What Is Range in Mathematics?

Professor explaining range to student in classroom
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You have two different ways to define range in math. If you're doing statistics, the "range" usually means the difference between the highest value and lowest value in a set of data. If you're doing algebra or calculus, the "range of a set of data" is understood to be the upper and lower bound for a set of possible results, or output values, of a function.

Range in Statistics

If you're asked to find the range in statistics, you're simply being asked to find the largest number and smallest number in your data set, and then find the difference between them. Any time you hear "difference," it's a clue that you're about to subtract. The range formula is as follows:

\text{highest value} - \text{ lowest value} = \text{ range}

Tips

  • Don't forget to include any units (feet, inches, pounds, gallons, etc.) that may be appended to your data set.

Range Example 1

Imagine that you snuck a peek at your teacher's notebook, and you saw that so far, the students' grade percentages in class are {95, 87, 62, 72, 98, 91, 66, 75}. Curly brackets are often used to enclose a set of numbers, so you know everything inside the curly brackets belongs together.

What's the range of this data set or, to put it another way, the range of the students' grades? First, identify the highest data point (98) and the lowest data point (62). Next, subtract the smallest value from the largest value:

98 - 62 = 36

So the range of this particular data set is 36 percentage points.

The Range of a Function

When you begin studying functions in mathematics, you'll run into a second definition of range. To understand range, it helps to think of functions as little math machines. The set of values you can put into the math machine are called the domain (another very important concept). The set of possible results, once you crank those values through the math machine, is called the ‌codomain‌ And the set of actual results or outputs you get is called the ‌range‌.

There are a couple of important relationships between range and domain that you need to understand. First, within domain and ranges of real number, each value in the domain corresponds to only one value in the range of your function. If any value(s) in the domain correspond to more than one value in the range, you might have a relationship between the two sets of data, but it's not technically classified as a classical function. However, it is possible for more than one domain value to correspond to the same value in that function's range.

Tips

  • This rule of one to one correspondence only applies to what are called injective functions in mathematics. There are many other types of functions that use different domain and range relationships (and even imaginary numbers).

One of the best ways to make sense of this is to imagine your very own math class. The students in the class represent the domain (or the information that goes into the function), while the class itself is the function or "math machine." Your final grades represent the range, or what you get after cranking the elements of the domain (students) through the function (math class).

When you look at that example, you can intuitively see that each student is going to receive only one final grade once class is over. Each value in the domain corresponds to only one value in the range. However, it is possible for more than one student to get the same grade. For example, there might be two or three students in your class that studied very hard and managed to get a 96 percent as their final grade. Multiple values in the domain can correspond to a single value in the range.

Range Example 2

Imagine that you're dealing with the function x^2, with a domain restricted to { −3, −2, −1, 1, 2, 3, 4}. What's the range of this function?

Although you'll learn more advanced ways of finding the range later on, for now, the simplest way to find the range of this function is to apply the function to each element of the domain, and track your results. In other words, insert each element of the domain, one at a time, a x in the function x^2. This gives you a set of results:

\{9, 4, 1, 1, 4, 9, 16\}

But as you can see, some elements are repeated there. Recalling the example of math grades as a function, that's okay; more than one student can end up with the same grade, or more than one element of the domain can "point" to the same element in the range. But you don't want to write down the repeated elements when you give the range. So, your answer is simply:

\{1, 4, 9, 16\}

Notice how the positive numbers and negative numbers in the domain play different roles in the range (in terms of which domain values correspond to the highest numbers and lowest numbers). These relationships are often unique to a function.

What do we use range for?

Descriptive statistics can use many metrics to describe data, and range is very useful for putting these other measurements in context. Arithmetic means and medians are considered measures of central tendency (or what value a set of data can be ‘found around’). Standard deviation and interquartile range are also useful to indicate the distribution of the data around these means and medians, but range provides additional crucial information about the overall spread of data and possible influences from outlying data.

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