Full disclosure: I’m a graphic designer, not a mathematician or statistician. My reason for posting this article is that I’ve needed to explain these ideas a few times in the past week. As our next election approaches, I’m guessing some may find it useful for interpreting/assessing campaign claims. It’s good info to understand even if you just have a credit card in your wallet!
An “average” is a statistical “middle figure” for any set of numbers. In mathematics and statistics there are a lot of different ways to find an average. The “mean” and the “median” are often used. The problem is that, although they are both an “average”, they each represent different information. For clear communication, t’s important to understand what each represents when reading claims made by politicians, banks, insurance companies, etc.
When people talk about averages, they’re usually referring to a “mean”. The “mean” of a number set is the “average” you learned about in grade school. It’s found by adding up all then numbers in a set and then dividing that total by the number of numbers in the set. For example:
Given set: 7, 3, 6, 18, 2, 2, 2, 4, 50
First add the numbers: 7 + 3 + 18 + 2 + 2 + 2 + 4 + 50 = 94
Then divide by the number of numbers: 94 ÷ 9 = 10
So 10 is one “average” you can use to represent this set, but it only tells part of the story. After all, in this set of nine numbers only two (22%) are actually above the mean. When a set of numbers contains several similar numbers with just a few outliers, using the mean as your average may not provide a fair representation of that set. It’s not a lie, but it’s not a clear picture either. This is often the case in economics, especially when discussing things like “average worth”.
To the right is a chart that shows both “average” (a.k.a. “mean”) and “median” U.S. household wealth by age group.
The reason for the big difference in the two counts is that using the mean doesn’t adjust for the very few ultra wealthy people living in the U.S. Although it accurately represents the average wealth of all U.S. households, it’s misleading in the way it makes the majority of Americans seem far wealthier than we actually are.
That’s why people may choose to use the “median”. It’s used to describe number sets where outlying data points can present a misleading picture.
The “median” is the number that falls in the middle of any given set of numbers. You find it by first putting the set in numerical order, count up the number of items in the set, then divide that total in half to find the middle number. Using the same set as before, here’s an example of finding the median:
Given set: 7, 3, 6, 18, 2, 2, 2, 4, 50
Numerical order: 2, 2, 2, 3, 4, 6, 7, 18, 50
Dividing: In a set with an even number of items, all you’ll need to do to find the midpoint is divide by two. The example here has an odd number of items in it, and that complicates things just a bit. To find the middle number if your set has an odd number of items, first add one to your count (to make an even total), then divide by two.
Add one (for an odd total count): 9 + 1 = 10
Divide by two: 10 ÷ 2 = 5
In our set with nine numbers, the middle number is the fifth. So the median for this number set is 4, the number that appears fifth in the list. Note that seven of the nine numbers (77.8%) deviate from 4 by a value of 5 or less. By using the median, more of the population of the set is being better represented.
To clearly understand the data people offer up, be sure to pay attention to whether someone says they’re providing an average, a mean, or a median.
If you’re interested in this topic, and want to learn more about averages, there are plenty of resources available on the internet. Purplemath.com has a very well written article that gives definitions for and explains how to find Mean, Median, Mode, and Range. At the same location, there’s also a link you can follow to practice finding the median.