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# The average is whichever thing has the most

I don’t have a picture for this, but every single one of my 4th Graders thought that “average” meant “the most common thing.”

(a) Where do they get this idea from?

(b) Is it a big deal misconception?

(c) How do you create a need for something besides “most common”?

(I think I have my own answers for (a) and (c), but I’m more curious to know what you guys all think.)

## 8 replies on “The average is whichever thing has the most”

a. Maybe it comes from how students have heard the term before.
b. After thorough teaching and practice it seems to go away, so I think it isn’t.
c. I haven’t worked at that level enough to give a good answer, but I think batting averages, shooting percentages or some other known average might help.

a. That sounds like mode.
b. I don’t think I can ponder this well without examples of how they were dealing with data. My initial reaction is ‘no’ as ‘most common’ doesn’t seem like a horrible start to thinking about average, it just has some flaws because mode isn’t always useful.
c. Dice? Though perhaps that got them into this as 7 is the most common roll of two dice and the mean. Something without an even distribution?

mpershansays:

The context was trying to figure out how many moves the game Mastermind usually takes to win. We had played a bunch of games and collected all the number of winning moves and made a line plot. Then I asked the kids to tell me how many moves Mastermind typically takes, and they said “5! Becaues it’s the average.”

Yeah, I pretty much agree with Andy and Ashli.

Mode is one type of average. Sounds like they don’t understand that there are others, too (median and mean). That’s kind of a big-deal misconception, but not a bad first-order approximation for understanding averages.

Maybe you could show them two data sets (say, number of pets that people have) with the same mode, but obviously different means (like 0,1,1,2,2,2 vs. 2,2,2,5,7), which might prompt them to see the need for different ways of talking about central tendency.

mpershansays:

I ended up making a bunch of different line plot distributions and asking them to find the typical number in each. It went really well — look for a post up on Rational Expressions over the next few days. (Or hours, if I can finish these stupid med-term reports…)

Michael Paul Goldenbergsays:

Seems pretty reasonable to me for the age/grade level. If you haven’t been explicitly taught “arithmetic mean” as either THE average or AN average, and if you in fact have really only heard the word used in non-mathematical contexts (or mostly so, or in ways that didn’t alert you to the possibility that there could be more to this than meets the eye), then what might you be inclined to think? “He’s an average guy” doesn’t really tell you much about average or guys, particularly not if you’re in 4th grade (which means you’re about 9 years old). “It’s a warmer than average day for July” might be better, if you already knew the idea of mean, but if not, the mode (regardless of whether you know that term) is a reasonable way to think about what’s being said. Even things like “What was your average for those quizzes?” requires that you already know in advance what is meant. I am too old to recall if at that age I knew what means were, knew how to compute them (if the first was true, then I’m confident I could deal with the second for non-weighted means).

Following sports can be a key way for some kids to learn about statistical ideas, and I was definitely a big sports fan, but I’m not sure at what age I really started following any sport very closely, including getting into things like batting averages seriously. I would guess somewhere between age 10 and 12.

Bottom line: I see nothing shocking here. Further, I doubt it’s a huge problem. Any given kid eventually will start getting some sort of formal instruction about measures of central tendency. She may at that point already know mean, median, and mode, know how to find them in a given set of data, They’re really different (at least in principle) and as suggested, it’s important for students to look at sets of data that bring out some of the properties, strengths, and weaknesses of each. Certainly knowing that mean is the least robust of the three is going to be important, and it’s very easy to demonstrate. But is that something 4th graders must master?

Voting situations (favorite flavor of jelly bean) are natural situations to illustrate where the mode is what matters. Situations where the mode salary at a company is far more informative than either the median or mean (the vast majority of entry workers make, say, \$35K, but because of some outliers, the mean salary is considerably higher (though probably the median would work here, too).

And at some point, Simpson’s Paradox is so fun to look at. There are many good real world examples, one of which is baseball batting averages, where one player’s two-year average may be higher than another player’s, even though the second player has a higher average than the first for each of the individual years (which shows a lot about the danger of trying to “average” averages and the importance of understanding weighted averages).

Dave Nsays:

I think I agree. Kids will encounter the word “average” in everyday use before they are familiar with the idea of division. So, when they ask an adult what it means, the adult will probably explain it in vague or non mathematical terms. “Average weather means the most common weather.” could definitely lead to a response like we saw here. Other informal uses of the word could lead to other responses: “Average ability means normal — not the best and not the worst.” “Average meal means the right amount for most people.” “Batting average means how many times they hit the ball.”

I support that anytime we introduce a math term that also is used commonly outside of math, it’s important to take some time and talk about how the math usage is specific and can’t be lumped together with their previous understanding of the word.

“Mode” is an average. So they are not wrong if the question asks for the “average”.