Learn about a third method for calculating average values, the mode, and find out exactly when and why it is useful.

In recent articles, we’ve talked about two methods of calculating average values, the mean and the median, and we’ve talked about some everyday uses for them such as how to use median averaging to get better photos. But that’s not all there is to say about average values. Today, we’re talking about one additional average statistic: the mode.

## Mean, Median, and Mode

Mean, median, and mode. This trio of names rolls easily off the tongues of people who spend time dealing with statistical quantities. In my case, being an astrophysicist, I use these statistical values in one way or another almost every day. In fact, while writing this series of articles about the mean, median, and mode, I’ve been internally referring to it as “MMM”—which also happens to be the name of a computer program used by many astrophysicists to automatically calculate how bright the sky is (a task made much harder by all those interloping bright stars). All this is to say I couldn’t have talked about the mean and the median without saying a few words about the mode. Besides, it’s useful in a way that the mean and median can never be—which we’ll get to shortly.

## Why Another “Average?”

So, why do we need another average value? The short answer is that the mean and median alone can’t provide answers for all the statistical questions you may be confronted with. But that sort of “you need it because you need it” answer isn’t very satisfying to me. So let’s look at the reasoning in a little more detail. To really get to the bottom of answering “Why another average?” let’s start by taking a look at what the various types of averages we’ve talked about so far are actually good for.

## When Should You Use the Mean?

First, the mean. In the article on how to calculate mean values we learned that the mean is the most natural way to describe average values. To find the mean, we take all of whatever we’re averaging, and we spread it evenly amongst the various “containers” we’re averaging. A little more concretely, if we’re calculating the average weight of tomatoes in several bowls, the tomatoes are the “whatever,” and the bowls are the “containers.” Or, if we’re talking about the average number of points students have earned in a class, the student’s scores are the “whatever,” and the students themselves are the “containers.” The net result is a number that tells us the average number of whatever per container—tomatoes per bowl, points per student, and so on. It’s very intuitive.

## When Should You Use the Median?

Next, the median. In the article on how to calculate median values we learned that the median steps up to the plate to help out when the mean is overwhelmed by problematic data. In particular, if a couple of values in your set of data are outliers, meaning they’re either much larger or much smaller than most of the other values, the mean value will be thrown off. But the median will not be since it is resistant to these types of outlying values. It’s a very powerful statistic as we discovered in the last article on using median averaging to remove tourists from your photos.

## When Should You Use the Mode?

And now we’ve arrived at the crux of our question: Why do we need another average? Well, let me answer that question with a question. What is the average color of a cat? To answer, we obviously need to calculate an average value of some sort—should we use the mean or the median? Do either actually make any sense? No, not really. First, it’s not at all clear how to calculate the mean value. Take a brown tabby, a calico, and a black cat. We can’t even start to calculate the mean of these cat colors since it’s totally different than doing something like calculating the average number of coffee beans in jars—with cat colors, there are no numbers so there’s nothing to add up! And, if you think about it for a minute, you’ll see that you can’t calculate a median value either since there’s no way to put cat colors in order and find the one in the middle!

## What is the Mode and How is it Calculated?

Problems like this clearly show that we need a new way to calculate averages. And that new way is called the mode. Unlike the mean and median, the mode can be used to find the average of non-numerical data—like the color of cats. Here’s how it works. Imagine asking everybody in your city to write down the color of their cats. Then, take all these submissions and create a list of all the unique cat colors. Next, go through all the submissions and tally the number of cats of each particular color. The mode is the color that occurs most frequently. In other words, if there are 100 black cats, 50 calicos, and 150 brown tabbies, the mode would be 150, since brown tabby occurs most frequently. In this case, brown tabby is the “average” color of a cat—in the sense that it’s the most popular or common.

Of course, the mode works for things that are entirely numerical from the get-go too. Remember, the cat color problem didn’t start with numbers—it started with cat colors—although we did turn it into a numerical problem by tallying up the totals in each category. In general, the mode is the value that occurs most frequently in a list of numbers. The mode of the list of numbers (3, 5, 5, 7) is 5 since it occurs twice. If the list of numbers were instead (3, 5, 5, 7, 7, 9), you’ll notice that there isn’t a unique mode. This list has two modes—5 and 7—and is therefore called bi-modal.

## Brain-Teaser Problem on Averages

So, now you’re up-to-date on the big three of averages: the mean, median, and mode. Hopefully you have a good intuitive grasp of what these three quantities represent, and when they’re useful. And to make sure you have a good practical grasp of how to use them too, here’s a brain-teaser problem for you to try courtesy of the Ask Dr. Math website (which is a great resource with lots of answers to common math questions). Here’s the question: “Find a set of five data values with modes 0 and 2, median 2, and mean 2.” Give the problem a shot, and then check out this week’s Math Dude “Video Extra!” episode on YouTube for an explanation of the solution.

Next week we'll talk about another subject related to averages: range and standard deviation.

## Wrap Up

Okay, that’s all the math we have time for today. Now, the next time you’re at a party and somebody asks you to calculate the “average” name of partygoers, you can use your knowledge of the mode to help out. (Okay, this’ll probably never happen—but you get the idea.)

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!