## SBR: On average, the average ‘average’ isn’t as average as the average person thinks

by Ted Carter

tcarter@kasb.org

In what some are calling the “Post-Fact World,” it is critical that we work to be informed consumers of information. To that end, I thought it might be useful to share some thoughts and information on the notion of “average” amounts and why you need to look at reported averages very closely to make sure you understand what they mean.

First, let’s talk about what the word “Average” means.

Merriam-Webster says Average is “a single value (such as a mean, mode or median) that summarizes or represents the general significance of a set of unequal values.”

Wikipedia says Average is “a middle or typical number of a list of numbers.”

Neither of these definitions is terribly helpful. Many statistics texts talk about “measures of central tendency,” which is what measures typically called “averages” really are getting at. As Merriam-Webster mentions, the three measures that are often referred to as averages are Mean, Mode and Median.

- The Mean is what most of us think of when we hear “Average.” It is calculated by adding up all the values in a list and then dividing by the number of values in that list.
- The Mode is the most frequently occurring value in a list of values.
- The Median is the middle value in a series of values sorted from lowest to highest or from highest to lowest.

Let’s take the following list of values:

- 1
- 2
- 2
- 2
- 3
- 4
- 5
- 5
- 6
- 7
- 7
- 8
- 9

The median for the above list would be 5, because that is the 7th number on the list of 13 going from smallest to largest.

The mode for the above list would be 2, because there are three 2’s in the list.

The mean for the above list would be 4.69, because 1+2+2+2+3+4+5+5+6+7+7+8+9 = 61 and 61/13 = 4.69.

So, as can be seen, even on a simple list of numbers, different values can be reported for “Average” depending on what statistic is used.

Why do we have these different statistics? And when should each be used? Typically, mean and median are the most commonly used statistics for calculating averages. And the biggest reason to choose median over mode is to control for what are called outliers.

In our example list above, the median and mode are still pretty close (5 versus 4.69). But what if the list looked like this:

- 1
- 2
- 2
- 2
- 3
- 4
- 5
- 5
- 6
- 7
- 7
- 8
- 9
- 100

In this case, the median is still 5, but the mean is 12.38. A single value that is very different from the rest of the list is enough to make the mean less meaningful.

People will sometimes “throw out” outliers to avoid producing averages that aren’t really representative of the data, so that is something to watch for as well. Typically this will be done by excluding a certain number of the highest and lowest values in a list.

On the list above, if the two highest and two lowest values were excluded from the calculation (to “control” for outliers), the median would still be 5, and the mean would also be 5.

On the other hand, what if our list looked like this:

- 1
- 1
- 1
- 1
- 1
- 1
- 1
- 100
- 100
- 1,000
- 1,000
- 10,000
- 10,000

In this case, our median is 1, and our mean is 1,708.23. If we exclude the two highest and lowest values, then the median is still 1, and the mean would be 245. Neither “average” really gives us a good feel for what is in the dataset. In fact, for this data the mode might be the better statistic to use.

Another consideration is the level of aggregation. In other words, when dealing with groups within groups, how are the statistics calculated?

For example, let’s say we are interested in the average income for a state. We have information on the average by county, so if we want to know the state average, we can just add up the county amounts and divide by the number of counties to get the mean for the state, right?

That would work if each county had the same number of income-earners in it. But they don’t. So to get an accurate average across the state by income-earner, you would have to know the number of people in each county, and “weigh” the average amount for each county based on the number of people in that county.

Let’s use another example. This time we are going to use grocery stores. The following list shows the store number, the average sale amount, and the number of customers.

- 1 $30 42
- 2 $25 20
- 3 $10 75

If we want to know the average (mean) sale amount by customer, we can’t just add $30 + $25 + $10 and divide by three to get $21.67. We have to do this:

( ($30 x 42) + ($25 x 20) + ($10 x 75) ) / (42 + 20 + 75) = ($1,260 + $500 + $750) / 137 = $2,510 / 137 = $18.32

If we want to know the average total store sales, on the other hand, we would do this:

( ($30 x 42) + ($25 x 20) + ($10 x 75) ) / 3 = $836.67

By the way, I apologize if I am causing you high school algebra flashbacks; sometimes working the math out is the best way to ensure you know what you are looking at.

Next, let’s talk about rounding. Best practice suggests that we should do all of our calculations using the most specific data, then the final results can be rounded for reporting.

For example, if I have a list of numbers that each have four decimal places, but I want to report out just using a single decimal place, I should still do all of my calculations using four decimal places and then round my final answer back up to one decimal place. You will often see people taking shortcuts and rounding their list of numbers before trying to calculate an average. Though this does not result in an average that is wrong, it does result in an average that is less accurate.

I’ve talked about the calculations above without mention of deliberate bias so far, but this has to be considered as well. In the examples above comparing mean and median, some folks would calculate both and then use whichever one was more consistent with their message. The same can be said for rounding, throwing out outliers and level of aggregation.

These are just a few considerations to keep in mind when reading reported averages. There are others, of course, but the important thing to know is that there are many ways to calculate averages, and when reading or hearing about a particular average, it is a good idea to try and determine how it was calculated and whether there was an ulterior motive in the choice of statistics used.