In our recent PPP Loan Data blog, we reference both the median and the average, or mean, loan amounts for geographies in Michigan. These are two statistical measures that are used to determine a middle value for a set of numbers.
But why are these numbers sometimes so different? And why present both? In this blog, we’re going to define mean and median, and provide a quick lesson on how to calculate them, and when it makes sense to use each one.
Mean Defined
The mean, also known as the average, is a method of finding the most common or most central value in a set of numeric values. It’s calculated by adding all of the values in a set together and then dividing that total by the number of values.
For example, if you had a set of three values that included 6, 11, and 7, you would find the mean with the following steps:
- Add the values together:
6 + 11 + 7 = 24 - Divide the total by the number of values:
24 ÷ 3 = 8
The end result is a mean of 8, which is an accurate calculation of a middle value for this set of numbers.
This method of calculating a middle value can be easily used for any number of values. The same steps are used to calculate the mean no matter if there is an even number of values or an odd number of values.
When do you want to find the mean?
The mean is thought of as the most accurate middle value for a given set of values because it considers the outliers—significantly higher or lower numbers—as part of the calculation. This makes it a useful number to present when you’re looking for a middle number that takes into account excellent or poor results. One common example is school grades, which are often calculated as the mean of all test scores during a term, and need to include the highest and lowest scores to demonstrate student performance.
Median Defined
The median is the middle value in a series of numbers. Unlike the mean, the median isn’t calculated. It’s found by arranging a set of values in ascending order, and then identifying the number that falls in the middle.
For example, let’s look at this set of values: 1, 6, 8, 10, 13, 3, 3.
You would identify the median with the following steps:
- Put the values in ascending order (lowest to highest):
1, 3, 3, 6, 8, 10, 13 - Identify the middle number:
1, 3, 3, 6, 8, 10, 13
The median of this series is 6.
This method of finding a middle value can be used for any number of values, but can quickly become more difficult to find manually the more values a series has.
Additionally, an extra step is needed to find the median of an even number of values. For example, let’s find the median of this even-numbered set of values: 1, 6, 8, 10, 13, 3.
You would use the following steps:
- Put the values in ascending order:
1, 3, 6, 8, 10, 13 - Identify the middle two numbers:
1, 3, 6, 8, 10, 13 - Calculate the mean (or average) of these two numbers:
(6 + 8) ÷ 2 = 7
The median of this series is 7.
When do you want to find the median?
The median is a good middle value for a set of numbers that has a fairly even distribution, without significantly large or small values, or for presenting a middle value that you don’t want to be skewed by outliers. One common example is finding the median income of a neighborhood to determine income or wealth trends, which you wouldn’t want to be skewed high or low by outliers.
Mean vs. Median
The mean and the median may both be methods for finding a middle value in a set of numbers. However, they are each determined differently, and have different use cases. In summary:
- Medians are used to find the middle set of a value of numbers and do not take outliers—significantly larger or smaller values in a dataset—into consideration.
- Means, by comparison, are easily skewed by substantial outliers.
As a result of these differences, while the mean and median both can be used to represent the middle value of a dataset, they can be substantially different. Take the following example.
Find the mean and the median for each of these sets of numbers:
Set 1: 8, 9, 10, 11, 12
Set 2: 8, 9, 10, 11, 100
Set 3: 8, 9, 10, 11, 1,000
You should have the following values:
As you can see, the median for each of these sets of numbers is the same—10—while the mean is very different, because of the increasing outliers for each set. The important takeaway from this exercise is that the mean can be easily and substantially skewed by large outliers, making it an less reliable measure for determining trends among a dataset, like income in a neighborhood, and a better measurement for numbers in which outliers should be factored in, like school grades.
In conclusion, understanding the difference between the median and the mean is an important aspect of responsible data consumption. With this knowledge, you can determine why the two figures may be wildly different for the same dataset, why an analysis might have used one method over the other, and what information is meant to be conveyed in any findings.