Interquartile Range (IQR) Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Interquartile Range (IQR).

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 interquartile range (IQR) is the range of the middle 50% of data, calculated as $Q_3 - Q_1$ . It measures spread while ignoring the top and bottom 25% of values, making it resistant to outliers.

IQR focuses on where most of the data lives, ignoring the extremes. If regular range is how far the outliers stretched, IQR is how wide the main crowd is. More resistant to outliers than range.

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: Interquartile Range (IQR) asks how tightly or loosely the values sit around the data set, not just where the middle is.

Common stuck point: Students often know a procedure related to interquartile range (iqr) but skip the recognition step: Do I need to describe how far the data values extend or vary, rather than where the middle is? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Do I need to describe how far the data values extend or vary, rather than where the middle is?

Worked Examples

Example 1

medium

Find the IQR of

5, 8, 11, 14, 17, 20, 23, 26

Box plot of 5, 8, 11, 14, 17, 20, 23, 26. Find the IQR.

Answer

12

First step

Eight values; median

=(14+17)/2=15.5

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Example 2

medium

A box plot shows

Q_1=22

Q_3=34

. The largest value is

80

. Does

80

qualify as a high outlier by the

1.5\cdot\text{IQR}

rule?

Q₁ = 22, Q₃ = 34. Upper fence = Q₃ + 1.5·IQR = 52. Is 80 an outlier?

Example 3

medium

For

10, 12, 14, 14, 15, 18, 20, 22

, find the IQR.

Box plot of 10, 12, 14, 14, 15, 18, 20, 22. Find the IQR = Q₃ − Q₁.

Example 4

hard

Find the IQR and the upper outlier fence for

4, 6, 8, 10, 12, 14, 16, 20, 30

Example 5

hard

Daily commute times (min):

20, 22, 25, 28, 30, 35, 38, 40, 95

. Find the IQR and explain why it is a better spread measure than the range here.

Commute times (min): 20, 22, 25, 28, 30, 35, 38, 40, 95. Q₁ = 23.5, Q₃ = 39, IQR = 15.5. The value 95 is an outlier.

Example 6

challenge

Two cities report median household incomes both equal to $60k. City A has IQR

\$10k

; City B has IQR

\$60k

. What do these tell you about income inequality?

Example 7

easy

Given the data set: 4, 7, 9, 12, 15, 18, 22, 25, 30, find the interquartile range (IQR).

Example 8

medium

Test scores: 55, 60, 65, 70, 72, 75, 78, 80, 85, 90, 95. Find the IQR and use it to determine the boundaries for outliers using the

1.5 \times IQR

rule.

Practice Problems

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

Example 1

easy

Q_1=5

and

Q_3=12

, find the IQR.

The box spans Q₁ = 5 to Q₃ = 12. What is the width of the box (the IQR)?

Example 2

easy

Q_1=20

and

Q_3=20

, find the IQR.

Example 3

easy

IQR measures the spread of what fraction of data?

Example 4

easy

For

Q_1=8, Q_3=18

, find the IQR.

Box spans Q₁ = 8 to Q₃ = 18. Find the IQR.

Example 5

easy

Is IQR more or less affected by outliers than range?

Example 6

easy

For a set with

Q_1=100, Q_3=140

, find the IQR.

Box spans Q₁ = 100 to Q₃ = 140. Find the IQR.

Example 7

easy

Which uses only the middle of the data: range or IQR?

Example 8

easy

If the IQR is

0

, what does that say about the middle 50%?

Example 9

medium

For

4,8,15,16,23,42

, compute the IQR.

Example 10

medium

For

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

, compute the IQR.

Box plot of 1, 2, 3, 4, 5, 6, 7, 8. Find the IQR.

Example 11

medium

A data set has range

50

but IQR

6

. What does this contrast suggest?

Example 12

medium

Outlier rule: a value is an outlier if above

Q_3+1.5\cdot\text{IQR}

. With

Q_3=20

, IQR

=8

, find the upper fence.

