Interquartile Range Examples in Math

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

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 interquartile range (IQR) is $Q3 - Q1$ — the spread of the middle 50% of the data, resistant to outliers.

The IQR ignores the extreme 25% on each end, capturing only the spread of the central bulk of data — making it robust when outliers inflate the regular 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: IQR throws away the extreme 25% on each end and measures only how wide the central 50% of the data is.

Common stuck point: The procedure for interquartile range is the easy part; the trap is subtracting min from max instead of $Q_3-Q_1$ . Asking "Am I measuring the width of the middle half of sorted data, not the full extent?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I measuring the width of the middle half of sorted data, not the full extent?

Worked Examples

Example 1

easy

Calculate the IQR for:

\{15, 22, 28, 35, 42, 50, 58, 65\}

and explain what it measures.

Answer

IQR = 29

First step

Find

Q_1

: lower half is

\{15, 22, 28, 35\}

;

Q_1 = \frac{22+28}{2} = 25

Full solution

2
Find $Q_3$ : upper half is $\{42, 50, 58, 65\}$ ; $Q_3 = \frac{50+58}{2} = 54$
3
Calculate IQR: $IQR = Q_3 - Q_1 = 54 - 25 = 29$
4
Interpretation: the middle 50% of values span a range of 29 units

The IQR measures the spread of the middle 50% of data. It is resistant to outliers (unlike the range) because it ignores the top and bottom 25% of values. A larger IQR indicates more variability in the central portion of the data.

Example 2

medium

Data set A:

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 100\}

and Data set B:

\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}

. Compare the range and IQR for both sets and explain why IQR is preferred as a measure of spread.

Example 3

medium

Find the IQR of test scores:

55,60,62,68,72,75,78,80,85,90

Example 4

hard

Data:

5, 10, 12, 14, 18, 22, 25, 30, 200

. Find the IQR and decide whether

200

is an outlier.

Practice Problems

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

Example 1

easy

A box plot shows

Q_1 = 30

and

Q_3 = 50

. Calculate the IQR and the lower and upper fences for outlier detection.

Example 2

hard

A data set has

Q_1 = 40

, median

= 55

Q_3 = 70

. A new value of 120 is added. Without recalculating quartiles, explain why the IQR may remain unchanged and identify whether 120 is an outlier.

Example 3

easy

For the ordered data with

Q1 = 12

and

Q3 = 20

, find the interquartile range.

Example 4

easy

A data set has

Q1 = 5

and

Q3 = 17

. What is the IQR?

Example 5

easy

Find the IQR for the ordered set

2, 4, 6, 8, 10, 12, 14, 16

(lower half

2,4,6,8

; upper half

10,12,14,16

Example 6

easy

A box plot shows the box edges at

30

and

48

. What is the IQR?

Example 7

easy

Is the IQR resistant to a single extreme outlier? Answer yes or no.

Example 8

easy

The range of a data set is

40

and the IQR is

15

. Which measures the spread of the middle half?

Example 9

easy

For the set

7,7,7,7,7

, all values equal

7

. What is the IQR?

Example 10

easy

Given

Q1 = 100

and

Q3 = 100

, what is the IQR, and what does it tell you?

Example 11

medium

For the ordered set

3,5,7,9,11,13,15

(7 values, median

=9

), find the IQR. Use the lower half

3,5,7

and upper half

11,13,15

Example 12

medium

A test has

Q1=72

, median

=80

Q3=88

. A new student scores

150

(the new maximum). Does the IQR change?

Example 13

medium

The IQR of a data set is

20

and

Q1 = 35

. Find

Q3

Example 14

medium

An outlier is often flagged if it lies more than

1.5 \times \text{IQR}

below

Q1

or above

Q3

. If

Q1=20

Q3=40

, find the upper outlier boundary.

Example 15

medium

Find the lower outlier fence for

Q1=20

Q3=40

using the

1.5\times\text{IQR}

rule.

Example 16

medium

Two classes have the same median test score. Class A has IQR

=4

; Class B has IQR

=18

. Which class has more consistent middle-half scores?

Example 17

medium

A five-number summary is min

=4

Q1=10

, median

=15

Q3=22

, max

=40

. Compare the IQR with the range.

Example 18

medium

Data:

10,12,12,13,14,15,16,18,19,40

(10 values). Lower half is the first five, upper half the last five. Find the IQR.

Example 19

challenge

A symmetric data set has IQR

=10

and median

=50

. Assuming the

1.5\times\text{IQR}

rule, give the full interval of values that are NOT flagged as outliers, given

Q1=45

and

Q3=55

Example 20

challenge

Set A:

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

. Multiply every value by

3

to get Set B. By what factor does the IQR change?

Example 21

challenge

Adding the same constant

c

to every value of a data set changes the IQR how? Justify briefly.

Example 22

medium

A data set has

Q1=14

and an IQR of

9

. A box plot is drawn. Where is the right edge of the box?

Example 23

easy

Q_1=8

and

Q_3=23

. Find the IQR.

Example 24

easy

A box plot shows the box from

40

70

. What is the IQR?

Example 25

easy

For the data

\{4,6,8,10,12,14,16,18\}

find the IQR. (Lower half:

4,6,8,10

; upper half:

12,14,16,18

Example 26

easy

True or false: the IQR can be negative.

Example 27

easy

Q_1=15

Q_3=45

. Find the IQR.

Example 28

medium

For the data

2, 5, 7, 8, 10, 13, 14, 16, 20

, find the IQR. (9 values; median

=10

; lower half

2,5,7,8

; upper half

13,14,16,20

Example 29

medium

Class A: range

=20

, IQR

=4

. Class B: range

=20

, IQR

=12

. Which class is more clustered around its center?

Example 30

medium

IQR

=12

and

Q_3 = 50

. Find

Q_1

Example 31

medium

With

Q_1=10

Q_3=22

, what is the upper outlier fence using the

1.5 \times \text{IQR}

rule?

Example 32

medium

Q_1=10

Q_3=22

. What is the lower outlier fence?

Example 33

medium

Data set has

Q_1=30

, median

=45

Q_3=58

. Is the value

95

an outlier using the

1.5\times

IQR rule?

Example 34

medium

Multiplying every value of a data set by

4

changes the IQR by what factor?

Example 35

medium

A five-number summary is min

=2

Q_1=12

, med

=18

Q_3=24

, max

=40

. Compare IQR and range.

Example 36

hard

For the data

3,7,8,5,12,15,9,11,2,10

, find the IQR after sorting.

Example 37

hard

Test scores

\{60,70,75,80,82,85,88,92,95,100\}

. After replacing

100

with

200

, does the IQR change?

Example 38

hard

A box plot has whisker endpoints at

5

and

95

, box from

30

70

. Identify the IQR and check if

95

is an outlier under the

1.5\times

IQR rule.

Example 39

hard

Two data sets have the same median but Class X has IQR

=2

and Class Y has IQR

=14

. Which class has scores more spread out around the median?

Example 40

hard

A symmetric data set has

Q_1=40, Q_3=60

. Give the interval of values NOT flagged as outliers.

Example 41

challenge

A data set of 200 values has

Q_1=30, Q_3=60

. Approximately how many values lie in

[30,60]

Example 42

challenge

After a linear transformation

y = 5x + 7

, the IQR of

x

4

. What is the IQR of

y

Example 43

challenge

Why is IQR generally preferred over standard deviation when a data set has heavy tails or extreme outliers?

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Related Concepts

Quartiles Box Plot

Background Knowledge

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

quartiles