Explainer: The Range of a Data Set

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

Definition: The Range

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

The range tells us how spread-out the data is.

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

Example 1: Finding the Range of a Data Set

The number of goals scored by twelve soccer players in a season are 13, 11, 12, 5, 5, 9, 6, 11, 8, 5, 6, and 19. State whether the following statement is true or false: The range of the data is 14 goals.

Answer

The range is the difference between the smallest data value and the largest. One way to identify these is to write the data in order from least to greatest.

Then, the range is 195=14.

Therefore, the statement is true.

Example 2: Finding the Range of Data in a Line plot Graph

The graph shows the weights, in kilograms, of emperor penguins at a zoo. Is the range of the weights 24 kilograms?

Answer

To find the range of the data, we need to identify the largest and smallest data values.

The smallest data value is the weight 23 kg and the largest data value is 49 kg.

Then, the range is the difference, 4923=26.kg

So, the range of the weights is not 24 kg.

The range of a data set tells us how spread-out the data is. For example, consider the following two data sets.

The data values in the first data set are much closer together than the data values in the second data set and the range of the first data set is much smaller than the range of the second.

But, this is not the only reason that one data set has a higher range than the other. Consider the following two data sets which are almost identical, but the second has one value which is much higher than the rest.

Even though 4 out of the 5 data values are the same, the range of the second data set is much higher than the range of the first because the second data set has a surprising large greatest value. Values like this are sometimes called outliers and they can greatly affect the range.

We will finish with some more examples.

Example 3: Finding the Range Given a Summary of the Data

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 know that the range of a set of data is the difference between the greatest and the smallest values in the set: rangegreatestvaluesmallestvalue=.

Therefore, smallestvaluegreatestvaluerange=.

Now we substitute for the greatest value and the range: smallestvalue=445254=191.

So, the smallest element is 191.

Example 4: Finding the Value of the Range When a Data Value Is Added

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

The range of the data is equal to the difference between the greatest and least data values.

Since the largest data value in the line plot is 5 glasses and the smallest is 0 glasses, the range is currently 50=5.glasses

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

Finally, we will look at a pair of examples which show how we can use the range to identify possibilities for unknown values in a data set.

Example 5: Using the Range To Find Missing Data Values

Matthew has the following data: 6,8,𝑘,8,8,9.

If the range is 3, which of the following numbers could 𝑘 be: 6, 5, 3, 4, or 13?

Answer

The range is equal to the difference between the greatest and least values.

Without 𝑘, the range of the data 6,8,8,8,9 is 96=3.

If when we add 𝑘, the range is still 3, this means that 𝑘 does not change the smallest or the largest value. So, the possible (whole number) values of 𝑘 are 6,7,8,9.

Looking at the list of answers we were given, we see that only 6 is a possible value for 𝑘.

In the above example, the range of the data set 6,8,𝑘,8,8,9 is equal to 96, which was the range of the data set without the unknown value 𝑘. Now, we will look at the same data set when the range is equal to something else.

Example 6: Using the Range To Find Missing Data Values

Benjamin has the following data: 6,8,𝑘,8,8,9.

If the range is 7, which of the following numbers could 𝑘 be: 5, 2, 9, 6, or 8?

Answer

The range is equal to the difference between the greatest and least values.

Without 𝑘, the range of the data 6,8,8,8,9 is 96=3.

If when we add 𝑘, the range increases to 7, this means that 𝑘 must be either the smallest or the largest value.

We will think about what would happen in both of these cases.

We have found two possible values for 𝑘: 2 or 13. Looking at the list of answers we were given to choose from, we see that 2 is the answer.

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