Mean, Median, Mode: Find Five Numbers

Date: 05/10/2001 at 22:24:39
From: Amanda
Subject: Describing Data

The median of five numbers is 15. The mode is 6. The mean is 12. What 
are the five numbers?

Date: 05/11/2001 at 13:41:00
From: Doctor Twe
Subject: Re: Describing Data

Hi Amanda - thanks for writing to Dr. Math.

Let's do a similar problem. 

Suppose we are told that the median of five numbers is 5, the mode is 
1, and the mean is 4. How could we find the five numbers? 

We can start by drawing a blank for each of the values. Then we'll 
try to fill them in, putting them in ascending order as we go. Here 
are the five blanks:

     __   __   __   __   __

Now, what do we know about the numbers? We know that "the median of 
five numbers is 5." The median is the middle number when arranged in 
ascending order, so let's put it there:

     __   __    5   __   __

What else do we know? We're told that "the mode is 1." That means that 
1 has to appear more than any other number. Since 1 is less than 5, 
all 1's will have to go to the left of the 5. We know we need at least 
two of them (otherwise, 5 would be a mode as well), so both blanks on 
the left will have to be 1's. Putting them in, we have:

      1    1    5   __   __

Now the last clue is, "the mean is 4." The mean is the "average" of 
the numbers. It is computed by taking the sum of the numbers and 
dividing it by the number of numbers. Algebraically, if we call our 
values A, B, C, D and E, we'd write:

     M = (A+B+C+D+E) / 5

Since we already know the first three numbers, let's plug them in for 
A, B, and C. We also know that the mean is 4, so we'll plug that in, 
too:

     4 = (1+1+5+D+E) / 5

     4 = (7+D+E) / 5

Now let's solve for D+E, the two numbers we don't know:

          4 = (7+D+E) / 5

      4 * 5 = (7+D+E)

         20 = 7+D+E

     20 - 7 = D+E

        D+E = 13

So the sum of the last two numbers must be 13. We also know that each 
of those two numbers must be greater than 5. What two numbers will 
work? Only 6 and 7 (can you think of WHY only 6 and 7 work?) So our 
five numbers must be:

      1    1    5    6    7

To check; median = 5 (check), mode = 1 (check), mean is:

     M = (1+1+5+6+7) / 5

       = 20 / 5

       = 4   (check)

Using this as a model, can you solve the original problem? Don't 
forget to check your answer!

I hope this helps. If you have any more questions, write back.

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