Mode Examples in Math: 46 Problems with Answers

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

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 mode is the value or values that appear most frequently in a data set — it is the most common or most popular data value.

The mode is the "most popular" value — if you had to guess one number and wanted to be right as often as possible, pick the mode.

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: The mode is the value that shows up most often — the data's most popular answer.

Common stuck point: The procedure for mode is the easy part; the trap is reporting the frequency instead of the value. Asking "Which value occurs more often than any other?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Which value occurs more often than any other?

Worked Examples

Example 1

easy

Find the mode of

\{2, 4, 4, 5, 7, 7, 7, 9\}

Answer

Mode = 7

First step

List the data set and tally the frequency of each value:

2 \to 1

4 \to 2

5 \to 1

7 \to 3

9 \to 1

Full solution

2
Identify the value with the highest frequency: $7$ appears $3$ times, more than any other value.
3
Therefore, the mode is $7$ .

The mode is the most frequently occurring value in a data set. Unlike the mean and median, the mode can be used with categorical (non-numeric) data as well.

Example 2

medium

Find the mode of

\{3, 5, 5, 8, 8, 12\}

Example 3

medium

Test scores:

80, 80, 85, 90, 90, 95, 100

. Compare mode, median, mean.

Example 4

medium

A histogram has bars of heights

3, 8, 5, 2

for intervals

[0,10),[10,20),[20,30),[30,40)

. What is the modal interval?

Example 5

hard

Find a data set of

5

positive integers with mode

4

, median

4

, mean

5

Practice Problems

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

Example 1

easy

Find the mode of

\{1, 3, 3, 3, 5, 5, 6, 9\}

Example 2

medium

The data set

\{3, 5, 5, 7, 8, x\}

has mean

5.5

and mode

5

. Find

x

Example 3

easy

Find the mode of

2, 3, 3, 5, 7

Example 4

easy

Find the mode of

4, 4, 4, 9, 9

Example 5

easy

Find the mode of

1, 2, 2, 3, 3, 4

Example 6

easy

Find the mode of

5, 6, 7, 8

Example 7

easy

In the data

10, 10, 20, 30, 30, 30

, find the mode.

Example 8

easy

Colors picked by students: red, blue, blue, green, blue, red. What is the modal color?

Example 9

easy

Find the mode of

7, 7, 8, 8, 9

Example 10

easy

Find the mode of

100, 1, 1, 1, 50

Example 11

medium

A data set is

4, 4, 5, 6, 6, 6, 7

. Compare its mode, median, and mean.

Example 12

medium

3, 5, 5, 7, 7, 7, 9, x

, the unique mode is

7

. What values of

x

are allowed?

Example 13

medium

Test scores:

70, 85, 85, 90, 70, 85, 95

. Find the mode and explain its meaning.

Example 14

medium

A data set

a, a, b, b, b, c

has mode

b

. After removing one

b

, what is the new mode?

Example 15

medium

Find the mode of

2.5, 2.5, 3.1, 3.1, 3.1, 4.0

Example 16

medium

A frequency table lists: value

1

(count 4), value

2

(count 9), value

3

(count 9), value

4

(count 2). Find the mode.

Example 17

medium

Heights (cm):

150, 152, 152, 152, 158, 160, 160

. Find the mode and median.

Example 18

medium

In a class, exam grades A, B, B, C, C, C, C, D appear. What grade is the mode and how often?

Example 19

medium

The data

9, 11, 11, 13, 15

— find the mode and state whether it is unique.

Example 20

challenge

A set of 7 positive integers has mode

4

(unique), median

5

, and the smallest possible sum. Find that sum.

Example 21

challenge

Explain why a perfectly uniform data set like

5,5,7,7,9,9

is ambiguous for the mode.

Example 22

challenge

A data set's mode is

10

, mean is

12

, and median is

11

. What does the ordering mode < median < mean suggest about shape?

Example 23

easy

Find the mode of

\{6, 8, 8, 9, 11\}

Example 24

easy

Find the mode of

\{12, 15, 18, 21\}

Example 25

easy

Find the mode of

\{4, 4, 7, 7, 7, 9, 11\}

Example 26

easy

Shoe sizes sold:

6, 7, 7, 8, 8, 8, 9, 10

. Find the mode.

Example 27

easy

Find the mode of

\{0, 0, 1, 2, 3\}

Example 28

easy

A survey:

5

people said yes,

5

said no,

2

said maybe. What is the modal response and is it unique?

Example 29

medium

The data

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

has unique mode

6

. Find

x

Example 30

medium

Frequency table:

1

(count

5

2

(count

7

3

(count

7

4

(count

2

). Find the mode(s).

Example 31

medium

For

\{a, a, a, b, b, c\}

, the mode is

a

. After adding one

b

, what changes?

Example 32

medium

\{2, 5, 5, 7, 9, 9, 9, 12\}

, find the mode and the gap between mode and mean.

Example 33

medium

\{10, 12, 12, 12, 15, 18, 18, 18, 20\}

, find all modes.

Example 34

medium

In a class of

30

12

students chose pizza,

9

chose pasta,

6

chose salad,

3

chose soup. Identify mode and its share.

Example 35

medium

True or false: changing one value in a data set always changes the mode.

Example 36

hard

A list of

6

positive integers has mode

3

(unique), median

4

, mean

5

. Find the largest possible value in the list.

Example 37

hard

A data set has values

\{5, 5, 5, 7, 9, 12\}

. If you append a value

v

so the new set is bimodal with modes

5

and

7

, what must

v

be?

Example 38

hard

A data set

\{3, 3, 7, 8, 8, 8, 10\}

has mode

8

. If we replace one

8

with

3

, what is the new mode?

Example 39

hard

For data set

\{x_1, x_2, \ldots, x_n\}

with unique mode

m

, prove that any data set you get by adding values

\ne m

keeps the mode if you add fewer than

(\text{count of } m) - (\text{next highest count})

extra of any single value.

Example 40

challenge

Among lists of

7

positive integers (with repetition) with unique mode

5

, what is the smallest possible mean?

Example 41

challenge

Construct a

10

-value data set where mode

=2

, median

=5

, and mean

=10

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Related Concepts

Mean Median Frequency