Mean Absolute Deviation (MAD) Examples in Statistics

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

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

Concept Recap

The Mean Absolute Deviation (MAD) is the average of the absolute distances between each data point and the mean of the dataset. It measures how spread out data values are from the center, with larger MAD values indicating more variability.

Find how far each number is from the mean (ignoring +/-), then average those distances. It tells you: on average, how far is a typical value from the center?

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: Mean Absolute Deviation (MAD) asks how far, on average, the data values sit from the center - it measures spread, not where the center is.

Common stuck point: Students often know a procedure related to mean absolute deviation (mad) but skip the recognition step: Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I need one number that represents the center of the data, and have I checked whether extreme values change that choice?

Worked Examples

Example 1

medium

Find the MAD of

3, 5, 7, 9

Answer

2

First step

Mean

= (3+5+7+9)/4 = 6

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

medium

A class has temperatures

68, 70, 72, 70, 70

(

^\circ

F). Find the MAD.

Example 3

medium

Data

1, 2, 3, 4, 100

has mean

22

. Find the MAD.

Example 4

hard

A data set has MAD

= 5

. After dividing every value by

5

, find the new MAD.

Example 5

hard

Find the MAD of

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

Example 6

medium

Rainfall (mm) over four days:

0, 6, 10, 12

. Find the MAD.

Example 7

challenge

Show that for any data set of size

n

, MAD

\le

range. Prove it for

n = 5

with

\{1, 3, 5, 7, 9\}

Example 8

medium

Compute the MAD of

4, 6, 8, 10, 12, 14

step by step.

Example 9

medium

Find the MAD of

3, 5, 5, 7, 10

Example 10

hard

Set

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

has mean

6

. Replace

10

with

20

. By how much does the MAD change?

Example 11

medium

Find the MAD of the daily highs

68, 70, 72, 74, 76

(degrees F).

Example 12

easy

Find the mean absolute deviation (MAD) of the data set: 4, 6, 8, 10, 12.

Example 13

medium

Two data sets: A = {10, 10, 10, 10, 10} and B = {2, 6, 10, 14, 18}. Both have a mean of 10. Calculate the MAD for each and explain what it tells you.

Practice Problems

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

Example 1

easy

Find the MAD of

2, 4, 6

(mean

=4

Example 2

easy

Find the MAD of

1, 3

(mean

=2

Example 3

easy

Find the MAD of

5, 5, 5

(mean

=5

Example 4

easy

The first step of MAD is to find what?

Example 5

easy

Why take absolute values in MAD?

Example 6

easy

Find the MAD of

0, 10

(mean

=5

Example 7

easy

Larger MAD means data is more what?

Example 8

easy

Find the MAD of

3, 7

(mean

=5

Example 9

medium

Find the MAD of

2, 4, 6, 8

Example 10

medium

Find the MAD of

10, 20, 30, 40, 50

Example 11

medium

Set A has MAD

2

, Set B has MAD

8

, same mean. Which is more consistent?

Example 12

medium

Find the MAD of

4, 4, 10, 10

(mean

=7

Example 13

medium

Add

5

to every value. How does the MAD change?

Example 14

medium

Multiply every value by

3

. How does the MAD change?

Example 15

medium

How does MAD differ from standard deviation conceptually?

Example 16

challenge

Find the MAD of

1, 2, 3, 4, 5, 6

Example 17

challenge

A data set of

4

values has mean

10

and MAD

0

. What must the data be?

Example 18

challenge

Show that the MAD is always less than or equal to the range.

Example 19

medium

Find the MAD of

6, 8, 10, 12, 14

Example 20

medium

Set A has MAD

1.5

and Set B has MAD

6

, both mean

20

. Which is tighter?

Example 21

easy

Find the MAD of

1, 2, 3, 4, 5

(mean

=3

Example 22

easy

Find the MAD of

10, 20, 30, 40

(mean

=25

Example 23

easy

Find the MAD of

7, 9

(mean

=8

Example 24

medium

Test scores

80, 85, 90, 95, 100

have mean

90

. Find the MAD.

Example 25

medium

Which data set has the larger MAD: A

= \{4, 5, 6\}

or B

= \{2, 5, 8\}

Example 26

medium

MAD is which type of statistic: measure of center or measure of spread?

Example 27

medium

Find the MAD of

4, 4, 4, 4, 12

(mean

=5.6

Example 28

hard

Compare two athletes' run times (s): A

= \{10, 12, 14\}

vs. B

= \{8, 12, 16\}

. Which is more consistent based on MAD?

Example 29

hard

A data set

\{a, b\}

with

a < b

and mean

m = (a+b)/2

has MAD equal to what expression?

Example 30

medium

If a data set has MAD

= 0

, what must be true?

Example 31

medium

Two classes have mean test scores of

80

. Class P has MAD

= 4

and class Q has MAD

= 10

. Which class is more variable?

Example 32

easy

Compute the MAD of

0, 0, 0, 4

(mean

=1

Example 33

challenge

For

n

data points, what is the smallest possible MAD, and when does it occur?

Example 34

easy

Find the MAD of

1, 2, 3, 4, 5

Example 35

easy

Find the MAD of

6, 8, 10, 12, 14

Example 36

easy

Find the MAD of

2, 4, 4, 4, 6

Example 37

easy

Find the MAD of

1, 1, 1, 5

Example 38

easy

Find the MAD of

7, 9

Example 39

medium

Find the MAD of

10, 10, 10, 10, 20

Example 40

medium

Each value in a set is increased by

5

. What happens to the MAD?

Example 41

medium

Each value in a set is multiplied by

4

. The original MAD was

2.5

. What is the new MAD?

Example 42

medium

Class A test scores have MAD

4

; class B has MAD

9

. Which class's scores are more consistent?

Example 43

medium

Find the MAD of

0, 0, 4, 4

Example 44

medium

The MAD of a set of temperatures is

3

degrees. Two sets have the same mean; one has MAD

1

and the other MAD

5

. Which set has temperatures that vary more day to day?

Example 45

medium

Find the MAD of

1, 2, 2, 3

Example 46

hard

A data set

\{a, b\}

has mean

m

and MAD

d

. Express

|a-b|

in terms of

d

Example 47

hard

Find the MAD of

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Example 48

hard

Why is MAD generally less sensitive to extreme outliers than standard deviation?

Example 49

hard

A set of

n

identical values

c

has MAD

0

. Add one new value

v \ne c

. What is the new MAD?

Example 50

hard

Set A:

\{4, 6, 8\}

, mean

6

, MAD

\tfrac{4}{3}

. Set B:

\{2, 6, 10\}

, mean

6

, MAD

\tfrac{8}{3}

. By what ratio is B's spread larger?

Example 51

hard

Set

\{x, x+2, x+4, x+6, x+8\}

. What is its MAD?

Example 52

challenge

Among data sets of

4

values from

\{0, 1, 2, \dots, 10\}

with mean

5

, what is the maximum possible MAD?

Example 53

medium

The heights (in cm) of 6 plants are: 12, 15, 14, 18, 13, 16. Calculate the MAD and interpret the result in context.

Example 54

hard

A teacher claims that adding the same constant to every value in a data set does not change the MAD. Test this claim with the data set {5, 10, 15, 20, 25} by adding 100 to each value and comparing the MADs.

← Back to Mean Absolute Deviation (MAD) Formula Explained Practice Mode

Related Concepts

Mean as Fair Share Standard Deviation Data Variability

Background Knowledge

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

mean fair share