Mean, Median & Mode: SAT Math Statistics Explained

How many statistics classes have you taken in high school? 3? 4? How about zero? Most students have never taken a statistics course, yet 30% of the SAT math section is made up of data and statistics. And guess what? This is good news.

Why, you ask?

Because any chance to pick up points simply by becoming more familiar with a concept is a good day in the test prep world. And mean, median and mode are easy concepts.

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Number Games: Middle vs. Average vs. Most Common

Before we jump straight into definitions, let’s start with an example.

Imagine you are trying to describe wealth in the United States. On one end, we (unfortunately) have some very, very poor people who have essentially no wealth. On the other end, we have someone like Elon Musk, who at the time of this writing, is worth $286 billion. Somewhere in the middle (okay, not the middle…but in between) is the median US family, which has a net worth of $121,000.

A few questions:

• Why isn’t the median family halfway in between the poorest and wealthiest?
• Is the median family the same thing as the average family?
• Where do most people fit in here?

These are all related but different concepts. Now for some definitions:

• Median is the middle data point. In our example, it would be the person right in the middle. If we started counting from the richest and poorest toward the middle, it would be the very middle person we would land on. If there are seven people in a set, the median person is number 4.
• Mean is the average. For reasons we’ll discuss in a minute, mean and median are not the same thing. We can calculate the mean by adding up the sum and dividing by the number of things we have. Think about your scores on a test: If you scored an 88, 97 and 94, your average would be a 93 (88+97+94 divided by 3).
• Mode is the number that occurs most often. This is totally unrelated to median and mode. It’s tempting to think that the mode should be at or near the median, but that isn’t necessarily true. If you scored an 88 on your fourth test, 88 would be the mode.

Normal Distributions: When Things Aren’t Weird

Let’s start by thinking about a very normal situation, or what’s called a normal distribution. Normal distributions look like this:

A few points about a normal distribution:

• The mean, median and mode all occur at the same spot (the very top)
• There is just as much “mass” on the left as the right. Another way of saying this is that there are just as many data points on the left as the right.
• The shape of the distribution is symmetrical. Not only are there an equal number of data points on each side, but these data points are distributed equally far from the mean and median. For every data point on the left, there is a corresponding data point on the right that is an equal distance from the middle.

If you think through what a normal distribution would mean in the real world, you can see how it differs from our wealth example. In a normal distribution, the smallest data point is an equal distance from the middle as the biggest data point. In the real world, this would mean that if the lowest amount of wealth was zero, and the median was $121,000, the very top would be $242,000 (121,000 x 2). Obviously, Elon Musk is worth more than $242,000.

Normal distributions are easy to think about, but there’s a good chance the SAT will throw you something a little funkier.

When Mean, Median and Mode Get Divorced and Holidays Get Awkward

Think back to our wealth example. We know the median is $121,000. But where would the mean fit in? Because we have very, very large outliers to the right (high wealth), the mean will end up further to the right than the median.

Let’s put some numbers to this to illustrate: Imagine we have a number set containing 1, 2 and 10. The median is 2 because it is the middle number. But the average is 4.33 (1+2+10 divided by 3). The 10 pulls a lot of “weight” and brings the mean closer to it than the median.

This sort of distribution would look like this:

An easy way to think of this is that extreme outliers (in this case, very wealthy people) “pull” the mean toward them because they affect the mean more than the median. Elon Musk only gets counted once in the median. But his wealth has a big impact on the mean.

What if the situation were reversed, and there were some really, really small outliers? This is the sort of distribution that might exist for easy exams in school. If the test is easy, most students might get a 90 or higher. But there’s always that one kid. And maybe he actually has a partner in underperformance. The distribution for this sort of test would look like this:

In this case, Todd is pulling the mean down because his extremely low score affects the mean disproportionately. He only counts one “vote” toward the median, but the mean is heavily skewed by his very low score.

Example of Mean, Median and Mode Problems

Before we jump into SAT-specific questions, let’s take a look at some foundational problems that illustrate these concepts.

