The median is an average of a set of numerical data. As opposed to mean or mode averages, this type of average is calculated by working out where the middle of a set of data is, also known as the midpoint.1
In this article, you will learn how to make such a judgment and apply it to your academic writing, a very useful skill when you are discussing statistical data, whether it is your own or drawn from another source.
The median is the middle number in a set of data.2
Note, too, that median averages can yield very different results.
In the aforementioned example, the median is six while the mean average would be seven, which is relatively close. However, in another data set – say 3, 5, 6, 91, 120 – it would still be six while the mean average would be 45.
|Type of Average||Mean Average||Mode Average||Median Average|
|Meaning||The sum of a set divided by the number of terms in that set||The most frequently occurring number in a set||The midpoint in a set|
Whether a set-up data is small or large, the calculation method is the same. You still just have to arrange all the data in order and pick the middle value. That’s straightforward for odd-numbered data sets.
However, if there is an even number of values in a given set of statistical data, then there will be no single value in the middle.
Let’s imagine a set of data based on UK shoe sizes.
From a survey of recent shoe sales, it is established that on a given day, pairs of shoes in sizes 3, 4 ½, 9, 9 ½, 7, 12, 5, 6, and 5 were sold. In other words, there were nine pairs of shoes sole with the most frequently occurring being size five.
To work out the midpoint average:
- Rearrange the data in ascending order, that is to say, 3, 4 ½, 5, 5, 6, 7, 9, 9 ½, and 12.
- Since the number five is midway between one and nine, it will the fifth number in the ordered set, which is the median.
- Count along the ordered dataset from left to right until you reach the fifth number. In this example, it is six which is the midpoint value.
To work out the midpoint average:
- Rearrange the data in ascending order simplified into seconds only, that is to say, 48, 56, 59, 60, 62, 69, 70, 71, 75 and 79.
- Since the fifth and sixth numbers in the ordered list could both be the midpoints, find them both. In this example, 62 and 69 are the correct ones to use.
- Add them together and divide them by two. In this example, that would be (62+69) ÷ 2 = 65.5. Therefore, 65.5 seconds is the midpoint value.
The median in a normal distribution
Normally distributed sets of data can be represented quite easily on a graph. When a normal distribution makes a bell curve on a graph, the midpoint between the two tips of the bell will be the median value.
Because bell curves formed by normal distributions are symmetrical, the midpoint on the x-axis of such a graph will also coincide with the highest point on the y-axis. As such, when data sets are normally distributed, the midpoint average value will also be the most frequently occurring. In other words, the median, mean, and mode averages will all be the same.3
Mean averages are calculated by adding up all the values in a set and dividing that sum by how many values there are, whereas the median is a straightforward midpoint between the highest and lowest values.4
Median averages that rely on midpoints are good for establishing averages that do not take account of extremes at either end of the scale. In short, they tend to ignore outlier statistics.
With an odd set of data, the midpoint formula to use is:
while the formula
should be applied to even sets of data.5
1 Dictionary.com. “Median.” Accessed December 13, 2022. https://www.dictionary.com/browse/median.
2 BiteSize. “Median.” BBC. Accessed December 13, 2022. https://www.bbc.co.uk/bitesize/guides/zwhgk2p/revision/2.
3 MathIsFun. “Normal Distribution.“ Accessed December 13, 2022. https://www.mathsisfun.com/data/standard-normal-distribution.html.
4 Khan, Sal. “Statistics intro: Mean, median, & mode.” Khan Academy. https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-data-statistics/mean-and-median/v/statistics-intro-mean-median-and-mode.
5 CueMath. “Median.” Accessed December 13, 2022. https://www.cuemath.com/data/median/.