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.2

Understand that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread, and overall shape.

6.SP.3

Recognize that a measure of center 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.5

Summarize numerical data sets in relation to their context, such as by:

6.SP.5.c

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations (for example, outliers) from the overall pattern with reference to the context in which the data was gathered.

7.05 Mean absolute deviation (MAD)