Mean Examples in Math

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

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 arithmetic mean (average) of a data set is the sum of all values divided by the number of values.

Imagine redistributing all the data equally — the mean is the value each person would get if everyone shared equally. It is the balance point of the data.

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 mean is what every value would equal if you poured the whole sum together and split it evenly among all of them.

Common stuck point: The procedure for mean is the easy part; the trap is forgetting to divide by the count after adding. Asking "Am I adding up all the values and dividing by how many there are?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I adding up all the values and dividing by how many there are?

Worked Examples

Example 1

easy

Find the arithmetic mean of the data set:

\{4, 8, 15, 16, 23\}

Answer

\bar{x} = 13.2

First step

Add all values:

4 + 8 + 15 + 16 + 23 = 66

Full solution

2
Count the number of values: $n = 5$ .
3
Divide the sum by $n$ : $\bar{x} = \frac{66}{5} = 13.2$ .

The arithmetic mean is the sum of all values divided by the number of values. It represents the central tendency of a data set and is sensitive to extreme values (outliers).

Example 2

medium

A student scored

72, 85, 90, 68

, and

95

on five tests. What score does she need on the sixth test to achieve a mean of

85

Example 3

medium

Find the mean, median, and mode of

\{4,4,6,8,10,12,14\}

Practice Problems

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

Example 1

easy

Find the mean of

\{12, 7, 3, 14, 9, 11\}

Example 2

medium

Five quiz scores have a mean of

18

. Four of the scores are

12

17

20

, and

19

. What is the fifth score?

Example 3

easy

Find the mean of

4, 8, 6

Example 4

easy

Find the mean of

10, 20, 30, 40

Example 5

easy

The mean of

5

numbers is

12

. What is their sum?

Example 6

easy

Find the mean of

7, 7, 7, 7

Example 7

easy

Find the mean of

2, 4, 6, 8, 10

Example 8

easy

A student scores

80, 90, 100

on three tests. What is the average?

Example 9

easy

Find the mean of

0, 0, 6

Example 10

easy

The mean of

3, 5, x

5

. Find

x

Example 11

medium

A student has a

85

average over

4

tests. What must they score on the 5th test for a

87

average?

Example 12

medium

The mean of

6

numbers is

10

. If one number (

4

) is removed, what is the new mean?

Example 13

medium

Class A (20 students) averages

80

. Class B (30 students) averages

90

. What's the combined average?

Example 14

medium

The mean of

4

numbers is

15

. Three of them are

10, 12, 20

. Find the fourth.

Example 15

medium

Add

5

to every number in a data set with mean

12

. What's the new mean?

Example 16

medium

Multiply every number in a data set with mean

8

3

. New mean?

Example 17

medium

Find the mean, median, and mode of

3, 5, 5, 7, 10

Example 18

medium

A set of

3

numbers has mean

10

. Adding a 4th number makes the mean

12

. What's the 4th number?

Example 19

medium

The average of

a

and

b

20

, and the average of

b

and

c

30

. Find

c - a

Example 20

challenge

Five numbers have mean

20

. If the largest is removed, the mean of the rest is

15

. What was the largest number?

Example 21

challenge

A list of

10

numbers averages

50

. Two numbers averaging

80

are added. New average?

Example 22

challenge

Can the mean of a data set be a value that doesn't appear in the data? Give an example and explain.

Example 23

easy

Find the mean of

\{6,9,12,15\}

Example 24

easy

Find the mean of

\{15,25,35,45\}

Example 25

easy

Find the mean of

\{0,0,0,8\}

Example 26

easy

Find the mean of

\{2,4,6,8,10,12\}

Example 27

easy

A student scores

70,80,90

on three quizzes. What is the mean?

Example 28

medium

Find the mean of the first

10

positive integers

1,2,3,\dots,10

Example 29

medium

A student's first four test means

85

. After the fifth test the mean rose to

87

. Find the fifth score.

Example 30

medium

A data set has mean

30

and sum

180

. How many values are in the set?

Example 31

medium

The mean of

5

numbers is

14

. If the smallest (

4

) is removed, what is the new mean?

Example 32

medium

Class A (15 students) averages

70

. Class B (25 students) averages

82

. Find the combined average.

Example 33

medium

Every number in a data set with mean

12

is increased by

3

. Find the new mean.

Example 34

medium

Every number in a data set with mean

7

is multiplied by

2

. Find the new mean.

Example 35

medium

Six numbers have mean

12

. If a seventh value of

19

is added, what is the new mean?

Example 36

medium

The mean of

a,b,c

12

and the mean of

a,b,c,d

15

. Find

d

Example 37

medium

Five test scores have mean

82

. After one

50

is replaced by an unknown score, the mean is

86

. Find the unknown score.

Example 38

hard

A group of

n

values has mean

50

. After adding a value equal to

80

, the new mean is

52

. Find

n

Example 39

hard

The mean of

a,b,c

20

and the mean of

b,c,d

24

. Find

d-a

Example 40

hard

Five integers average

30

. The least possible value of the largest integer occurs when the others are as large as possible (but less than the largest). What is the largest if all are positive integers and the others are distinct positive integers below it?

Example 41

medium

A student's weighted grade is

40\%

exam (

88

) and

60\%

homework (

95

). Find the weighted mean.

Example 42

medium

Find the mean of

0.4, 0.6, 0.8, 1.0, 1.2

Example 43

challenge

The mean of

20

numbers is

40

. By removing two numbers

50

and

60

, find the new mean of the remaining

18

Example 44

challenge

\overline{x}=10

for data

\{x_1,\dots,x_n\}

, prove that the new mean of

\{ax_1+b,\dots,ax_n+b\}

10a+b

← Back to Mean Formula Explained Practice Mode

Related Concepts

Addition Division Median Mode Standard Deviation

Background Knowledge

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

additiondivision