Range, Mean, Median, and Mode

Date: 11/17/98 at 18:16:32
From: Stephanie R. Wallace
Subject: How do you understand "Range, Mean, Median, and Mode?"

I have some questions that you may want to answer for me:

1. Why do we have to study range, mean, median, and mode?
2. Could you help me understand them more?
3. How is it going to help me later in life?

Date: 01/08/99 at 14:16:21
From: Doctor Stacey
Subject: Re: How do you understand "Range, Mean, Median, and Mode?"

Hi Stephanie!  Thanks for writing Dr. Math. 

I'm going to wait to talk about range for a moment and concentrate on 
mean, median, and mode. Mean, median, and mode are all types of 
averages, although the mean is the most common type of average and 
usually refers to the _arithmetic mean_ (There are other kinds of means 
that are more difficult).

The arithmetic mean is a simple type of average. Suppose you want to 
know what your numerical average is in your math class. Let's say your 
grades so far are 80, 90, 92, and 78 on the four quizzes you have had.  
To find your quiz average, add up the four grades:

   80 + 90 + 92 + 78 = 340

Then divide that answer by the number of grades that you started with, 
four: 340 / 4 = 85. So, your quiz average is 85! Whenever you want to 
find a mean, just add up all the numbers and divide by however many 
numbers you started with.

But sometimes the arithmetic mean doesn't give you all the information 
you want, and here is where your first and third questions come in.  
Suppose you are an adult looking for a job. You interview with a 
company that has ten employees, and the interviewer tells you that the 
average salary is $200 per day. Wow, that's a lot of money! But that's 
not what you would be making. For this particular company, you would 
make half of that. Each employee makes $100 per day, except for the 
owner, who makes $1100 per day. What? How do they get $200 for average 
then?!  

Well, let's take a look:

Nine employees make $100, so adding those up is 9 x 100 = 900. Then 
the owner makes $1100, so the total is $1100 + $900 = $2000. Divide by 
the total number of employees, ten, and we have $2000/10 = $200. 
Because the owner makes so much more than everyone else, her salary 
"pulls" the average up.

A better question to ask is, "What is the _median_ salary?" The median 
is the number in the middle, when the numbers are listed in order. For 
example, suppose you wanted to find the median of the numbers 6, 4, 
67, 23, 6, 98, 8, 16, 37. First, list them in order: 4, 6, 6, 8, 16, 
23, 37, 67, 98. Now, which one is in the middle?  Well, there are nine 
numbers, so the middle one is the fifth, which is 16, so 16 is the 
median.  

Now, what about when there is an even number of numbers? Look at the 
quiz grade example again: 90, 80, 92, 78. First list the numbers in 
order: 78, 80, 90, 92. The two middle ones are 80 and 90. So do we have 
two medians? No, we find the mean of those two: 80 + 90 = 170, and 
170 / 2 = 85. So 85 is the median (and in this case the same as the 
mean)!

Now look at those salaries again. To find the median salary, we look at 
the salaries in order: 100, 100, 100, 100, 100, 100, 100, 100, 100, 
1100. This is an even number of salaries, so we look at the middle 
two. They are both 100, so the median is $100. That's much better at 
telling you how much you'll make if you accept the job.

But the median doesn't always give you the best information either. 
Suppose you interview with a company that has 10 general employees, 7 
assistants, 3 managers, and 1 owner. For this company, the mean salary 
is $400, and the median is also $400. But you are applying for the 
position of general employee, whose starting salary is $100!  Why are 
the mean and median so far away?

Well, the 10 general employees each make $100. The 7 assistants each 
make $400, the 3 managers each make $900, and the owner makes $1900.  
If you do the math to find the median or mean, $400 is the answer (try 
it!). So what can you do?  

The mode is the type of average you want to know in this situation. 
The mode is the number the occurs most frequently. In the example for 
median, 6 would be the mode because it occurs twice, while the other 
numbers each occur once. In our employee example, the mode is $100 
because that number occurs ten times, which is more than any other 
number occurs.

Now, mean, median and mode are all good types of averages, and each 
works best in different types of situations. Knowing all three is a 
good way to know what kind of data you're looking at. But another good 
thing to know is the range. For that first company, if the interviewer 
had only told you that the salary _range_ was from $100 to $1100, you 
might have figured out that you would be making $100. Similarly with 
the second company example.

I hope this gives you some good information about why we use all these 
different words, and how they can be important to us. Feel free to 
write back with any further questions.  

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