Posted by: Gary Ernest Davis on: March 22, 2013

Karl PearsonÂ described histograms – and gave them their name – in 1895. As the cartoonÂ below suggests, peopleÂ still have troubleÂ interpretingÂ them today:

*Cartoon courtesy of www.whatthegregg.com*

In our experience, even universityÂ mathematicsÂ majors, studying statistics, have trouble with the meaning ofÂ histograms.

But histograms can be explained to children in elementaryÂ school.

For example, suppose the children know theirÂ heightÂ or have it measured for them in class – Â let’s say in inches, as in a U.S. school, where metric hasn’t really caught on yet.

Imagine the children areÂ enthusiasticÂ at knowing how the heights of all the children in grades 4, 5, and 6 in the school will pan out. They go from classÂ toÂ class measuring andÂ recordingÂ theÂ heightsÂ of all theseÂ childrenÂ – let’s say 200 of them.

The teacher asksÂ themÂ the smallest of all theÂ recordedÂ heightsÂ – let’s say it was 50 inches – and the largest – let’s say it was 64 inches.

The teacher asks the students what is the differenceÂ betweenÂ the largest and smallest heights, and they quickly figure it to be 14 inches.

The teacher says: “Let’s divide the 14 inch differenceÂ betweenÂ the largest and smallestÂ heightsÂ into nice even andÂ equalÂ lots. What should we choose?”

After some argument the consensus is 2 inch lots.

“Let’s see”, says the teacher. “That would be 7 lots of 2 inches. ” She asks one of the students to collect 7 plasticÂ containersÂ from theÂ storageÂ room, and then adds: “You had better get one more.”

The teacher writes 50-51 on a piece of paper andÂ sticky-tapes itÂ toÂ one of the containers.

“That’s our first bin”, she says. “I want you to look through your list of 200 heights, and every time you Â see aÂ heightÂ of 50 or 51 inches I want you to put a marble in this bin.”

One of the students goes to the store room to collect a bag of marbles.

“The next bin will have a note with 52-53 on it”, theÂ teacherÂ says.

“That’s for people whose height was 52 inches or 53 inches”, says one of the girls.

‘That’s right”, says the teacher. “Now can you finishÂ labelingÂ the bins andÂ puttingÂ marbles in them?”

The children go through the list and finally cross off everyone on it.

The teacher draw theÂ labelsÂ of the bins on the board like this:

**50-51 Â 52-53 Â 54-55 Â 56-57 Â 58-59 Â 60-61 Â 62-63 Â Â 64-65**

She asks theÂ childrenÂ how many marbles are in the binÂ labeledÂ 50-51. Â A boy looks in that bin and says there is only 1 marble. The teacher draws 1 cross above 50-51 on the board, like this:

“Now how many marbles are in the bin labelled 52-53?” she asks. The answer comes back: “Four”. The teacher draws 4 crosses above 52-53 on the board, like so:

“Do youÂ thinkÂ you can finish this?” she asks the class. “YES”, they exclaim, and set to, toÂ produceÂ the following chart on the board:

“Do you think you canÂ calculateÂ theÂ averageÂ heightÂ forÂ all those 200 names on your list?â€ she asks.

Two of the girls work onÂ theirÂ calculatorÂ and give her an answer of 57.985.

The teacher says she thinks that’s close enough to 58.

“This picture is called aÂ histogramÂ , she says. “It shows us Â in a picture how theÂ heightsÂ you recorded are spread out. What do you notice about the picture?”

The children raise their hands and sing out answers, and one of the girls says: “There are very few heights at either end. They mainly bunch up in the middle, around the average.”

“Yes they do, don’t they?” says the teacher. ” And theÂ histogramÂ allows us to see that very clearly, doesn’t it?”

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So if grade 4 students canÂ understandÂ histograms why do even educated adults have problems withÂ theirÂ interpretation? Is itÂ becauseÂ we’reÂ teachingÂ by formulas and rote procedures in college, withoutÂ understanding? Have we forgotten, or never knew, the simple human roots of histograms?

I meant to say that the kids were right about the difference. Unfortunately, that’s not exactly what we need to look at. If the heights had been just 50, 51, and 52 inches, we’d get largest minus smallest = 2. But clearly there are 3 different heights. With discrete data (usually whole number choices, instead of a continuum), we always need to use largest minus smallest plus one to get the right number of items.

If heights had been measure with fractions, so the first bin was 50-52 and the next 52-54, then the last would have been 62-64. (We’d still have to figure out what to do with heights exactly at the edge, like 52.)

1 | Sue VanHattum

March 23, 2013 at 1:30 pm

Reading a histogram seems simple to me, but you’ve demonstrated some of the issues that can come up in

creatingone. The kids said the difference between largest and smallest was 14, but there are 15 different heights. (This issue is sometimes talked about in relation to fenceposts, because for 10 feet of fencing with a post every foot, you’ll need 11 fenceposts.)When you divided 14 up into 7 equal groups, you got a group for 64-65 inches, which has an artificially small number of students in it. If you had used 5 groups, your histogram would be more accurate at the ends.