My first bit of advice for questions involving any set of hypothetical numbers is to create an actual number set. Let’s use 1, 2, 3, 4, 5, 6, 7.

If we double the largest number (7), our set is now 1, 2, 3, 4, 5, 6, 14.

Now, let’s think about what’s changing:

• Mean: The mean changed because the sum of all of the numbers changed. If the sum changes, the mean changes.
• Median: The median did not change because the middle number is still 4. This is an important point: The median is unchanged by small or big numbers getting bigger or smaller. It only changes when the middle value changes.

Looking at our answer choices, option A is correct.

Let’s change things up just a bit:

Just like with the previous question, it makes sense to actually go through the motions and change our number set based on what the question is asking for. When we change the original smallest number (3) into the new largest number, it becomes 16 or larger.

Here’s our new set: 7, 9, 12, 15, 16

All of the answer choices are asking about what our new mean and median would be, so let’s see if we can calculate those:

• Median can be calculated because we don’t need a specific value for the largest figure. The largest figure doesn’t matter as long as we know the middle value. In this case, the median of our new set is 12.
• Mean cannot be calculated because we don’t know the actual value of our new largest number. We simply know that it is 16 or larger. In order to calculate mean, we have to know all of the values in the set.

Therefore, our answer is B.

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Let’s take a look at one more foundational problem before moving on to SAT-specific question types:

This question is tougher than the first two, and if you can answer it, you almost certainly have a strong understanding of these concepts.

First, is this an Elon Musk scenario or a Todd the Underachiever™ scenario? We know that the mean is greater than the median, so there are big outliers that are bringing the mean up. Therefore, Elon is our prime suspect.

Now that we have a picture of what is going on, let’s try to test each answer choice. A good approach on SAT math questions that ask you which statement is true is to try to disprove each answer choice. Try to imagine a scenario in which the condition in the answer choice would not work.

• A: A set of 100, 101 and 110 would give us an Elon Musk scenario, but 110 is not twice as large as the smallest number. This option is eliminated.
• B: A set of 100, 100 and 110 would also give us an Elon Musk scenario, but the median (100) and the smallest number (100) are equal.
• C: This is a defining trait of any Elon Musk scenario if we have only three numbers. In order for the mean to be “pulled up” above the median, the largest number must be further away from the median than the smallest number is. Just as Elon is further away from the median net worth than a person with no wealth is, our number set must feature a largest number that is a bigger outlier than the smallest number. Note that this isn’t necessarily true if we have larger number sets. The set 1, 9, 10, 17, 18 would have a median of 10 and a mean of 11, which means it is an Elon Musk scenario. However, the largest number (18) is actually closer to the median (10) than the smallest number (1). Data sets can get complicated as they get big.
• D: In the set of 100, 101 and 110, 110 is not twice as large as the mean.

How the SAT Math Section Tests Mean, Median & Mode

The SAT math section will typically test mean, median and mode by asking you to do one of two things:

• Calculate the mean, median and mode from a given data set, which is typically pretty easy
• Figure out either what the new mean, median or mode would be after a proposed change to the data set or what the nature of the change would be

Let’s take a look at the second sort of question:

Quick, is this a Musk or Todd situation? Our mean is higher than our median, so Elon is at it again. Looking through our answer choices, we can say the following:

• A: We dont’ have enough information to determine this. A set of three houses could be worth $1, $135,000 and $449,999 and satisfy our requirements.
• B: The opposite is actually true. Our outliers are much more expensive than the rest.
• C: This is a defining trait of an Elon Musk situation, so this is our answer.
• D: We don’t have enough informaton to determine this, and it wouldn’t cause the scenario we have here.

Mean, Median and Mode: Final Thoughts

Mean, median and mode are tested less often than core concepts like linear equations but more often than truly rare concepts like the discriminant or complex numbers. You’re probably going to see a question relying on these concepts on your test, though, so it’s important to feel comfortable with the things we’ve talked about here. The good news is that these questions are rarely really, really hard. If you can do the practice questions we’ve done here, and you can tell Elon from Todd, you should be fine.