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Q&A #6065


Measures of central tendency; median

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From: Joshua (for Teacher2Teacher Service)
Date: Apr 04, 2001 at 19:48:53
Subject: Re: Measures of central tendency; median

Hi Jennifer,

It's always nice that students care about how you would use the math.
On the other hand, I do often ask my students "how do you use knowledge
of Shakespeare in real life?" or "how do use knowledge of impressionist
art in real life?" ... hoping to get them to realize that part of
the purpose of education is to get some breadth of perspective, new
ways of looking at things, learn how to appreciate beauty in forms
that I might not otherwise recognize, and so on.

Still, this particular question is one that has plenty of useful
real-life applications and doesn't need any reference to abstract beauty.

The first question you could ask is whether the mean or median would
be a fairer measure of a student's grade in your class.  That should
get some interesting debate going, since almost every teacher I know
uses the mean.  If everyone is thinking that the mean is the fairest,
ask whether someone who scores a 90 on every test, knowing each topic
at barely the A- level, should get the same, better, or worse grade
than someone who scores 100, 100, 100, 100, 50, knowing 4/5 of the
topics in the course perfectly and having some serious trouble with
the last 1/5.  Then ask the same question, if the student's mom died
at the end of the year, accounting for the one low score ...

Next, you can look at things like the 'average home price'.  Should
it be the mean or the median?  Well, in my neighborhood, there are
(let's imagine) 49 houses that sell for $250,000 and 1 house that sells
for $12,500,000.  What would you say is a good description of the
cost of a house in this neighborhood: $250,000 or $500,000?  That's
why the newspapers talking about the cost of a house use the median
home price instead of the mean: they want a good representation of
what the "typical" person pays.

Similarly, ask about the average salary in Seattle.

Suppose for simplicity that there are 250,000 people making minimum
wage, $13,000 per year; 500,000 people making a moderate amount,
$40,000 per year; and Bill Gates, making $10,000,000,000 per year.
Now, the mean of all that would be $44,333 per year.  Would it be
fair to say that "people in Seattle make, on average, $44,333 per year"
when in fact 750,000 people make less than that and 1 person makes more?

For an example that goes the other direction, consider these
sets of test scores for an algebra class:
Chapter 6: 50, 50, 80, 90, 100, 100, 100, 100, 100, 100
(median 100, mean 87) and
Chapter 7: 80, 80, 80, 90, 90, 90, 90, 100, 100, 100
(median 90, mean 90).

In which chapter would you say the class as a whole showed better
understanding?  To me, the median better captures the overall performance.
In the first case, chapter 6, the two very low scores dragged down the
mean too much when over half the class had it perfectly!  Some of
your students may want to argue differently, though, and that's OK;
then they will understand the reasons why you might want to choose
one of these measures over the other (in fact, the set of data I
provide here is one of few that might suggest the mode is the best
thing to look at.  If you give your kids these numbers, they might
end up suggesting it themselves!).

Obviously these examples are artificial and exaggerated, but the point
that one whopping huge number can distort the mean a lot is the reason
that the median is so often used in real life; when you want to
get a feel of what the "typical" person is like, it's a better measure
than the mean.

I think teaching the elements of statistics is an incredibly useful
thing in everyday life, and answering this particular question for
them (not "how do you calculate the median", but "how do you decide
whether the mean or the median is a better measure to use") will serve
them very well.  Of course, the latter question is much more abstract
and difficult and ambiguous than the first; perhaps at the age level
of your students, they won't quite be ready yet.  I have more experience
with high school and some middle school students.  I still think it'll be
good for them if you can get them to spend some time arguing about
which measure is better in addition to the time they spend calculating
each of the different measures.

I seem to have gone on at quite some length!  I hope my rambles were
useful to you.  Thanks for a fun question!

 -Joshua, for the T2T service

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