Defending the Mean

Date: 09/07/2003 at 16:52:59
From: leigh
Subject: Mean

I've been asked by one of my students to explain why it makes sense to
add the sum of the numbers in a data set and then divide by the
quantity of numbers when finding the average or mean.  

Date: 09/07/2003 at 19:57:20
From: Doctor Achilles
Subject: Re: Mean

Hi Leigh,

Thanks for writing to Dr. Math.

That's an excellent (and thoughtful) question.  I believe that when 
you take time to work through the math that is going on, it is 
intuitive, but I'm not sure if I can give a more principled 
argument.  Let me work through what I mean.

Let me explain what I mean by this using students and apples.  Let's 
say that we have a set of students, and each student has some number 
of apples.  Some students may have more apples than others.  But I 
want to know how many apples a typical student might have.

One way to think about this "typical" student is this: If another 
student walks into the room, how many apples would you guess she 
has?  Let's take a very simple case: we have 20 students and 
every single one of them has 8 apples.  If I say another student just 
arrived, how many apples would you guess she has just based on what 
you know about the 20 students who are already there?  You'd probably 
guess she has 8, wouldn't you?  You might not be shocked if I told 
you she has 7 or 9, but if it turns out she has 17 or 20, you would 
probably say that there must be something different about her that 
caused her to bring more apples.

There are three intuitive ways I can think of to define 
this "typical" student.  The first is to look and see if a lot of the 
students have the same number of apples.  For instance, if we have 4 
students and 3 of them all have 7 apples and the 4th student has 9 
then it might make sense to say that a typical stduent has 7 apples.  
This, as you probably know, is called the "mode".  For a little more 
information about modes which is not really relevant to this 
discussion but may be interesting to you or your students, check out 
this page:

  More Than One Mode?
  http://mathforum.org/library/drmath/view/61375.html 

Another way you might define the typical student is to line up all 
the students in order from the one with the fewest apples to the one 
with the most, and then say that the one in the middle of the line is 
the "most typical".  I personally find this the least intuitive of 
the 3 ways to definte a typical student, but it is certainly a way 
you could do it.  As you probably know, this is called the "median".  
This is really, by definition, not so much an assessment of what is 
typical of the population as a whole as just what is actually true of 
the one individual in the middle of the population.

So what about the mean?  The advantage that the mean has over the 
median is that it tells you somthing about the population beyond just 
what is at the middle.  The advantage that the mean has over the mode 
is that the mode only works if there is one number which occurs more 
than any other.  Also, modes only tell you about the largest part of 
the population but ignore any minority groups.

Let's start with an easy case for why we use the mean.  Let's say 
that there are 2 students, one with 7 apples and the other with 9. 
What is a typical student in that group?  Well, it seems 
intuitive to me (before I really even do any math), that a typical 
student should be halfway between the two students in our 
population.  For me, it just plain makes sense to say that a typical 
student has 8 apples in this set.  No other number represents the set 
well.  You could say 7 is typical, but that ignores the student with 
9 apples.  You could say 9 is typical, but then you have an identical 
problem.  You could say 10 or even 0 or 20, but none of these numbers 
are even close to 9 or 7 and you would be downright nuts if you said 
any of them were typical of this set.  You could maybe try saying 
7.5, that doesn't completely ignore the student with 9 apples, but it 
means you're paying more attention to the student with 7.  Similarly 
8.5 doesn't completely ignore the student with 7, but it pays more 
attention to the student with 9.

Let's move to a slightly more difficult case.  Let's say we have two 
students with 7 apples each and one student with 10 apples.  What is 
typical in this group?  Well, you could say 7 is typical (the mode) 
but that ignores completely the student with 10 apples.  You could 
say 8.5 is typical (halfway between 7 and 10) but there are two 
students with 7 apples and only one student with 10 apples, so 
shouldn't we pay more attention to the 7 apples?  What about 8, is 
that a reasonable compromise?  Sure, it is only 1 away from 7 and it 
is 2 away from 10; it make perfect sense for it to be twice as far 
from 10 as it is from 7 because there are twice as many students with 
7 apples as there are with 10 apples.

You can make up more examples, but if you work through what seems 
like a fair way to find a typical student in each group, I bet you'll 
always come up with the mean.

Another way to think of the mean is this: if every student put his or 
her apples into one big pile (take the sum of all the apples) and 
then we passed out an equal number of apples to each student (divide 
by the number of students), how many apples would each student have?  
Exactly the mean!  So the mean is a good way to think about what the 
group would look like if everyone redistributed their apples so that 
all the students had the same number.

I hope this helps!

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

Date: 09/07/2003 at 21:10:05
From: leigh
Subject: Thank you (Mean)

Thank you very much for your quick responce.  That will be 
helpful tomorrow in my math class and to answer my 
student's questions.  In college, we are taught how to 
teach to calculate the mean, but not why, so thank you very 
much.

Leigh