22 Apr How to Find the Range in Math

What is the Range in Math?

The range is a mathematical tool that’s used in finding the spread in a data set. It’s easy to calculate. If you’re trying to find the range of a data set (or you’re being asked to find it), all you need are the highest and lowest value in a set.

The Range in Statistics

Statistics is a discipline that involves the analysis and collection of data. It helps people make predictions about future events to a great degree, as well as describe large masses of data. This latter part is where range comes in. To find the range in a data set, simply identify the highest and lowest number, find the difference, and viola—you have the range.

Let’s take a look at an example. You have a class of 12 students, and after you give them their weekly exam on Friday, you look at the scores. Overall, they did pretty well, scoring the following:

{85,81,88,90,90,91,95,92,92,82,100,89}

We could rearrange them if we wanted from smallest to largest, but that’s not necessary. All we want to do is pick out the smallest and largest numbers, which in this case are 81 and 100. To find the difference, simply subtract the smallest from the largest. So:
100-89=19

The range in this example is 19.

Pretty easy, right?

Let’s look at one more example. Pretend you are timing amateur race car drivers in the quarter mile at your local track. You look at the times of the last six cars (recorded in seconds) and get:

{10.5,11,10.2,9.4,12.3,13.0}

What’s the range of times? Simple enough: 

13.0s-9.4s = 3.6s

A couple of quick points before we move one. First, you have to have at least two data points for there to be a range. Second, you could have a range of zero if your data is filled with the same number (e.g. {3,3,3,3}).

The Range with Functions

The other time you’ll encounter what’s labeled as a range in mathematics is when it comes to functions. Functions are those wonderful things that we plug numbers into in order to pop out other numbers, hopefully with the goal of accurately describing what’s taking place in a certain scenario.

For example, in physics, we have a function that describes force as a product of mass and acceleration. You might recognize this as:

F = m*a

If you run several experiments, let’s say, calculating the force after you took an object weighing 2kg and accelerating it at 5 m/s2, 10m/s2, and 20 m/s2, you’d have different points of data to work with. Now, the data points you have to work with, your inputs, are considered the domain. This is something you need to remember. In this case the domain we have is {5 m/s2, 10m/s2, and 20 m/s2}

With that in mind, we can use our domain to calculate the range quickly by plugging our numbers into the formula. When doing that, we get:

F = 2kg * 5 m/s2

F = 2kg * 10 m/s2

F = 2kg * 20 m/s2

And solving for F gives us {10N,20N,40N}

What’s the N, you ask? Those are just Newtons, a unit in physics describing force. If we were just using a vanilla formula of y = 2x and plugged {5,10,20} into x, we could ignore the units. But we want to do this right.

So then, much like the range in statistics, the range in functions deals with the difference between the smallest and largest number in our answers. Going back to our physics problem, we see that our range can be found with the following:

Range = 40N – 10N

Therefore…

Range = 30N

That wasn’t too hard now, was it? It’s important to note here that we have to actually solve our function within our domain to find the range of a function. That is to say, we’re not looking at the range of our starting data—the input. We’re looking at the range of the output. This, as you can see, can be wildly different than the domain.

There’s also another point to consider in all of this. When you start getting into exponents, your range of a function can be very surprising.

Let’s look at a straightforward exponential function: y=x2.

Now let’s restrict our range finding to the domain of {-4,4}.  At first glance, you might think the range is 8 (4 – -4). But then you remember we’re not looking at the range of the domain, but the function. Once we plug in our numbers now, we get (16,16) because when you square both -4 and 4, you get the same number. Therefore, the range of the function x2 with the domain of {-4,4} is 0.

However, all we have to do is throw a single extra number in to change it completely. What if our domain is {-4,1,4}? What’s the range?

Did you say 15? Awesome! If not, reread the above and see where you went wrong.

How is Range Used in Real Life?
Range is used in hundreds of ways. For instance, you could determine the range between college majors with the highest and lowest unemployment rates. Or you could use it as a tool when analyzing which college majors have the worst return on investment. However, the applications are hardly limited to college and money. Scientists from biologists to chemists to physicists use range daily, as do accountants, programmers, and probably even you, yourself, without knowing to a certain degree.