Quartiles Examples in Statistics

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

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

Concept Recap

Quartiles are values that divide ordered data into four equal parts: $Q_1$ (25th percentile) marks the boundary below which 25% of data falls, $Q_2$ (the median, 50th percentile) splits the data in half, and $Q_3$ (75th percentile) marks the boundary below which 75% falls.

If you line up 100 people by height and divide into 4 equal groups, quartiles mark the dividing points. $Q_1$ is where the shortest 25% ends, $Q_2$ is the middle, $Q_3$ is where the tallest 25% begins.

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: Quartiles asks how a value or feature behaves inside the full distribution.

Common stuck point: Students often know a procedure related to quartiles but skip the recognition step: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I interpreting the whole distribution or a value position inside it, rather than just computing a single summary?

Worked Examples

Example 1

medium

Find

Q_1, Q_2, Q_3

for

2, 4, 5, 8, 9, 10, 12

Find Q₁, Q₂, Q₃ — each marked as ? on the box.

Answer

Q_1=4,\ Q_2=8,\ Q_3=10

First step

Seven values;

Q_2

is the 4th value

=8

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Example 2

medium

Test scores:

55, 62, 68, 75, 80, 88, 92, 95

. Find

Q_1, Q_2, Q_3

Find Q₁, Q₂, Q₃ from the test-score distribution.

Example 3

hard

Annual rainfalls (in inches) over 9 years:

20, 22, 25, 28, 30, 32, 35, 40, 50

. Find all three quartiles.

Find Q₁, Q₂, Q₃ for the rainfall data.

Example 4

hard

Class quiz scores:

5, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10

. Find the three quartiles.

Example 5

challenge

A college reports 25th-percentile SAT score

1180

and 75th-percentile

1410

. What does this tell admitted students?

Example 6

easy

Find the quartiles (

Q_1

Q_2

Q_3

) of the data set: 3, 5, 7, 8, 12, 14, 16, 18, 21.

Example 7

medium

Find the quartiles of this data set with an even number of values: 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65.

Practice Problems

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

Example 1

easy

Find the median (

Q_2

) of

2, 4, 6, 8, 10

Example 2

easy

For ordered

1,2,3,4,5,6,7

, find

Q_1

Example 3

easy

For ordered

1,2,3,4,5,6,7

, find

Q_3

Example 4

easy

What percentile does

Q_1

represent?

Example 5

easy

Order

9, 3, 7, 1

before finding quartiles. Give the ordered list.

Example 6

easy

For ordered

2,4,6,8

, find

Q_2

Example 7

easy

How many parts do quartiles divide data into?

Example 8

easy

Which quartile equals the median?

Example 9

medium

For

4,8,15,16,23,42

, find

Q_1

and

Q_3

Read Q₁ and Q₃ off the box plot.

Example 10

medium

For

5,7,8,9,10,12,13,14,15

, find

Q_1

Find Q₁ — shown as ? on the box.

Example 11

medium

For

5,7,8,9,10,12,13,14,15

, find

Q_3

Find Q₃ — shown as ? on the box.

Example 12

medium

Q_1=10

and

Q_3=22

, where do the middle 50% of values lie?

The shaded box shows where the middle 50% of values lie.

Example 13

medium

For

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

, find all three quartiles.

Find all three quartiles Q₁, Q₂, Q₃ — each shown as ? on the box.

Example 14

medium

A student finds

Q_1=15

but the data minimum is

20

. What error occurred?

Example 15

challenge

For

100

ordered distinct values, which positions approximate

Q_1

and

Q_3

using the

\frac{n+1}{4}

rule?

Example 16

challenge

Data is symmetric about

50

with

Q_1=40

. Find

Q_3

Use symmetry about 50 to find Q₃, shown as ?.

Example 17

challenge

Show that for any data,

Q_1\le Q_2\le Q_3

Example 18

medium

For

3,6,7,11,14,18,20,22

, find

Q_1

Find Q₁ — shown as ? on the left edge of the box.

Example 19

medium

For

3,6,7,11,14,18,20,22

, find

Q_3

Find Q₃ — shown as ? on the right edge of the box.

Example 20

medium

Q_1=30, Q_2=45, Q_3=60

, is the data symmetric about

Q_2

Are the gaps Q₂ − Q₁ and Q₃ − Q₂ equal? That tells you if the data is symmetric.

Example 21

easy

For ordered

3, 5, 7, 9, 11

, find

Q_2

Example 22

easy

For ordered

3, 5, 7, 9, 11

, find

Q_1

Example 23

easy

For ordered

3, 5, 7, 9, 11

, find

Q_3

Example 24

easy

Order

12, 5, 9, 3, 7

from least to greatest.

Example 25

easy

For ordered

10, 20, 30, 40

, find

Q_2

Example 26

easy

For ordered

10, 20, 30, 40

, find

Q_1

Example 27

medium

For ordered

6, 9, 11, 14, 17, 20

, find

Q_1

Example 28

medium

For ordered

6, 9, 11, 14, 17, 20

, find

Q_3

Example 29

medium

For ordered

6, 9, 11, 14, 17, 20

, find

Q_2

Example 30

medium

Q_1 = 12

and

Q_3 = 30

, in what range do the middle 50% of values lie?

Example 31

medium

If your test score is at the 75th percentile, what fraction of students scored at or below you?

Example 32

medium

Ordered ages:

14, 15, 16, 17, 18, 19, 20

. Find

Q_1

Example 33

medium

Ordered ages:

14, 15, 16, 17, 18, 19, 20

. Find

Q_3

Example 34

hard

For

\{3, 7, 8, 12, 14, 21, 22, 27, 30, 35\}

, find

Q_1, Q_2, Q_3

Find all three quartiles for this 10-value data set.

Example 35

hard

A data set has

Q_1 = 18

Q_3 = 42

. What can you say about the value

50

Example 36

hard

A box plot shows

Q_1 = 50

Q_2 = 60

Q_3 = 64

. Is the data skewed? Which direction?

Compare the two box halves: which side is more spread out?

Example 37

hard

If a data set has

20

values, between which two ordered positions is

Q_1

found (inclusive method)?

Example 38

challenge

A teacher curves the test so every score increases by

5

points. What happens to

Q_1, Q_2, Q_3

Example 39

medium

The ages of 15 members of a sports club are: 18, 19, 20, 21, 22, 23, 24, 25, 27, 30, 32, 35, 40, 45, 50. Find

Q_1

Q_2

, and

Q_3

, and determine how many members are between

Q_1

and

Q_3

Example 40

hard

Two data sets have the same median (

Q_2 = 50

). Data set A:

Q_1 = 45

Q_3 = 55

. Data set B:

Q_1 = 20

Q_3 = 80

. Compare the distributions and explain what the quartiles reveal about each data set.

← Back to Quartiles Practice Mode

Related Concepts

Median Box Plot Interquartile Range (IQR)

Background Knowledge

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

median intro