Choosing between Median and Mode as the Best Representation

Date: 03/02/2009 at 15:46:52
From: Henry
Subject: A question about measures of central tendency

I recently did a quiz relating to measures of central tendency (mean,
median, mode, range).  On it was this question:

The set of numbers are: 1, 3, 9, 10, 13, 15, 25, 39, 58, 63.  Which
measure of central tendency best represents the data?

I thought it was the mean but my teacher thought it was the median.

The mean of the numbers is 23.6
The median of the numbers is 14
There is no mode

I am unsure as to what measure of central tendency best represents the
set of numbers.  I believe it is the mean because it factors in all of
the numbers, including the very low and very high numbers, therefore
it is not distorted by any very high or very low numbers.

My teacher disagrees and thinks is the median.  My teacher believes
that the mean is distorted by the bigger numbers, however, I believe
that the median is distorted by the smaller numbers.

Date: 03/02/2009 at 16:40:34
From: Doctor Peterson
Subject: Re: A question about measures of central tendency

Hi, Henry.

I don't think there is a correct answer to the question.  It depends 
on your point of view and the purpose of your measurement, as you 
stated nicely in your last paragraph.  The question is, what do you 
mean by "represent"?

For example, suppose you listed the annual pay for all the employees 
of a company.  The mean would represent how much each would be paid 
if the total payroll were divided evenly among the employees.  That 
might be the natural way for the manager to summarize the pay, since 
the total payroll is important to him.  It well represents the effect 
of all the salaries on the bottom line.  But if a few have very high 
salaries while most are not paid much, the mean would make it look 
as if everyone earned a lot of money, "on the average".  The mean 
would be too greatly influenced by those "outliers".

The employees, on the other hand, might consider the median to 
better represent their average pay, since it would show how much 
an "average" (typical) employee made (focusing on the individual 
rather than the bottom line). 

So which is most useful or important depends on your point of view; 
and each contains different information.  The employees and the 
manager are both right; they just have different interests, and 
different ideas of what it means for an "average" to represent the 
data.  (It just happens that the mean also favors the manager in 
making the company look generous, while the median favors the 
employees by making them look underpaid.  But I don't think that's 
why they would tend to make the choices they would!)

In your case, your teacher seems to be more like the employees, 
focusing on the individuals, while you are more like the manager, 
thinking of the whole.

One way to clarify this is to diagram the whole distribution to get 
a better sense of how the numbers relate to one another.  If there 
were more data I'd use a histogram, but I'll just use a "dot plot":

   o o   oo  oo       o          o              o    o
  +-------+-------+-------+-------+-------+-------+-------+
  0      10      20      30      40      50      60      70
              ^      ^
            median  mean

Both measures are very much in the middle; the median is more "in 
the middle" of the individuals, while the mean is more "in the 
middle" of the whole.

My inclination is to agree with the teacher, if I had to take sides; 
the median also seems closer to what the mode would be if you were 
to group the data, since it is densest around 9-15.  But I also 
notice that this is not quite the kind of situation I described for 
employees, where a very few numbers are far above all the others. 
It's hard to say that the many numbers clustered toward the low 
end "distort" the median (which is where most of the numbers are, 
anyway), or that the numbers smeared out toward the high 
end "distort" the mean (which is not so far away from the median). 

So I still don't really know what it means to "best represent the 
data" without a context!  This sort of question works better as an 
essay topic than a multiple choice.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/