Median Examples in Statistics

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 Statistics.

Concept Recap

The median is the middle value when all data points are arranged in order from smallest to largest. Half the values lie above it and half below. For an even number of values, the median is the average of the two middle values.

If you lined up your whole class by height, the median height is the person standing exactly in the middle. It's not affected by whether the tallest kid is 5'5" or 7 feet - the middle person stays the same.

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: Median asks what single value best stands for the center of the data, then checks whether that value is fair for the situation.

Common stuck point: Students often know a procedure related to median 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?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Mean vs Median Mistakes

Worked Examples

Example 1

medium

Eight scores:

60, 65, 70, 75, 80, 85, 90, 95

. Find the median and explain which positions you used.

Eight evenly-spaced scores — the median falls between positions 4 and 5.

Answer

77.5

First step

n=8

; the middle pair are positions

4

and

5

75

and

80

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Example 2

hard

Twelve test scores have median

80

. After the teacher adds

5

bonus points to each score, what is the new median?

Example 3

challenge

Seven distinct positive integers have median

10

. What is the smallest possible value of their sum?

Example 4

easy

Find the median of: 12, 5, 8, 3, 15, 7, 10.

Example 5

medium

Find the median of: 4, 9, 2, 7, 11, 6.

Practice Problems

20, 5, 15, 30, 25, 10

Find the median of this data set: 20, 5, 15, 30, 25, 10.

Example 11

medium

A data set has median

12

and the values

8, 12, x

in order. What can

x

be?

Example 12

medium

Find the median of the nine values

5,5,6,7,8,9,10,11,12

Example 13

medium

Eight homes have prices (in thousands)

200,210,220,230,240,250,260,2000

. Find the median.

Example 14

medium

The median of

3, 7, x, 15

(already increasing) is

10

. Find

x

Example 15

medium

Find the median of

14, 22, 14, 30, 22, 14

Six values — the median falls between the two middle values. Find it.

Example 16

challenge

A set of

5

positive integers has median

4

, and all values are distinct. What is the smallest possible sum?

Example 17

challenge

Seven distinct positive integers have median

10

and mean

10

. The largest is

20

. What is the largest possible value of the third-smallest integer (i.e., position

3

in sorted order)?

Example 18

challenge

Data

1,2,3,\dots,n

has median

25

for some odd

n

. Find

n

Example 19

medium

Find the median of

30, 10, 40, 20, 50, 60

Find the median of 30, 10, 40, 20, 50, 60.

Example 20

medium

The median of

5, 9, 11, x

(increasing) is

medium

Compare mean and median: data set

\{2, 3, 4, 5, 100\}

. Which is closer to the bulk of the data?

Example 31

medium

A data set is

\{4, 7, 9, x\}

with median

8

. Find

x

x>9

Example 32

medium

Add a value to

\{2, 4, 6\}

so the median becomes

5

. What value should be added?

Example 33

medium

Find the median of the first ten positive integers

1, 2, \ldots, 10

Example 34

medium

Three students score

80, 90, x

. Their median equals their mean. Find

x

Example 35

medium

Five home prices in

\$1000

250, 260, 280, 300, 2000

. Which is a more honest 'typical' price, mean or median?

Example 36

hard

Find the median of

\{1, 3, 3, 6, 7, 8, 9\}

Example 37

hard

If every value in a data set is increased by

7

, what happens to the median?

Example 38

hard

If every value in a data set is multiplied by

3

, what happens to the median?

Example 39

hard

A data set has

\{3, 5, x, 11\}

with median

7

. Find

x

Example 40

hard

Five integers have median

10

and the smallest two are

3

and

5

. What is the smallest possible value of the largest?

Example 41

easy

Find the median of: 20, 35, 15, 40, 25.

Example 42

medium

The ordered data set is 4, 7, 8, 9, 12, 14, 18. What is the median, and if 30 is added to the set, does the median change?

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Related Concepts

Mean as Fair Share Quartiles Box Plot

Background Knowledge

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

mean fair share