Data Set With Specified Mean, Median, Range

Date: 12/11/2003 at 18:52:58
From: Anita
Subject: How do I find the mean, median and range?

The problem is:  Make up a set of 5 numbers with a mean of 20, a
median of 10, and a range of 50.

So this is what I came up with for a mean:  

  1 + 3 + 4 + 5 + 7 = 20

Now my median is supposed to be 10 but if you look at my selections my
median is 4.  I cannot get a mean of 20 with 5 numbers and then get a
median of 10.  I don't get it.

Date: 12/11/2003 at 23:52:08
From: Doctor Jason
Subject: Re: How do I find the mean, median and range of a number

Hi Anita,

Thanks for writing to Dr. Math!

I would like to address the 4 parts of your problem in the following
order:

1.  A set of 5 numbers
2.  the median is 10
3.  the range is 50
4.  the mean is 20

The first part of the problem is pretty straightforward.  We can
create 5 blanks to provide a place to record our 5 numbers.


                _____   ______   ______   ______   ______

For the second part of the problem, we need a median of 10.  What this
means is that the middle value of the set, when the numbers are in
order from least to greatest, must be 10.  We can enter the number 10
in the middle position.

                _____   ______   __10__   ______   ______
 
Having a range of 50 means that when we subtract the smallest number
from the largest number, the difference must be 50.  For example, we
can make the smallest value 5, which means the largest value must then
be 55.  

                __5__   ______   __10__   ______   __55__ 

To satisfy the last requirement, we need have a mean of 20.  The mean
of a set of numbers is the sum of the values divided by the number of
values.  Since we know that we have 5 numbers and the mean must be 20,
we can calculate what the sum of our 5 digits must be:

                            sum of digits
                   mean = ----------------
                          number of digits

                                  ?
                     20 = ----------------
                              5 digits

Based on the equation above, the sum of our 5 digits must be 100.  So
far we have a sum of 70 (5 + 10 + 55 = 70).  The difference between
100 and 70 is what we have left to share between the two remaining
blanks.  

Remember that one of the blanks needs a value between 5 and 10, while
the other needs a value between 10 and 55.  I believe that there are
four different pairs of numbers that may be used to finish the problem
as we have started it.  There are many other possible solutions if you
start with numbers other than 5 and 55.

Can you take it from here?  

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