# 7.05 Mean absolute deviation (MAD)

Lesson

Measures of distribution tell us how far the scores in a data set are spread out.

We've already looked at one measure of spread, the range, which is the difference between the greatest and least score in a data set.

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$2324=1

$18-24=-6$1824=6

$31-24=7$3124=7

$28-24=4$2824=4

$20-24=-4$2024=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.

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:

1. Calculate the mean.
2. Find the absolute value of the differences between each value in the set and the mean.
3. 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.

1. First, calculate the mean of

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

2. 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 $\editable{}$
3. 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?

1. 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

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 center, spread, and overall shape.

#### 6.SP.A.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.B.5

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

#### 6.SP.B.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 from the overall pattern with reference to the context in which the data was gathered.