In this explainer, we will learn how to find the range of a data set.

Throughout this explainer, we will only be considering data sets involving numbers, which will allow us to preform calculations on the members of the data set. The range of a data set is one way of measuring the spread or dispersion of a data set. It is essentially the largest possible difference between two entries in the data set, thus giving us a measure of this spread. We can define this formally as follows.

### Definition: The Range of a Data Set

The range of a data set is the difference between the largest and smallest values:

The range gives us an indication as to how spread out the data is; it is the maximum difference between elements in the data set.

Let’s see how to apply this definition and why the range may be useful by considering an example. Imagine at a shop there are bunches of grapes, all costing the same price. The weights of each bag will differ slightly and can be represented by the following data set:

The range of this data set is the difference between the greatest weight and the lowest weight of the bags. We can calculate this as

This gives us some useful information about the spread of weights of the bags. First, it tells us that the maximum difference in the weights of the bunches is 7 g. Second, it allows us to use the weight of the heaviest or lightest bunch to determine the other extreme. For example, if we know that the range is 7 g and the heaviest bunch has a weight of 254 g, then

Finally, if we know the range of two different data sets, we can use these to compare them. For instance, suppose a second group of apples had a range of 15 g. This indicates that the spread of the weights in the second group is larger than in the first.

Let’s look at some examples to practice finding and using the range.

### Example 1: Verifying the Range of a Given Data Set

True or False: If the numbers of goals scored by twelve soccer players in a season are 13, 11, 12, 5, 5, 9, 6, 11, 8, 5, 6, and 19, then the range of the data is 14 goals.

### Answer

We recall that the range is the difference between the smallest data value and the largest. One way to identify these values is to write the data in order of size from smallest to largest. This gives us the following:

The range is then the difference between the smallest data value and the largest, which is

Therefore, the statement is true.

In our next two examples, we will use the range and a given extreme value to determine the other extreme value of the dataset.

### Example 2: Finding the Smallest Element in a Data Set When Given the Largest Value and the Range

Let the greatest element in a set be 445 and the range of the set be 254. What is the smallest element of this set?

### Answer

We recall that the range of a set of data is the difference between the largest and the smallest values in the set:

Rearranging this equation, we have

Now, we substitute the greatest value and the range into the equation to get

Hence, the smallest element is 191.

### Example 3: Finding the Largest Element in a Data Set When Given the Smallest Value and the Range

Given that the range of a set of data is 86 and the lowest value is 53, determine the highest value in the set.

### Answer

We recall that the range of a set of data is the difference between the largest and the smallest values in the set:

Rearranging this equation, we have

We substitute in the values of the range and smallest data value to get

Hence, the highest value in the set is 139.

In our next example, we will see how the range of a data set can give us information about the spread of the data values.

### Example 4: Comparing Data Using the Range

The table below shows the number of points scored by two basketball teams during the 8 games they played this month.

Team A | 112 | 107 | 99 | 101 | 92 | 96 | 89 | 99 |
---|---|---|---|---|---|---|---|---|

Team B | 70 | 66 | 75 | 74 | 84 | 67 | 78 | 93 |

- The range of the scores for team A is 23. Find the range of the scores for team B.
- Which of the following statements can be used to compare the spread of the scores for team A and team B?
- The spread of the scores for team A and that for team B are identical.
- The spread of the scores for team A is less than the spread of the scores for team B.
- The spread of the scores for team A is greater than the spread of scores for team B.
- We cannot draw any comparisons between the spread of the scores for team A and team B.

### Answer

**Part 1**

We recall that the range of a set of data is the difference between the largest and the smallest values in the set:

We can therefore find the range of team B’s scores by finding the difference in the largest score and the smallest score. From the table, we see that the largest score is 93 and the smallest is 66. Hence,

**Part 2**

We are told that the range of Team A’s scores is 23 and in the previous part we found that the range of team B’s scores was 27. Since 27 is larger, we can say that their scores are more spread out.

In particular, we can say that the spread of the scores for team A is less than the spread of the scores for team B, which is option B.

In our next example, we will determine the range of a data set given in a line plot.

### Example 5: Describing the Effect of an Additional Data Value on the Range of a Data Set

The following figure demonstrates the number of glasses of water a group of people consume per day. Describe how the range would change if an additional data value of 1 was added to the data set.

### Answer

We recall that the range of a data set is equal to the difference between the largest and smallest data values. We are given a line plot of this data set; we recall that each mark tells us that the value is included in the data set. For example, since there is one mark next to 4, we can conclude that 4 appears only once in the data set.

We can use this to determine the largest and smallest elements in the data set. The largest element with a mark is 5 and the smallest is 0. Hence, the range of the data set is the difference between these two values:

If an extra data value of 1 was added, this would not change the largest or the smallest data value, so the range will not change.

Therefore, the range would remain unchanged at 5.

In our final example, we will determine the possible missing value of a data set by using its range.

### Example 6: Using the Range to Find the Missing Value in a Set of Data

Sameh has the following data: .

If the range is 7, which number could be?

- 5
- 6
- 2
- 9
- 8

### Answer

We recall that the range of a data set is the difference between the largest and smallest values of the data set.

Let’s start by considering the data set with the value of removed. Then the data set is

The greatest value is 9 and the lowest value is 6, so the range is

There are then two possible ways we can add a value of to the data set so that the range increases to 7. Either we add a new greatest value or a new least value. We can consider each option separately.

First, if is the greatest element of the data set, then the range is given by

Since the range is 7, we have

Similarly, if is the least element of the data set, then

Since the range is 7, we have

Hence, or 13.

Looking at the list of answers we were given, we see that only 2 is a possible value for , which is option C.

Let’s finish by recapping some of the important points of this explainer.

### Key Points

- The range of a data set is the difference between the largest and smallest values.
- The range gives us an indication as to how spread out the data is; it is the maximum difference between elements in the data set.
- We can use the range and one of the maximum or minimum values of the data set to determine the other extreme value.
- We can compare the spread of two data sets by comparing their ranges.