Introduction to How to Calculate Medians

Dealing with the stats part of the AP® Statistics exam can be intimidating enough on its own, but on top of that, you are probably discovering that there’s enough confusing vocabulary to make your head spin! Take the median, for example – along with its siblings mean and mode; they make up the “measures of central tendency,” and like any family members, they can be tough to tell apart.

However, once you get a handle on putting names to (mathematical) faces, you’ll find that calculating the median is easy, and it’s a simple concept that will be very crucial on the exam. This AP® Statistics review will explain what the median is and how to tell it apart from similar concepts, show you how to calculate the median, and help you learn how to conquer median-related questions on the AP® exam.

What is the Median and Why do We Use it in AP® Statistics?

Before we learn how to calculate the median, it helps to understand it in the context of related terms. The median is the second of the “measures of central tendency” (see our other blog post for a review of the mean), which means it is one way of showing us the central value within a set of numbers. Outside of statistics, you’ve probably heard the word “median” used to refer to the strip of land that divides the two halves of the road, like in the picture below. Similarly, the mathematical median is the point that divides the two halves of a distribution. In stats, we refer to a set of numbers as a “distribution.” A distribution, unlike a physical object, doesn’t have just one center. Rather, there are different ways to think about what is the central value in a distribution, and, therefore, there are different ways to calculate it.

One way to think about the “center” of a distribution is to take the mathematical average, or mean. For example, when your teacher returns a test, your class will have a distribution of scores – some students may score high, the ones who didn’t study will have low scores, and the rest will fall somewhere in the middle. If you found out that the mean score on a test was 60%, you might guess that it was a difficult test and most students did poorly (however, the brightest students probably still scored well above the mean!). If the mean score on the next test was 90%, you might guess that most students did very well (though the worst students probably still scored well below the mean). In either case, there is still a spread of scores from the lowest to the highest, but the center of that spread appears to be moving around from test to test.

However, there are a few times when the mean does not give us an accurate picture of the center of a distribution. That’s because the mean is pulled towards extreme values. For example, if most people have a very low score on the test but a few smarty-pants get 100’s, the mean score will be pulled up. As a result, the majority of the class will fall below the mean, while only a few people will be above the mean. If there are significantly more people on one side than the other, we clearly haven’t divided our distribution very well! Let’s illustrate this with a simple example of 5 students:

To calculate the mean, we add up all the scores and divide by the number of students, and we discover that the mean is 52%. This calculation means that 4 out of 5 students (80%) fall below the mean, while only 1 student (20%) is above the mean. In this case, the mean isn’t a very good way of measuring the center of the scores.

That’s where the median comes in. The median divides the scores evenly in half. It’s the middle value, such that 50% of the distribution falls below it, and 50% above it. The median is a useful descriptive statistic for cases in which the distribution is asymmetrical, or skewed, such as in the picture below. As you can see, in this image, the mean has been pulled slightly to the right of the median (obviously not as drastic of a shift as in the previous example of 5 students, but pulled nonetheless). Conversely, when a distribution is symmetrical, the mean and median will be the same value! As a result, we can compare the mean and median to figure out the shape of a distribution. When the mean is above the median, the distribution is positively skewed meaning the long tail is to the right. When the mean is below the median, the distribution is negatively skewed (for more details on this topic, review skewness).

Keeping Your Terms Straight

One of the initial challenges you have to overcome before you can master calculating the median and using it on the test is simply making sure you don’t mix it up with related concepts! The measures of central tendency – mean, median, and mode – are obviously similar, but they all serve different functions, so it’s important to remember which term is which. You can accomplish this by creating memory associations or other mnemonic devices. Let’s talk about some of these options.

Memory associations link the name of something to its meaning through ideas or images that are easy to remember. For example, to remember that “mean” is the “average,” we might come up with a phrase like “the average crocodile is very mean.” For “median,” you might think of the image of the road median from the beginning of this post, and remember that “the median divides evenly in half.” It doesn’t matter what image you choose – in fact, you may want to make up your own! The stranger and more personal it is, the easier it will be to remember (just make sure it connects back to the definition somehow!).

Other useful mnemonic devices are songs, poems, or rhymes. A teacher of mine taught us to remember median by adapting an old nursery rhyme: “Hey diddle diddle, the median’s the middle…” Again, you don’t have to use one that already exists. If you spend a little bit of time now thinking of your own way to remember these terms, I guarantee you’ll never mix them up again!

How to Calculate the Median for AP® Statistics

Now that we know what it is, let’s talk about how to calculate the median. The first step to calculating the median is to arrange our scores in numerical order. Let’s try this with our example above of the five students. When we rearrange their scores from smallest to largest, they look like this:

From there, it’s simple – we just look to see which value is right in the middle, and divides the set in half. So the median in this example is 41%!

It’s pretty obvious when you have an odd number of scores, but how do you determine the median when there is no clear middle? Let’s try an example with an even number of scores. Imagine there are 6 people, and we’re looking at their heights in inches.

Again, let’s arrange them in numerical order.

In this case, there is no middle score. However, another way to think about it is that there are two middle scores. So, we need to find the number that is directly between those two. To do that, we’ll make use of our old friend mean to find the value that falls right between Person 3 and Person 6.

First we add up the two middle scores: 61 + 65 = 126

Then, we divide by the number of scores we’re looking at: \dfrac{126}{2}=63

In this case, our median is 63 inches. It’s important to note that the median doesn’t have to be a number that is actually found in any of the scores in your distribution!

Mastering Median-Related Questions on the AP® Stats Exam

Since the median, like the other measures of central tendency, is one of the most basic concepts in AP® Stats, you’re unlikely to encounter any questions on the exam that simply ask you to calculate a median. Rather, this calculation is much more likely to be combined with other concepts such as the mean. For example, you may be asked to determine whether the mean or the median is a better measure to use in a specific case. That’s why it’s very important to remember the differences between the two and when they are each useful: mean is good for symmetrical distributions with no outliers, median is good for asymmetrical distributions or ones with outliers. You could also be asked to use the mean and median to determine the shape of the distribution, so make sure to review the properties of skewness mentioned previously.

From this AP® Statistics review, you should now know how to calculate the median, when to use it relative to other measures, and how to keep your terms straight! Have you come up with any fun ways to remember the difference between mean and median? Drop your tips in the comments below to help out your fellow AP® Stats crammers!

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