Apples and Oranges

We compared apples and oranges in statistics today.

The teacher’s resource manual for the textbook comments that standard deviation lets us compare apples and oranges, because we can use it as a “ruler” to figure out how a particular data value compares with the mean. I took that idea and ran with it – to the grocery store, during my 4th period prep. I got 9 apples and a bag of clementines (there were 20 of them in the bag).

We put each piece of fruit on a paper plate with a letter on it (we had to use three Greek letters). We borrowed a balance from a science teacher. Before weighing the fruit, the class made conjectures about which apples would be biggest and smallest, and which oranges would be biggest and smallest. (They did know that our method of measurement would be using the balance.) I told them that in making those decisions, they were comparing apples with apples and oranges with oranges, but what if they wanted to determine which was relatively bigger, the biggest apple or the biggest orange? That, I said, was what they were going to figure out how to do today.

After they made their conjectures, they wrote the name of each piece of fruit on the board. (They decided the fruits needed names, so instead of boring old A, B, and C, we had Amy, Billy, and Caroline as our first three apples. I’ve studied Greek, so I was able to come up with some Greek names for those extra letters.) Then they recorded the mass of each piece of fruit.

The next part was easy: For each type of fruit, which piece was the biggest, and which the smallest? It turned out that the biggest apple was Francis (212.1 g), which had been their guess, but that was the only one they guessed correctly. Caroline was the smallest apple (144.4 g), Wilhelm the biggest orange (84.6 g), and Violet the smallest orange (52.1 g).

Then I told them to figure out the mean and standard deviation for each type of fruit. Doing this REALLY helped some of the students to better understand what standard deviation is (a measure of spread); I was able to point out that the standard deviation of apple masses (19.72) was more than twice that of orange masses (8.26), and the students were able to look at the actual fruits and see that yeah, there’s a lot more variation in size in the apples than in the oranges. They had been pretty confused by it before, so I was really glad to have that visual for them.

Once we had that concept a little more firmly understood, I said that we should look at Francis (biggest apple) and Wilhelm (biggest orange). I asked how we could figure out how many standard deviations above the mean Francis is, and they knew right away how we could do it. We did the same for Wilhelm, and we discovered that Francis was 0.981 standard deviations above, but Wilhelm was 1.746 standard deviations above. So even though Wilhelm is SMALLER than Francis when we look at the measurements themselves, Wilhelm is bigger as a big orange than Francis is as a big apple. We did the same thing with the smallest ones, and I pointed out that since we were subtracting the mean from smaller data values, we were ending up with negative answers…which just show us that we’re that many standard deviations below the mean. They got it.

And then I told them that these “how many standard deviations away” things they were coming up with are called z-scores. They were excited that “it has a cool name.” :) And I’m excited that I think they will actually remember it because of how we got there – they figured it out rather than having me throw a formula at them.

And then? We had a healthy snack to conclude the class. :) (My homeroom may or may not have juggled the leftover clementines right after the statistics class left.)

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