Mean Absolute Deviation Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mean Absolute Deviation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The average distance between each data value and the mean of the data set. Calculated by finding the mean, computing the absolute value of each deviation from the mean, and averaging those absolute deviations.

Standard deviation can feel abstract with its squaring and square roots. MAD is simpler: just ask 'on average, how far is each data point from the center?' If the mean test score is 80 and the MAD is 5, a typical student scored about 5 points away from 80—some above, some below.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: MAD is the mean of the absolute distances of each value from the data's mean.

Common stuck point: The procedure for mean absolute deviation is the easy part; the trap is forgetting the absolute value. Asking "Am I averaging the ABSOLUTE distances of each value from the mean (not squared distances, not signed ones)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I averaging the ABSOLUTE distances of each value from the mean (not squared distances, not signed ones)?

Worked Examples

Example 1

easy

Calculate the Mean Absolute Deviation (MAD) for

\{2, 5, 7, 10, 6\}

and explain what it measures.

Answer

MAD = 2

. On average, each data point deviates from the mean by 2 units.

First step

Mean:

\bar{x} = (2+5+7+10+6)/5 = 30/5 = 6

Full solution

2
Absolute deviations: $|2-6|=4$ , $|5-6|=1$ , $|7-6|=1$ , $|10-6|=4$ , $|6-6|=0$
3
$MAD = \frac{4+1+1+4+0}{5} = \frac{10}{5} = 2$
4
Interpretation: on average, each value is 2 units away from the mean

MAD measures the average absolute distance of data points from the mean. Unlike variance (which squares deviations), MAD keeps deviations in the original units. It is more interpretable than variance and more resistant to extreme values than variance.

Example 2

medium

Compare MAD and standard deviation for the data

\{1, 5, 5, 5, 9\}

. Calculate both and explain when MAD is preferred.

Example 3

medium

Calculate the MAD of

\{10,12,14,16,18\}

step by step.

Example 4

medium

Two students' quiz scores: A:

\{8,8,8,9,7\}

(mean

8

); B:

\{5,9,8,11,7\}

(mean

8

). Compute both MADs and decide who is more consistent.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Calculate MAD for daily temperatures (°F):

\{68, 72, 65, 75, 70\}

Example 2

hard

Data set A:

\{4, 5, 5, 6\}

and Data set B:

\{1, 5, 5, 9\}

. Both have mean 5. Calculate MAD for each and explain which has greater variability and why.

Example 3

easy

Find the mean of the data set

2, 4, 6, 8

(first step toward MAD).

Example 4

easy

The mean of a set is

10

. A data value is

7

. What is its absolute deviation from the mean?

Example 5

easy

For the data

5, 5, 5, 5

, what is the MAD?

Example 6

easy

Why must MAD use absolute values of deviations?

Example 7

easy

A data set has 5 values. To compute MAD, you divide the sum of absolute deviations by what number?

Example 8

easy

The absolute deviations of a 4-value set are

1, 3, 2, 2

. Find the MAD.

Example 9

easy

Does MAD measure center or spread of a data set?

Example 10

easy

Between MAD and standard deviation, which uses squared deviations?

Example 11

medium

Find the MAD of the data set

4, 6, 8, 10, 12

Example 12

medium

Find the MAD of

3, 3, 4, 8, 12

Example 13

medium

Two basketball players average 20 points. Player A has MAD

=2

; Player B has MAD

=9

. Who is more consistent?

Example 14

medium

A data set has mean

50

and MAD

5

. Roughly interpret what MAD

=5

tells you.

Example 15

medium

Find the MAD of

10, 20, 30, 40

Example 16

medium

Data:

7, 9, 9, 11, 14

. Compute the MAD.

Example 17

medium

If every value in a set is increased by 7, what happens to the MAD?

Example 18

medium

A 6-value set has absolute deviations

0, 1, 2, 3, 4, 2

. Compute the MAD.

Example 19

challenge

Data set

2, 4, 4, 4, 5, 5, 7, 9

has mean

5

. Compute the MAD.

Example 20

challenge

If every value in a set is multiplied by 4, how does the MAD change? Justify.

Example 21

challenge

Set A has values

10, 10, 10, 10

(MAD

0

). Set B has the same mean

10

but values

4, 8, 12, 16

. Compute B's MAD and state what differs between the sets.

Example 22

medium

Find the MAD of

6, 8, 10, 12, 14, 16

Example 23

easy

Find the MAD of the data set

\{3,5,7,9\}

Example 24

easy

A data set has mean

20

. One value is

17

. What is its absolute deviation from the mean?

Example 25

easy

Compute the MAD of

\{8,8,8,8,8\}

Example 26

easy

A data set has mean

25

and MAD

4

. Roughly interpret MAD

=4

Example 27

easy

Find the MAD of

\{1,2,3,4,5\}

Example 28

medium

Find the MAD of

\{12,15,18,21,24\}

Example 29

medium

Compute the MAD of

\{2,4,6,6,8,10\}

Example 30

medium

Two teams' daily wins both average

7

. Team A's MAD is

1

; Team B's MAD is

4

. Which team is more consistent?

Example 31

medium

Find the MAD of the test scores

\{75,80,85,90,95\}

Example 32

medium

If every value in a data set is decreased by

5

, what happens to the MAD?

Example 33

medium

If every value in a data set with MAD

=3

is multiplied by

5

, what is the new MAD?

Example 34

medium

Compute the MAD of

\{5,5,5,5,15\}

Example 35

medium

Daily rainfall (in mm) for five days:

\{2,4,6,8,10\}

. Find the MAD.

Example 36

hard

A four-value set

\{x,3,5,9\}

has mean

5

. Find

x

and the MAD.

Example 37

hard

A data set of

5

values has mean

20

and sum of absolute deviations equal to

25

. Find the MAD.

Example 38

hard

Set

A=\{2,4,6,8,10\}

and set

B=\{0,3,6,9,12\}

both have mean

6

. Which has the greater MAD, and by how much?

Example 39

hard

For a data set

\{a,b,c\}

with mean

0

, prove that the MAD equals

\tfrac{|a|+|b|+|c|}{3}

Example 40

medium

Compute the MAD of

\{1,1,1,1,6\}

Example 41

challenge

For a data set

\{x_1,x_2,\dots,x_n\}

, show that the value

m

that minimizes

\sum |x_i-m|

is the median, not the mean.

Example 42

challenge

A data set has mean

\mu

and MAD

d

. After appending one new value

\mu

, what happens to the MAD?

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Related Concepts

Mean Absolute Value Standard Deviation Variance

Background Knowledge

These ideas may be useful before you work through the harder examples.

meanabsolute value