Quartiles Examples in Math

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 Math.

Concept Recap

Quartiles divide an ordered data set into four equal parts: Q1 is the 25th percentile, Q2 is the median (50th), and Q3 is the 75th percentile.

Q1 = 25th percentile, Q2 = median (50th), Q3 = 75th percentile.

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 are the three cut points (Q1, Q2, Q3) that split ordered data into four equal-count quarters.

Common stuck point: The procedure for quartiles is the easy part; the trap is splitting by equal value width instead of equal count. Asking "Am I splitting ordered data into four groups that each contain the same fraction of the values?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I splitting ordered data into four groups that each contain the same fraction of the values?

Worked Examples

Example 1

easy

Find the quartiles

Q_1

Q_2

, and

Q_3

for the data set:

\{4, 7, 9, 11, 14, 18, 22, 25, 30\}

Answer

Q_1 = 8

Q_2 = 14

Q_3 = 23.5

First step

The data is already sorted;

n = 9

Full solution

2
$Q_2$ (median): middle value at position 5 → $Q_2 = 14$
3
Lower half (below median): $\{4, 7, 9, 11\}$ ; $Q_1$ = median of lower half $= \frac{7+9}{2} = 8$
4
Upper half (above median): $\{18, 22, 25, 30\}$ ; $Q_3$ = median of upper half $= \frac{22+25}{2} = 23.5$

Quartiles divide ordered data into four equal parts. Q1 is the 25th percentile, Q2 (median) is the 50th percentile, and Q3 is the 75th percentile. For even-sized halves, average the two middle values.

Example 2

medium

A dataset of 10 values (sorted):

\{2, 5, 8, 12, 15, 18, 21, 25, 30, 40\}

. Find all quartiles and the five-number summary.

Example 3

medium

Find

Q_1

Q_2

, and

Q_3

for the dataset:

4, 7, 8, 12, 15, 18, 21, 25, 30

Example 4

medium

For the dataset

11, 13, 14, 16, 18, 19, 21, 23, 25, 27

, find

Q_1

Q_2

Q_3

, and the IQR.

Example 5

hard

Compare two box plots: Plot A has

Q_1 = 20, Q_3 = 40

; Plot B has

Q_1 = 25, Q_3 = 50

. Which dataset has greater variability and by what margin (in IQR)?

Practice Problems

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

Example 1

easy

Find

Q_1

and

Q_3

for:

\{10, 20, 30, 40, 50, 60, 70\}

Example 2

hard

A student scores at the 75th percentile on a standardized test with scores

\{45, 52, 60, 67, 72, 78, 83, 89, 95, 98\}

. Confirm that

Q_3

challenge

For 100 sorted values, which positions estimate Q1 and Q3 using rank

=p(n+1)

Example 23

easy

Find the median of

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

Example 24

easy

Find

Q_1

and

Q_3

1, 3, 5, 7, 9, 11, 13, 15

(excluding the median when splitting).

Example 25

easy

Find

Q_2

2, 5, 7, 9, 12, 14, 18, 20

Example 26

easy

Why must we sort the data before finding quartiles?

Example 27

medium

A student is at the 90th percentile. What does this mean?

Example 28

medium

Find

Q_1

and

Q_3

for

6, 9, 12, 12, 15, 18, 21, 22, 25, 30, 33, 35

Example 29

medium

The five-number summary of a dataset is

\{10, 18, 25, 35, 50\}

. State

Q_1

, the median,

Q_3

, and the IQR.

Example 30

medium

A box plot shows

Q_1 = 24

and

Q_3 = 50

. Find the upper fence used to detect outliers.

Example 31

medium

How many data values out of

200

should you expect to lie at or below

Q_1

Example 32

medium

Compute the IQR of

7, 12, 15, 21, 24, 30, 35, 38, 42

Example 33

medium

Given

Q_1 = 15, Q_2 = 25, Q_3 = 32

, in which quartile range (Q1-Q2 or Q2-Q3) is the data more spread out?

Example 34

hard

A dataset has

Q_1 = 35

and

Q_3 = 55

. Determine whether

80

is an outlier by Tukey's rule.

Example 35

hard

Quiz scores:

4, 6, 7, 9, 10, 12, 13, 14, 15, 17, 18, 20

. Find the IQR and identify any outliers.

Example 36

hard

Given the sorted dataset

3, 5, 7, 8, 9, 11, 12, 14, 17, 18, 20, 25

, find

Q_1

Q_3

, and the upper outlier (if any).

Example 37

hard

The IQR of a class's test scores is

14

points, with

Q_3 = 88

. What is the lower fence?

Example 38

hard

For the dataset

2, 4, 4, 5, 6, 7, 7, 8, 9, 10

, find

Q_1, Q_2, Q_3

Example 39

hard

If you add a new value of

100

to a dataset whose original

Q_3 = 45

and IQR

= 18

, does

100

qualify as an outlier?

Example 40

hard

A dataset of

40

values has

Q_2 = 50

. How many values are above

50

at most?

Example 41

challenge

Two datasets have the same range but Set A has IQR

= 30

while Set B has IQR

= 12

. Which set has heavier tails (more extreme values)?

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Related Concepts

Median Interquartile Range Percentages

Background Knowledge

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

median