In this explainer, we will learn how to find and interpret the mean absolute deviation.

You should already know that the mean of a data set is calculated by adding up all the values and dividing by the number of data values in the set.

The mean absolute deviation is an example of a measure of variability. These give us some information about how spread out the data in a data set is. Another example of a measure of variability is the range. When used together with measures of center like the mean and median, they can give us a useful overview of the data set.

Let us do an example. A chef wants to hire someone to work in his kitchen. There are two candidates: Natalie and Benjamin. To test their abilities, he times how long it takes them to make each of four meals and records the data in two tables.

Natalie | ||||

Meal | A | B | C | D |
---|---|---|---|---|

Time Taken (minutes) | 2 | 8 | 5 | 9 |

Benjamin | ||||

Meal | A | B | C | D |
---|---|---|---|---|

Time Taken (minutes) | 5 | 6 | 5 | 7 |

He decides to calculate the mean time it took each candidate to make the four meals; he is planning to hire the candidate with the lower (quicker) mean time.

The mean time it takes Natalie to make a meal is

The mean time it takes Benjamin to make a meal is

Since the mean times are the same, he does not know which candidate is better. He needs to use another statistical measure to help him decide who to hire. He decides that another way to judge the candidate would be to see who is the most consistent. He would prefer a candidate who can reliably prepare all the meals in about the same amount of time, rather than one who might prepare some very fast but others too slowly.

To test this, he decides to calculate the differences between the mean time and the actual time it took to prepare each dish. The bar graphs show the results.

From the bar graphs, we can see that Natalie’s times are more spread out from
the mean; there is less consistency. We say that her
times are *more variable*. On the other hand, we can see that Benjamin’s times are much more
consistent, he produced most of the dishes in a time that is close to the mean.
We say that his times are *less variable*.
To measure this variability with a number, we can use the mean absolute deviation.

First, the chef calculates the absolute value of each of the differences between the times taken and the mean time and summarizes the data in a table.

Natalie | ||||

Meal | A | B | C | D |
---|---|---|---|---|

Time Taken (minutes) | 2 | 8 | 5 | 9 |

Distance from Mean | 4 | 2 | 1 | 3 |

Benjamin | ||||

Meal | A | B | C | D |
---|---|---|---|---|

Time Taken (minutes) | 5 | 6 | 5 | 7 |

Distance from Mean | 1 | 0 | 1 | 2 |

To get an idea of the typical difference (or distance from the mean), he then calculates the mean of these distances; this is the mean absolute deviation.

For Natalie, the mean absolute deviation of her times is the mean of the distances,

This suggests that when Natalie cooks one of the meals, it is expected to take 2.5 minutes more or less than the mean time.

For Benjamin, the mean absolute deviation of his times is

This suggests that when Benjamin cooks one of the meals, it is expected to only take 1 minute more or less than the mean time.

The chef sees that the mean absolute deviation of Benjamin’s times is lower than the mean absolute deviation of Natalie’s times. This shows that Benjamin is more consistent in his cooking times and agrees with what we observed from the bar graphs. Based on the mean absolute deviation, the chef decides that Benjamin is a better candidate because his work is more consistent.

Now, we will summarize the definition of the mean absolute deviation before we do another example.

### Definition: The Mean Absolute Deviation

The mean absolute deviation is the mean of the distances between each data value and the mean.

This is the mean of the absolute values of the differences between each data value and the mean.

### Example 1: Calculating the Mean Absolute Deviation of a Numerical Data Set

Calculate the mean absolute deviation of 15, 5, 17, 7, 14, 5, 15, and 20. Round your answer to the nearest tenth if necessary.

### Answer

To find the mean absolute deviation, we have to find the mean of the absolute values of the differences between the data values and the mean.

First, we calculate the mean to be

Next, we have to calculate the absolute values of the differences between the data values and the mean. We can also think of these as measuring the distance of each data value from the mean. For example, if we draw the data points on a line plot or think of them on a number line, then each point has a distance from the mean. These distances are always calculated to be nonnegative numbers.

For example, the data value 15 is a distance of from the mean. We can calculate the other distances in the same way. We summarize the results in a table.

Finally, we calculate the mean absolute deviation by finding the mean of these distances. Hence, the mean absolute deviation is

We were asked to round our answer to the nearest tenth, and that is 4.9.

Let us summarize the method for finding the mean absolute deviation of a data set.

### How to Find the Mean Absolute Deviation

- Find the mean of the data set.
- Calculate the distance between each data value and the mean.
- Calculate the mean of the distances.

The mean absolute deviation is a measure of how spread out the data is. A small mean absolute deviation tells us that most of the data values are very close to the mean (since the expected distance from each data value to the mean is small). A high mean absolute deviation tells us that many of the data values are spread out further from the mean. We can use the mean absolute deviation to compare data sets. We will see how in this final example.

### Example 2: Using the Mean Absolute Deviation to Compare Data Sets

A farmer is testing two different types of plant food in his garden. He used Ultra Feed on five of his plants and Mega Feed on another five. He measured the growth increase of each of the plants and recorded the results, rounded to one decimal place, in the table.

Increase in Growth (cm) | |||||

Ultra Feed | 1.2 | 1.4 | 0.9 | 1.8 | 1.7 |
---|---|---|---|---|---|

Mega Feed | 1.5 | 0.6 | 1.1 | 1.3 | 1.0 |

- Find the mean absolute deviation of the growth of the plants fed Ultra Feed.
- Find the mean absolute deviation of the growth of the plants fed Mega Feed.
- Which feed produces less variability in the growth?

### Answer

To calculate the mean absolute deviation, we have to first find the mean, then calculate the distance of each data value from the mean, and finally calculate the mean of these distances.

- The mean increase in growth of the plants fed Ultra Feed
is

Now we calculate the distance of each data value from the mean. We do this by finding the absolute value of the difference between the mean and the data value. We calculate the following distances:

We can summarize this in a table.

Then, the mean absolute deviation is the mean of the distances 0.2, 0, 0.5, 0.4, and 0.3. This isUltra Feed Increase in Growth (cm) 1.2 1.4 0.9 1.8 1.7 Distance from Mean 0.2 0 0.5 0.4 0.3 - We can follow the same method to find the mean absolute deviation of the growth of the plants fed Mega Feed. We summarize the results in the table below. Hence, the mean absolute deviation is
- To see which feed produces less variability in the growth of the plants, we can compare the mean absolute deviation. Since , we see that Mega Feed produces less variation.