprisingly good domain: The length of children’s vs. adult (or, at least older children’s) books. You’d think that it would be obvious that children’s books would be shorter, but, as it turns out, it’s not that simple. (Nothing ever is!)
We drew books from Leo and Ada’s bookshelves, where “adult” was a Leo book, and “child” was an Ada book. We also recorded whether the book was a “technical” (science or math) book, or not. (An example of an Ada technical book might be a counting book, or a book on butterflies. More on this later.) I tried to draw the book more-or-less at random, but lots of Ada’s books are so short and “child”-oriented that they don’t even have page numbers, and I wasn’t about to count them all up, so that almost certainly introduced a bias.
The experimental apparatus:
Anyway, we tabulated the child vs. adult books and computed some EDA (means and SDs):
I explained the t-test, and we looked at the equations, but we actually ran it on my go-to online quick stats site: Vassarstats:
Since we had a directional hypothesis (that children’s books were shorter), we can use the 1-tailed result of p=0.054, and claim the usual cheat of “marginal significance”.
So, this turned out to be way less clear-cut than either Leo or I thought it would be. We talked about some reasons that this might be the case. The top idea is the bias introduced by many of the shorter children’s books not having page numbers. Also, some of the longer children’s books actually had a number of stories under one cover. We actually thought about this when we were conducting the experiment, but decided not to change anything based on this because, after all, lots of the adult books have multiple chapters as well.
Oh….right, the technical book thing from above…I had the idea that we would “pivot” the data to look at the lengths of technical vs. non-technical books, and maybe even run a more interesting statistic, like a Chi-Square. Unfortunately, there weren’t enough technical books in the children’s category, so the statistics would have been badly skewed. Next time! 🙂