Q₃ = 20, IQR = 8. The dashed line marks the upper outlier fence. What value is it?

Example 13

medium

With

Q_1=10

, IQR

=8

, find the lower outlier fence.

Q₁ = 10, IQR = 8. The dashed line marks the lower outlier fence. What value is it?

Example 14

medium

Multiply all data by

4

. If the original IQR was

5

, find the new IQR.

Example 15

medium

Add

7

to every value. If the IQR was

9

, find the new IQR.

Example 16

challenge

A set has

Q_1=12

Q_3=24

. Is

45

an outlier by the

1.5\times

IQR rule?

Q₁ = 12, Q₃ = 24. Upper fence = Q₃ + 1.5·IQR = 42. Is 45 an outlier?

Example 17

challenge

Two sets have equal IQR

=10

but ranges

12

and

60

. What distinguishes them?

Example 18

challenge

Show that IQR

\le

range for any data set.

Example 19

medium

For

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

, compute the IQR.

Box plot of 5, 7, 8, 9, 10, 12, 13, 14, 15. Find the IQR.

Example 20

medium

With

Q_1=14

Q_3=26

, find the upper outlier fence.

Q₁ = 14, Q₃ = 26. IQR = 12. Where is the upper outlier fence?

Example 21

easy

Q_1 = 10

and

Q_3 = 25

, find the IQR.

Example 22

easy

Q_1 = 0

and

Q_3 = 9

, find the IQR.

Example 23

easy

Q_1 = 50

and

Q_3 = 70

, find the IQR.

Example 24

easy

For ordered

1, 3, 5, 7, 9

, find

Q_1

and

Q_3

, then the IQR.

Example 25

easy

A box plot's box stretches from

40

55

. What is the IQR?

The box spans from Q₁ to Q₃. What is the IQR (the width of the box)?

Example 26

medium

Find the IQR of

2, 3, 5, 7, 7, 8, 10

Example 27

medium

If you add

5

to every value, what happens to the IQR?

Example 28

medium

If you double every value, what happens to the IQR?

Example 29

medium

Q_1 = 30

, IQR

= 10

, find

Q_3

Example 30

medium

Q_3 = 80

and IQR

= 20

, find the lower fence (cutoff for low outliers).

Example 31

medium

A data set has range

200

but IQR

4

. What does this suggest?

Example 32

medium

Q_1=15

Q_3=27

, IQR

=12

. What is the upper outlier fence?

Q₁ = 15, Q₃ = 27, IQR = 12. Find the upper outlier fence.

Example 33

hard

Two data sets: A has IQR

4

and B has IQR

20

. Both have median

50

. Which set is more reliable?

Example 34

hard

A new value

200

is added to a set with

Q_1=20

Q_3=40

, and many values. The IQR barely changes. Why?

Example 35

hard

If every value of a set is multiplied by

3

and then

7

is subtracted, what happens to the IQR?

Example 36

hard

A box plot has

Q_1=10

Q_3=20

, and a maximum of

35

. Is

35

an outlier by the

1.5\cdot

IQR rule?

Q₁ = 10, Q₃ = 20. Upper fence = Q₃ + 1.5·IQR = 35. Is the maximum value 35 an outlier?

Example 37

challenge

A teacher reports median

=80

and IQR

=4

on a test. About what fraction of students scored between

78

and

82

Example 38

medium

Two classes took the same test. Class A:

Q_1 = 60

Q_3 = 80

. Class B:

Q_1 = 70

Q_3 = 90

. Which class has greater consistency in scores? Explain using the IQR.

Example 39

hard

Daily temperatures (°C) for two weeks: 15, 16, 14, 18, 20, 22, 19, 35, 17, 16, 18, 21, 20, 19. Find the IQR and use the

1.5 \times IQR

rule to identify any outliers. Then recalculate the mean with and without the outlier(s).

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Related Concepts

Quartiles Range Box Plot Outlier Detection

Background Knowledge

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

stat quartilesstat range