## The Lesson

The range of a set of numbers is the difference between the highest number and the lowest number in the set.

## Dictionary Definition

The Oxford English Dictionary defines the range as "the difference observed in a sample between the smallest and largest values of a variable."

## Understanding the Range

The range of a set of numbers is the difference between the highest number and the lowest number in the set. It is a measure of how spread out a set of numbers is.

## Finding the Range

Imagine we wanted to find the range of the numbers given below:

To find the range, find the largest number in the set (5) and the lowest number in the set (1). Subtract the lowest number from the highest number:
Range = 5 − 1 = 4

## A Formula to Find the Range

The formula for finding the range is shown below:

In this formula,
• xmax is the highest number in the set.
• xmin is the lowest number in the set.

## Why Is the Range Useful?

There are a lot of differences in the things we choose to measure. People have many different heights or ages or incomes or test scores. The range is useful because they can summarize how spread out the data is. Rather than recording the height of every adult in a group, it is useful to know that the difference between the tallest and the shortest height is 1 foot, or 31 cm.

## The Range As a Measure of Spread

The range is a measure of spread. It is one way of measuring how spread out numbers in a set are. In the example above, we found that the range of 1, 2, 3, 4, 5 is 4. The range is quite small, so the numbers are not that spread out. The range of 10, 20, 30, 40, 50 is 40 (= 50 - 10). The range is larger, so the numbers are more spread out. Note: The other useful summary is a 'measure of location' (also called a 'measure of central tendency'): where most of the numbers are bunched around. An average is a measure of location.

The range is the simplest measure of spread, but not a very good one. It only uses two pieces of information, the largest number and the smallest number. It ignores all other information. Consider the heights of the two groups of people below. The range (and the average - the middle height) is the same in both sets.

• In the top group, the heights are much more bunched around the middle (the average) height.
• In the bottom group, the heights are further from the middle height. The heights are more spread out.
A good measure of spread should tell us that the bottom group is more spread out than the top group, but the range tells us that both groups are equally spread out.

## The Range Is a Distance on a Number Line

The range is distance between the highest number and the lowest number in a set. Consider the numbers below:
1, 2, 3, 4, 5
We can plot these numbers on a number line (with the highest and lowest numbers in red).

The range is the distance between the highest and lowest numbers on the number line.

## The Range and Outliers

Sometimes in a group of numbers, a few of them are much larger...
3, 5, 2, 7, 150
...or much smaller than the rest:
1, 502, 847, 564, 980
These relatively large (150) or small (1) numbers are called outliers. Because the range is the difference between the largest and smallest numbers in a set, the range is affected by outliers. Consider the numbers:
1, 2, 3
The range is 2 (= 3 − 1). If instead the numbers have a large outlier:
1, 2, 300
The range is now 299 (= 300 − 1).