Finding the Median Value

The median is the middle value of a list of data values. To find the median of a list of data values:

1. Place the numbers in increasing or decreasing order by value.
2. Identify the number that appears in the middle position in the sorted list.

Example 1

Find the median of the list

Solution: Start by sorting these scores into increasing order. The sorted list is:

As there are five values in the list, the middle value is the third one in the list. So, the median is 6.

Example 2

Find the median of the list

Solution: Start by sorting these scores into increasing order. The sorted list is:

Now identify the middle value. There are 17 numbers in this list, so the median is the 9th value, or 19.

The repetition of values in the list (e.g. 15) does not affect the way we calculate the median. We simply include the repeated values in our sorted list.

What About Lists With Even Numbers of Elements?

If a list has an even number of elements, we can still find the median, even though we can't identify the middle element exactly.

When a list has an even number of elements, it has two middle numbers. The median is defined to be the average of these values (add the two numbers together and divide by 2).

Example 3

Find the median of the following list of values:

This list has an even number (18) of entries, so it has a pair of middle values. Let's write the list in increasing order:

The two middle entries are the 9th and 10th entries (19 and 21). So, to find the median of this list, we add them together and divide by 2 to give

The median of our list is \(20\).

Finding the Median of Longer Lists

Often, the most challenging part of finding the median is identifying the middle value in the list. However, there's an easy trick to get you out of trouble. Simply count the number of elements in your list, add one, and then divide the result by 2.

For example, if a list has \(253\) elements, the middle element will be the \(\dfrac{253 + 1}{2} = \dfrac{254}{2} = 127\)th element in the list after it has been sorted into increasing order.

But, what if the list has an even number of elements? That's OK. Let's suppose we have a list with \(366\) elements. When we apply our little trick, we end up with the middle element being element number \(\dfrac{366 + 1}{2} = \dfrac{367}{2} = 183.5\), which doesn't exist. However, 183.5 is halfway between 183 and 184, and we know that we have to find the average of the corresponding list elements to find our median. So, to find the median of the numbers in this list, we sort the list into increasing or decreasing order, identify the 183rd and 184th elements, add them together and divide the result by 2.

When is the Median Useful?

Mrs Fitzsimmons asks Sam's class to evaluate her teaching of Shakespeare. She receives an average score of 5.36 out of 10 from her 11 students, which makes her pretty happy until she decides to look at the data more closely. Then she sees that the 11 scores were 1,3,3,3,3,3,5,9,9,10,10. The median score of this list is a not very impressive 3 out of 10. This tells her that at least half of her students were not very happy with her Shakespeare teaching. The average hides this fact because the unexpected 9s and 10s (possibly from students who were trying to get a higher mark on their exam, or maybe they really liked Shakespeare) skewed the data, making the average higher than it should have been. In this case, the median is a much better choice for data analysis.

If you've ever looked at the property pages of a newspaper, you might have noticed that median house prices are more commonly reported than average house prices. This is because there are always a few house prices that are completely over the top, and which could over-inflate the apparent value of property if the mean was used as the reported measure.

The median is a better choice than the mean when you are looking for a more representative value for data sets that contain extreme values.