# Probability and Statistics

# Median

The **median** of a list of numbers is the "middle" number. To find the median, first we put our list of numbers in order. Then we cross off pairs of numbers (one from the top of the list and one from the bottom of the list, or one from each end of the list, depending on how we've written it out) until we're left with one number. That number is the median. If the median was a contestant on Survivor, he would totally be taking home that $1,000,000 prize.

### Sample Problem

Find the median of the list

3, 4, 6, 5, 7, 10, 11.

First, we put the numbers in order:

3, 4, 5, 6, 7, 10, 11.

Then we take off one number from each end of the list:

4, 5, 6, 7, 10.

We do it again:

5, 6, 7.

and one more time:

6.

We're left with the number 6, so that's the middle number in the list. The median is 6. The tribe has spoken.

So far, all the lists we've looked at have had an odd number of entries. We crossed off one number from each end of the list, and repeated until we were left with one number, and we were always left with *exactly* one number. However, if our list has an even number of entries to start with, we get two middle numbers. Keeping with our *Survivor* analogy, they'd probably need to fight each other to the death to determine a clear winner. But since this is a family example, let's instead find the median between these two middle numbers.

To find the median, we take the mean (or average) of these two numbers.

### Sample Problem

Find the median of the list

2, 3, 4, 8, 9, 10.

We cross off the first and last number:

3, 4, 8, 9.

Then we do it again:

4, 8.

Now we have only two numbers left. If we cross them both off we'll have no numbers left, which won't be useful for anything, aside from getting rid of all those noisy numbers so we can finally get a few minutes of shut-eye. Instead, we take the average of the two numbers we have left:

The median of the list is 6.