Introduction

When analyzing data, we often want to know how "spread out" the data is. In other words, how far away are the data points from the center?

One way to represent the spread of the data is to use the mean absolute deviation (MAD) of the data. To compute the mean absolute deviation, we

  • compute the mean of the data,

  • subtract the mean from each number in the data set, find the absolute value of the results, and add all these together, then

  • divide the sum by the total number of data points.

To demonstrate, let's compute the mean absolute deviation of the data set below: 1, \: 6, \: 5

First, we find the mean of the data: {\color{blue}\text{mean}} = \dfrac{1+6+5}{3} = \dfrac{12}{3} = {\color{blue}4}

Next, we subtract the mean from each number in the data set, find the absolute value of the results, and add all these together:

\begin{align*} |1-{\color{blue}4}| + |6-{\color{blue}4}| + |5-{\color{blue}4}| &= \\[2pt] |{-3}| + |2| + |1| &= \\[2pt] 3 + 2 + 1 &= \\[2pt] {\color{red}6} \end{align*}

Finally, to calculate the mean absolute deviation (MAD), we divide the sum obtained above ( {\color{red}6} ) by the total number of data points ( 3 ): \text{MAD} = \dfrac{\color{red}6}{3} = 2

Therefore, the mean absolute deviation is 2.

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Example: Computing the MAD Given the Mean

Given that the mean of the data set below is 5 , what is the mean absolute deviation of the data?

9, \: 2, \: 3, \: 6

EXPLANATION

We are given that the mean is {\color{blue}5}.

So, we subtract the mean from each number in the data set, find the absolute values of the results, and add all these together:

\begin{align*} \vert 9-{\color{blue}5} \vert + \vert 2-{\color{blue}5} \vert + \vert 3-{\color{blue}5} \vert + \vert 6-{\color{blue}5} \vert &= \\[5pt] \vert 4 \vert + \vert {-3} \vert + \vert {-2} \vert + \vert 1 \vert &= \\[5pt] 4 + 3 + 2 + 1 &= \\[5pt] {\color{red}10} \end{align*}

Finally, to calculate the mean absolute deviation (MAD), we divide the sum obtained above ( {\color{red}10} ) by the total number of data points ( 4 ): \text{MAD} = \dfrac{\color{red}10}{4} = 2.5

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Practice: Computing the MAD Given the Mean

Given that the mean of the data set below is $3$, what is the mean absolute deviation of the data?

\[ 1, \: 3, \: 2, \: 8, \: 1 \]

a
 $0.6$
b
 $2$
c
 $1.8$
d
 $3$
e
 $1$
Practice: Computing the MAD Given the Mean

Given that the mean of the data set below is $8.4$, what is the mean absolute deviation of the data?

\[ 4, \: 12, \: 6, \: 13, \: 7 \]

a
 $4.28$
b
 $3.28$
c
 $3.3$
d
 $4.3$
e
 $3$
Example: Computing the MAD

Find the mean absolute deviation of the data set below. 3, \: 1, \: 4, \: 8

EXPLANATION

First, we find the mean of the data: {\color{blue}\text{mean}} = \dfrac{3 + 1 + 4 + 8}{4} = \dfrac{16}{4} = {\color{blue}4}

Next, we subtract the mean from each number in the data set, find the absolute values of the results, and add all these together:

\begin{align*} \vert 3-{\color{blue}4} \vert + \vert 1-{\color{blue}4} \vert + \vert 4-{\color{blue}4} \vert + \vert 8-{\color{blue}4} \vert & = \\[5pt] \vert {-1} \vert + \vert {-3} \vert + \vert 0 \vert + \vert 4 \vert &= \\[5pt] 1 + 3 + 0 + 4 &= \\[5pt] {\color{red}8} \end{align*}

Finally, to calculate the mean absolute deviation (MAD), we divide the sum obtained above ( {\color{red}8} ) by the total number of data points ( 4 ): \text{MAD} = \dfrac{\color{red}8}{4} = 2

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Practice: Computing the MAD

\[ 6, \: 3, \: 10, \: 5 \] The mean absolute deviation of the data set above is

a
b
c
d
e
Practice: Computing the MAD

Find the mean absolute deviation of the data set below. \[ 4, \: 1, \: 2, \: 8, \: 5 \]

a
 $5$
b
 $2.2$
c
 $2$
d
 $1.5$
e
 $2.5$
Example: Word Problems

The total times of five racers during a Grand Prix were: 14 \,\text{minutes}, \quad 16 \,\text{minutes}, \quad 13 \,\text{minutes}, \quad 21 \,\text{minutes}, \quad 19 \,\text{minutes} Rounded to one decimal place, what is the mean absolute deviation of the times?

EXPLANATION

First, we find the mean of the data: {\color{blue}\text{mean}} = \dfrac{14 + 16 + 13 + 21 + 19}{5} = \dfrac{83}{5} = {\color{blue}16.6}

Next, we subtract the mean from each number in the data set, find the absolute values of the results, and add all these together:

\begin{align*} \vert 14-{\color{blue}16.6} \vert + \vert 16-{\color{blue}16.6} \vert + \vert 13-{\color{blue}16.6} \vert + \vert 21-{\color{blue}16.6} \vert + \vert 19-{\color{blue}16.6} \vert &= \\[5pt] \vert {-2.6} \vert + \vert {-0.6} \vert + \vert {-3.6} \vert + \vert 4.4 \vert + \vert 2.4 \vert &= \\[5pt] 2.6 + 0.6 + 3.6 + 4.4 + 2.4 &= \\[5pt] {\color{red}13.6} & \end{align*}

Finally, to calculate the mean absolute deviation (MAD), we divide the sum obtained above ( {\color{red}13.6} ) by the total number of data points ( 5 ): \text{MAD} = \dfrac{\color{red}13.6}{5} = 2.72

Rounding to the nearest tenth gives: 2.72 \rightarrow 2.7

Therefore, the mean absolute deviation of the times of the racers is 2.7 minutes.

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Practice: Word Problems

A farmer has four tanks whose capacities are as follows: \[ 21 \,\text{liters}, \quad 23 \,\text{liters}, \quad 25 \,\text{liters}, \quad 29 \,\text{liters} \] What is the mean absolute deviation of tanks' capacities?

a
 $2.2$ liters
b
 $2.5$ liters
c
 $1.6$ liters
d
 $1.5$ liters
e
 $2.3$ liters
Practice: Word Problems

Michael has five brothers whose ages are as follows: \[ 7 \,\text{years}, \quad 3 \,\text{years}, \quad 5 \,\text{years}, \quad 6 \,\text{years}, \quad 2 \,\text{years} \] Rounded to one decimal place, what is the mean absolute deviation of Michael's brothers' ages?

a
 $1.7$ years
b
 $1.9$ years
c
 $2.2$ years
d
 $2.5$ years
e
 $1.5$ years
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