The Three Averages

The Three Averages

We need to be careful when we use the “average.” Depending on the context, we could mean any one of a number of different averages. And the average that you should use depends on the information that you want to convey.

In everyday usage, there are three main types of averages, or as mathematicians like to call them, “measures of central tendency.” Let’s examine each of them in turn.

When people think average, this is the first thing that usually pops into their mind. It is the first type of averaging that you learn about in school. Take list of items that you are averaging, add them together, and divide by the number of items. That is the Mean of the list. It is used in school to find the average grade on a test, or in sports to find the average number of points a team scores in a game.

Using the Mean can be deceptive.

“Twenty of us are in a room having a party. Bill Gates walks in. On average, the worth of everybody in the room is over a billion dollars.”

Large outliers can have dramatic effects on the Mean value.

An alternative way of looking at average is the Median. The Median value is the “middle” value in a set of data. To find the median of a list of numbers you re-write the numbers in an ordered list and find the one in the middle. If there are an odd number of items in the list then there will be an exact middle.

What do you do if there is an even number? That depends on the situation. The usual method will be to find the mean of just those two numbers and declare that to be the median.

The median is less sensitive to random outliers than the mean is. It is a good measure of average if the results tend to skew to one side of the range. Of course, sometimes we want outliers to have weight, as in the case of test scores.

Now sometimes neither of these are what we really want. Sometimes we want to know what the most likely result. This may be neither the mean nor the median. The Mode of a list refers to the item that is the most common result. Look at the possible outcomes. Which is most likely to happen? That is the Mode.

There may be more than one possible Mode. In fact, in many practical data sets it is common to have more than one Mode.

Computing the Mean, Median, and Mode
Let’s take a look at the following list of numbers:

2, 8, 10, 2, 3, 2, 8, 5, 3

First, we’ll compute the mean.

The sum of the number is 43 and there are 9 of them, so the Mean is 43/9 which is about 4.78.

Now, let’s find the median. We will re-order the list like this:

2, 2, 2, 3, 3, 5, 8, 8, 10

Now we notice that there are 9 items, so the middle item is the 5th item. This is 3, which is the median.

Let’s see an example where all three of them are the same.

Consider the sum of two pair of dice. The probability distribution graph is shown below. A probability distribution graph shows the probabilities of all of the possible events occurring. You may have heard of a Normal Distribution, more commonly referred to as a Bell Curve. This looks very similar to a Bell Curve.

What is the most likely result? 7. 7 is the Mode of the distribution. The Median value is also 7. What about the mean? Turns out it is also 7.

The correct average to use depends on the situation. There are times when each is best. The mean is usually the easiest to compute on the fly. Just keep a running sum and divide by the number of items. Reordering the list to find the median can be time consuming, especially if the list of values is large.

Mean, Median, and Mode aren’t the only ways to think about averages in statistics, but they are the most common and the easiest to talk about.