Now we are going to learn about a new measure of spread called the Mean absolute deviation (MAD). The MAD of a set of data is the average distance between each data value and the mean.

Let's use an example to help explain this.

Worked example

Question 1

Find the mean absolute deviation of $23,18,31,28,20$23,18,31,28,20.

Think/Do:

1. Find the mean.

$\frac{23+18+31+28+20}{5}$23+18+31+28+205$=$=$24$24

2. Find the difference between each individual score and the mean.

$23-24=-1$23−24=−1

$18-24=-6$18−24=−6

$31-24=7$31−24=7

$28-24=4$28−24=4

$20-24=-4$20−24=−4

Take the absolute value of each difference.

$\left|-1\right|=1$|−1|=1

$\left|-6\right|=6$|−6|=6

$\left|7\right|=7$|7|=7

$\left|4\right|=4$|4|=4

$\left|-4\right|=4$|−4|=4

3. Find the mean of these differences.

$\frac{1+6+7+4+4}{5}$1+6+7+4+45$=$=$4.4$4.4

Therefore, the mean absolute deviation is $4.4$4.4 units.

Reflect: This means that, on average, scores in this data set are $4.4$4.4 units above or below the mean.

The box below summarizes our steps.

Mean absolute deviation (MAD)

The mean absolute deviation (MAD) of a set of data is the average distance between each data value and the mean.

To calculate the mean absolute deviation of a set of data:

Calculate the mean.
Find the absolute value of the differences between each value in the set and the mean.
Find the average of those values.

Practice questions

Question 2

Calculate the mean absolute deviation of the values below, by answering each question.

$2$2, $8$8, $6$6, $3$3, $10$10, $15$15, $6$6 and $6$6.

State your answer to 2 decimal places if necessary.

First, calculate the mean of

$2$2, $8$8, $6$6, $3$3, $10$10, $15$15, $6$6 and $6$6.

Complete the table of values, finding the distance of each value from the mean.

Value	Distance from $7$7
$2$2	$\editable{}$
$8$8	$\editable{}$
$6$6	$\editable{}$
$3$3	$\editable{}$
$10$10	$\editable{}$
$15$15	$\editable{}$
$6$6	$\editable{}$
$6$6	$\editable{}$

Using your values from the table above, calculate the mean of the differences.

Question 3

Which of the following is true concerning the mean absolute deviation of a set of data?

It describes the average distance between each data value and the mean.
A
It describes the variation of the data values around the median.
B
It describes the absolute value of the mean.
C
It describes the variation of the data values around the mode.
D

Outcomes

6.SP.A.2

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its measures of center (mean, median, mode), measures of variation (range only), and overall shape.

6.SP.A.3

Recognize that a measure of center (mean, median, mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

6.SP.B.5

Summarize numerical data sets in relation to their context.

7.05 Mean absolute deviation (MAD)