The more logic puzzles you do, the more easily you parse the clues. Tackling statistical word problems is no different...

Reading to Solve

Sara Stoudt
Bucknell University

It’s summertime, and, at least for me, it’s prime time for nostalgia. Do you remember those logic puzzles you may have played as a kid? Like this one about matching job candidates to bad interviews:

These puzzles came with a series of clues, and you had to go through them, one at a time, to help you fill in this grid with Xs and Os. Take the first clue for this puzzle as an example:

Of Carlos and the person with the last appointment of the day, one was unfortunate enough to fall asleep during the interview and the other was applying for the IT Team Leader position.

There is a lot to unpack here. Let’s take it a little chunk at a time.

“Of Carlos and the person with the last appointment of the day”

That tells us that Carlos is not the person with the last appointment of the day, so that gives us an X at the intersection of Carlos and 2:00 PM.

“one was unfortunate enough to fall asleep during the interview and the other was applying for the IT Team Leader position.”

This tells us that Carlos did not both fall asleep during the interview and apply for the IT Leader position (and neither did the other person, once we finally know who that is). This is a little trickier to denote on the grid. My strategy might be to not yet cross off this clue because I still have some unused information left. As I fill in the grid, this leftover information might become more actionable. Solving these puzzles requires iteration!

Let’s try another clue.

“The five unique candidates are the person applying for the Production Lead role, the person who had the 11:00am appointment, the person who forgot their own name during the interview, the person applying for the Executive Assistant role, and Sally.”

This tells us a lot:

1. Sally did not apply for the Production Lead role.
2. Sally did not apply for the Executive Assistant role.
3. Sally did not have the 11:00am appointment.
4. Sally did not forget her own name during the interview.
5. The person who applied for the Production Lead role did not have the 11:00am appointment.
6. The person who applied for the Executive Assistant role did not have the 11:00am appointment.
7. The person who applied for the Production Lead role did not forget their own name.
8. The person who applied for the Executive Assistant role did not forget their own name.
9. The person who had the 11:00am appointment did not forget their own name.

These findings are all immediately actionable, so we can cross off this clue. We now have 10 Xs on the board.

And so it goes. Now… what does this have to do with statistics?

Well, often people learning statistics for the first time find that there is more reading and writing than they expect in a mathematical subject. Mastering the mechanics of introductory statistics concepts like $z$-scores, confidence intervals, and hypothesis tests is a hurdle in itself, but then one must be able to sift through a paragraph of text, translate it to a mechanical problem, and then often translate that number solution into a wordy answer. What does that number mean? And better yet, what does it mean in context?

That can seem like a lot to ask. And I get it. But some of the same skills you need to parse a word problem are the skills you use to solve logic puzzles. Annotation is an active reading strategy that can help.

An Interlude - What is Annotation?

Remi Kalir and Antero Garcia define annotation in their book of the same name, Annotation:

Annotation—the addition of a note to a text—is an everyday and social activity that provides information, shares commentary, sparks conversation, expresses power, and aids learning. It helps mediate the relationship between reading and writing.

If you only have a pencil or pen, you can underline, underline as a squiggle, circle, cross out, or put a box around parts of the text to distinguish elements. If you have multiple colors of pens, pencils, or highlighters, you can set up your own color code (I love a pack of sparkly gel pens for this exact purpose).

One of my favorite examples of annotation that is discussed in Kalir and Garcia’s book is annotation being used as commentary, here on moves in chess games.

! Good move

!! Brilliant move

? Bad move

!? Interesting move

?! Dubious move

I often find myself trying to map this simple but effective annotation strategy onto writing I’m evaluating in some way, whether when grading or in my own reading life.

Annotation can help you turn a block of text in the question into a series of “clues” that you need, not only to solve the problem, but also to understand what type of problem you are even dealing with.

Let’s try it out

Now let’s look at a word problem that I use as an example in my introductory statistics course and see one way of approaching it.

I was listening to my running playlist, and it made me wonder if it had the same pump-up energy on average as a workout mix curated by Spotify. It turns out that Spotify has a bunch of data on each song, including a measure of “energy”. I decided to compare a random sample of 50 songs from my playlist to Spotify’s gold standard. They report that all of the songs on all of their workout playlists have an average energy score of 0.75 (larger means more energy) with a standard deviation of 0.15. The songs in my sample have a mean energy score of 0.71 and a standard deviation of 0.81.

We don’t initially know what type of problem this is, because it doesn’t come out and say “Calculate a confidence interval” or “Complete a hypothesis test”. Instead, we need to see what kind of information we have and which words within connect to what we know about ways to do statistical inference.

I might start with a simple annotation strategy: circle any numbers that might come in handy, underline any phrases that might signal what type of question this is, and put a box around any phrase that gives me information that might help me check the conditions for inference.

I would start by circling 50, 0.75, 0.15,  0.71, and 0.81. Then I’ll want to label those with symbols, but to do that I will need to decide which go with my data (my sample) and which go with Spotify’s data (the population). [Bonus: how do I know it’s the population? All of those “all”s.]

• $n = 50$
• $\mu = 0.75$
• $\sigma = 0.15$
• $\overline{x} =  0.71$
• $s = 0.81$

We’re getting somewhere. Now what are we going to do with these numbers? Let’s see what I underlined.

wonder if it [my playlist] had the same pump-up energy on average as a workout mix curated by Spotify.

It looks like I’m comparing means: my sample mean with a “gold standard” known mean (with a known standard deviation). I could do this with a confidence interval (Does the interval around my sample mean contain the claimed mean?) or a hypothesis test (Do I reject the null hypothesis that the true mean energy level on my playlist is equal to Spotify’s gold standard mean in favor of the alternative hypothesis that they are not equal?). [Note the “not equal” decision comes from seeing “the same” appear in the underlined phrase above.]

And now I’m in business, because often the computation is not the hard part. I even get to pick which type of inference to do. Maybe I feel more confident in computing confidence intervals (the pun at least a little intended).

But wait, let’s make sure we are even allowed to do inference. What did I put a box around?

I decided to compare a random sample of 50 songs from my playlist to Spotify’s gold standard.

I don’t know anything about the distribution of energy levels across songs, but I do know the true population standard deviation. That means I can take advantage of the large sample size of 50 and trust the Central Limit Theorem to do its thing. The random sample also helps ensure that my sample is representative (not just my most or least energetic songs) and will help me generalize findings to my whole playlist.

Now, this is a pretty bespoke example, but the more examples you see, the more you’ll get the hang of it. Here’s another annotated practice problem:

How might you use this annotation and logic puzzle intuition in more general situations? Start by thinking about the types of problems you know how to do and the mechanics for each of those types. What components does each require (both in the computation and for checking conditions)? Those components inform what you should circle or box.

The underlining part can, admittedly, be the most challenging. The more logic puzzles you do, the more easily you parse the clues. Tackling statistical word problems is no different. It takes practice! Look at practice problems and examples. What types of phrases do the problems with a particular solution strategy have in common? Make a list of those “signal” words. They won’t always be exactly the same in each problem, but these lists should give you an idea of what to keep an eye out for.

And hey, doing some logic puzzles for fun can’t hurt either. They just might be the low-stakes way to practice close reading and connect words to findings.