The arithmetic mean, median, and mode are all different ways to describe a group or set of numbers. The three concepts are related, and some questions on the SAT will test your knowledge of two or even three of them in conjunction. Here, we will cover each individually and then look at how they overlap.

The arithmetic mean, which also goes by the names average and mean, is the most important and most commonly tested of these three mathematical concepts. The basic rules of finding an average are not very complicated. To find an average of a set of n numbers, you need to find the sum of all the numbers and divide that sum by n.

For example, the average of the set 9, 8, 13, 10 is equal to the sum of those four numbers divided by 4:

Occasionally, the SAT will test your knowledge of averages in a straightforward manner, giving you a bunch of numbers and asking you to find their average. More often, the SAT will find some roundabout way to test your knowledge of averages. The SAT might give you three numbers of a four-number set, the average of that set, and then ask you to find the fourth number in the set:

There are two ways to solve this type of problem, and both are fairly simple. To use the first method, you have to realize that if you know the average of a group and also know how many numbers are in the group, you can calculate the sum of the numbers in the group. In the question asked above, you know that the average of the numbers is 22 and that there are four numbers. The four numbers, when added together, must equal From the information given in the problem and our own calculations, we know three of the four numbers in the set, and the total sum of the numbers in the set:

Solving for the unknown number is easy. All you have to do is subtract the sum of 7, 11, and 18 from 88:

All average problems on the SAT cover these, and basically only these, fundamental points. Difficult problems simply cover them in a trickier manner.

For example:

This question seems really tough, since it keeps splitting apart this theoretical set of seven numbers and you have no idea what the numbers in the set are. Often, when students can’t say exactly what numbers are in a set, they panic. But for this problem you don’t have to know the exact numbers in the set. All you have to know is how averages work. So let’s solve the problem.

There are 7 numbers in the entire set and the average of those numbers is 54. The sum of the seven numbers in the set is therefore: Now, as the problem states, if we take three particular numbers from the set, their average is 38. We can calculate that the the sum of those three numbers is: Suddenly, we can calculate the sum of the four remaining numbers, since that value must be the total sum of the set of seven minus the sum of the mini-set of three, 378 – 114 = 264. Now, since we know the total sum of the four numbers, to get the average of those numbers, all we have to do is divide that by 4:

The median is the number whose value is in the middle of the numbers in a particular set. Take the set: {6, 19, 3, 11, 7}. If we arrange the numbers in order of value, we get:

When we list the numbers in this order, it becomes clear that the middle number in this group is 7, making 7 the median.

The set we just looked at contained an odd number of items, but in a set with an even number of items it’s impossible to isolate a single number as the median. Let’s add one number to the set from the previous example:

In this case, we find the median by taking the two most middle numbers and finding their average. The two middle numbers in this set are 7 and 11, so the median of the set is ⁽⁷⁺¹¹⁾/ ₂ = 9.

As we said earlier, some SAT questions might test your knowledge of mean and median in conjunction. For example, a question might show you five sets and ask you to pick the set in which the average is greater than the median. For these questions, there are a few things you should know:

The mode is the number within a set that appears most frequently. In the set {10, 11, 13, 11, 20}, the mode is 11 since it appears twice and all the others appear just once. In a set where more than one number appears at the same highest frequency, there can be more than one mode: the set {2, 2, 3, 4, 4} has modes of 2 and 4. In a set such as {1, 2, 3, 4, 5}, where all of the numbers appear an equal number of times, there is no mode.