How do I find the median when there are an even number of values?
Quick answer
Put the values in order from least to greatest. If there are an even number of values, find the two values in the middle and calculate their average: add them, then divide by 2. For example, in 2, 4, 7, 9, the middle values are 4 and 7, so the median is (4 + 7) ÷ 2 = 5.5. The median is not the mean, which uses all the values. [S1]
The question, the details, the next step.
I have an even number of values. How do I know which ones count as the middle?
First arrange the values from least to greatest. Then count how many values there are. With an even count, there is no single value in the exact middle: the middle consists of the two values on either side of the halfway point. The median is the average of those two values. [S1]
For a small example, suppose the values are 9, 2, 7, and 4. They are not yet ordered, so sort them first: 2, 4, 7, 9. There are four values. The two middle positions are the second and third, containing 4 and 7.
What do I do with 4 and 7 to get the answer?
Add the two middle values and divide their sum by 2:
(4 + 7) ÷ 2 = 11 ÷ 2 = 5.5
So the median of 2, 4, 7, and 9 is 5.5. The median does not have to be one of the original values; here, it falls between the two middle values. The essential steps are to sort first, identify the central pair, and average that pair. [S1]
Why isn't the median 5.5 the same as the mean? I thought an average used every number.
“Average” can refer to different ways of describing a set. The median is found from the central position or pair in the ordered values. The mean is calculated using all the values: add them together and divide by how many values there are. [S1]
For this example, the mean is (2 + 4 + 7 + 9) ÷ 4 = 22 ÷ 4 = 5.5. It happens to match the median here, but that does not make the methods interchangeable. Consider 2, 4, 7, and 13. The median is still (4 + 7) ÷ 2 = 5.5, while the mean is (2 + 4 + 7 + 13) ÷ 4 = 6.5. Use the method the question asks for.
Can you give me a quick way to check my work on a new set?
Use this checklist: sort every value; count the values; locate the two central positions; take the values in those positions; add them and divide by 2. In a list of six values, for instance, the middle positions are third and fourth. You can check that you did not simply choose one middle value or average the entire list when the question asks for the median. [S1]
Copyable working format:
Values: [enter your values] Ordered values: [least to greatest] Number of values: [count] Middle positions: [the two central positions] Middle values: [first value] and [second value] Median: ([first value] + [second value]) ÷ 2 = [result]
For a numerical-list exercise, writing the ordered values and the two central positions makes the calculation easy to inspect.
A practical checklist
Arrange all values from least to greatest.
Count the values and confirm the count is even.
Identify the two central positions in the ordered list.
Add the values in those positions and divide by 2.
Check that you averaged the middle pair, not the whole set.
A few more useful answers
Does the median have to be one of the original values?
No. For an even number of values, the median is the average of the two middle values, so it can fall between them, as 5.5 does for 2, 4, 7, and 9. [S1]
What if the two middle values are the same?
Add the two equal middle values and divide by 2. Their average is that same value.
Should I sort the values before finding the median?
Yes. Arrange them from least to greatest first; otherwise, the values in the middle positions may not be the central values of the set. [S1]
Sources
References used for this guide.
- [S1] OpenStax Statistics: Measures of the Location of the Data openstax.org · Checked