# 4.8: Spread or Dispersion - Range (Range of Spread or Dispersion)

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Range of Dispersion ClkerFree Vector Images - pixabay.com/en/thermometer-temperature-instrument-29533/

Bob the weatherman records the high temperature each day for two weeks in June. He wants to see how much the temperature changes throughout those two weeks. Here is the data he collected for the 14 days he was keeping track of temperature highs:

72, 75, 68, 70, 67, 76, 71, 72, 68, 75, 71, 75, 69, 70

In order for Bob to know the difference in high temperatures throughout the two weeks, he needs to find the range of the data.

In this concept, you will learn how to find the range of a set of data.

## Range

The range of a set of data simply tells where the numbers fall and whether data in set is close together or spread apart. A set of data with a small range tells us something different than a set of data with a large range.

To find the range of a set of data:

1. Put the values in the data set in numerical order. This will show which is the greatest number in the set (the maximum), and which is the smallest number (the minimum).
2. Subtract the minimum from the maximum. This is the range of the data.

Take a look at the data set below.

11, 9, 8, 12, 11, 11, 14, 8, 10

First, arrange the data in numerical order.

8, 8, 9, 10, 11, 11, 11, 12, 14

Next, identify the minimum and maximum of the set. The minimum is 8 and the maximum is 14.

Then, subtract the minimum from the maximum to find the range.

14−8=6

The range of the data set is 6. This means that all of the numbers in the data set fall within six places of each other. All of the data results are fairly close together.

The context of a set of data and its range can reveal important information about the data. For example, if a data set showing plant growth using special soil is 4, that shows that the plants all grew in close range to each others. In other words, the special soil had the same impact on all of the plants. Suppose though, the data showed a wider range of 15. That would signify that perhaps the special soil did not have as equal of an impact on the plants as some grew a lot and some grew little to make that range so wide.

## Examples

### Example 1

Earlier, you were given a problem about Bob and his weather statistics.

Bob kept track of the high temperatures for two weeks to see the difference in temperature. Here was is data:

72, 75, 68, 70, 67, 76, 71, 72, 68, 75, 71, 75, 69, 70

Bob needs to find the range of his data to determine the range in temperature from these two weeks.

First, Bob puts the data in numerical order.

67, 68, 68, 69, 70, 70, 71, 71, 72, 72, 75, 75, 75, 76

Next, Bob identifies the minimum and maximum of the data set. The minimum is 67 and the maximum is 76. These data points mean that both were the high temperatures one day in those two weeks, but the weather clearly changed course at some point.

Then, Bob subtracts the minimum from the maximum.

76−67=9

The answer is 9. There was a 9 degree change in temperature at some point throughout the two weeks that Bob kept track of high temperatures.

### Example 2

The following is the number of patrons at a local movie theater. What is the range of the data?

26, 22, 40, 45, 46, 18, 30, 80, 60, 75

To figure this out, we need to find the difference between the highest number of patrons and the lowest number of patrons.

First, put the data in order from least to greatest.

18, 22, 26, 30, 40, 45, 46, 60, 75, 80

Next, identify the minimum and maximum of the data set. The minimum is 18 and the maximum is 80.

Then, subtract the minimum from the maximum to find the range.

80−18=62

The answer is 62. This shows there is a wide range of people who attend the movies in that data set. There is not consistent attendance at the movies.

### Example 3

Find the range of the following data set.

4, 5, 6, 9, 12, 19, 20

First, put the numbers in numerical order. This data set is already in order.

Next, identify the minimum and maximum of the data set. The minimum is 4 and the maximum is 20

Then, subtract the minimum from the maximum to find the range.

20−4=16

### Example 4

Find the range of the following data set.

5, 2, 1, 6, 8, 20, 25

First, put the numbers in numerical order.

1, 2, 5, 6, 8, 20, 25

Next, identify the minimum and maximum of the data set. The minimum is 1 and the maximum is 25.

Then, subtract the minimum from the maximum to find the range.

25−1=24

### Example 5

Find the range of the following data set.

65, 23, 22, 45, 11, 88, 99, 123, 125

First, put the numbers in numerical order.

11, 22, 23, 45, 65, 88, 99, 123, 125

Next, identify the minimum and maximum of the data set. The minimum is 11 and the maximum is 125.

Then, subtract the minimum from the maximum to find the range.

125−11=114

## Review

Find the range for each set of data.

1. 4, 5, 4, 5, 3, 3
2. 6, 7, 8, 3, 2, 4
3. 11, 10, 9, 13, 14, 16
4. 21, 23, 25, 22, 22, 27
5. 27, 29, 29, 32, 30, 32, 31
6. 34, 35, 34, 37, 38, 39, 39
7. 43, 44, 43, 46, 39, 50
8. 122, 100, 134, 156, 144, 110
9. 224, 222, 220, 222, 224, 224
10. 540, 542, 544, 550, 548, 547
11. 2, 3, 3, 3, 2, 2, 2, 5, 6, 7
12. 4, 5, 6, 6, 6, 7, 3, 2
13. 23, 22, 22, 24, 25, 25, 25
14. 123, 120, 121, 120, 121, 125, 121
15. 678, 600, 655, 655, 600, 678, 600, 600