Median Examples in Math

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

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 median is the middle value of an ordered data set — half of the values are above it and half are below it.

Half the values are below, half are above. The true 'middle.'

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: Sort the data and the median is the value standing in the exact middle, with half below and half above.

Common stuck point: The procedure for median is the easy part; the trap is finding the middle of the unsorted list. Asking "After sorting the data smallest to largest, what value sits exactly in the middle?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: After sorting the data smallest to largest, what value sits exactly in the middle?

Worked Examples

Example 1

easy

Find the median of

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

Answer

\text{Median} = 5

First step

Sort the data in ascending order:

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

Full solution

2
Since $n = 5$ (odd), the median is the middle value at position $\frac{5+1}{2} = 3$ .
3
The third value is $5$ .

For an odd number of data points, the median is the middle value after sorting. The median is robust against outliers, unlike the mean.

Example 2

medium

Find the median of

\{12, 4, 7, 19, 3, 15\}

Example 3

easy

In a class of 11 students, what position (rank) does the median occupy in sorted order?

Example 4

medium

Data set

\{x, 4, 6, 9, 12\}

is already sorted with median

6

. What constraint does this place on

x

Example 5

medium

A data set has 9 values with median

50

. What is the median of

\{x/2 : x \in \text{data}\}

Example 6

medium

What is the median of the first 10 positive integers?

Example 7

hard

Three students have ages

12, 14, 16

(median

14

). A fourth student joins, making the group's median

13.5

. Find the fourth student's age.

Example 8

hard

A class has 21 quiz scores sorted as

s_1 \le s_2 \le \dots \le s_{21}

with median

s_{11} = 80

. The teacher adds 4 perfect scores of

100

. Express the new median in terms of the original scores.

Example 9

hard

A data set of 7 distinct positive integers has median

10

and the smallest value is

3

. Is the mean bounded above?

Example 10

challenge

Show that among any

n

distinct real numbers

x_1, \dots, x_n

, the median minimizes the sum of absolute deviations

f(c) = \sum_i |x_i - c|

Practice Problems

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

Example 1

easy

Find the median of

\{22, 8, 15, 31, 10, 18, 27\}

Example 2

medium

The ordered data set

\{2, 5, 8, x, 14, 18\}

has median

10.5

. Find

x

Example 3

easy

Find the median of

3, 7, 5

Example 4

easy

Find the median of

10, 2, 8, 4, 6

Example 5

easy

Find the median of

4, 8

Example 6

easy

Find the median of

1, 2, 3, 4, 5, 6

Example 7

easy

Find the median of

9, 9, 9

Example 8

easy

Find the median of

12, 15, 11, 14

Example 9

easy

Find the median of

20, 5, 13

Example 10

easy

Find the median of

7, 1, 7, 3, 9

Example 11

medium

A data set sorted is

4, 6, x, 10

and its median is

7

. Find

x

Example 12

medium

Scores are

82, 90, 76, 88, 94, 79

. Find the median.

Example 13

medium

A list has values

5, 8, 12, 15, 20

. Add the value

100

. How does the median change?

Example 14

medium

The median of

3, 7, 11, x

9

(set is already sorted). Find

x

Example 15

medium

Find the median of the first eight positive even numbers.

Example 16

medium

A class of 7 students has ages

11,12,12,13,13,13,40

(a teacher included). Find the median age.

Example 17

medium

Two data sets are combined:

\{2,4,6\}

and

\{8,10\}

. Find the median of the combined set.

Example 18

medium

5, 9, 9, 12, 15, 18

, find both the median and state whether it equals any data value.

Example 19

medium

A data set of 9 numbers has median

20

. If every value is increased by

5

, what is the new median?

Example 20

challenge

A sorted set of 6 distinct integers has median

10

. The two middle values are consecutive integers. What are they?

Example 21

challenge

Five distinct positive integers have median

6

and mean

8

. What is the largest possible value in the set?

Example 22

challenge

A set of

n

numbers (

n

even) has median

m

. You append the single value

m

. Is the new median still

m

? Explain.

Example 23

easy

Find the median of

\{4, 1, 7\}

Example 24

easy

Find the median of

\{15, 11, 13, 19, 17\}

Example 25

easy

Find the median of

\{100, 50, 75\}

Example 26

easy

Find the median of

\{-3, -1, 0, 2, 5\}

Example 27

medium

Find the median of the data set

\{12, 4, 18, 9, 7, 14, 2\}

Example 28

medium

Eight test scores are

72, 85, 91, 68, 77, 88, 94, 80

. Find the median.

Example 29

medium

A data set of 10 numbers has median

15

. If every value is decreased by

7

, find the new median.

Example 30

medium

Six house prices in a neighborhood are (in thousands of dollars):

250, 290, 310, 305, 280, 1500

. Compute median and mean; which better represents the typical price?

Example 31

medium

A set of 12 numbers has median

30

. You append the value

30

to the set. What is the new median?

Example 32

hard

A set of 5 positive integers has median

7

and mean

7

. Give one example of such a set and explain why the median and mean being equal is consistent here.

Example 33

hard

In the sorted set

\{3, 5, 7, x, 11, 13\}

with median

9

, find

x

Example 34

hard

What is the median of the integers from

1

100

Example 35

challenge

Two data sets each of size 5 have medians

8

and

12

. When combined (size 10), what range of medians is possible?

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