Quartiles and box plots | Year 11

01 / LEARN

The idea, step by step

Split the ordered data consistently

Order the data. Here Q1 is the median of the lower half and Q3 the median of the upper half; exclude a single overall median from both halves. Software can use different conventions, so state the method when small samples are compared.

The five-number summary is minimum, Q1, median, Q3, maximum. $I Q R = Q_{3} - Q_{1}$ measures the spread of the middle half and is less sensitive than range to isolated extreme observations.

Read proportions and distances

A box runs from Q1 to Q3 with a line at the median. These unmodified plots use the minimum and maximum as whisker endpoints. A modified box plot may show flagged values separately, so always check the convention.

Each quarter represents approximately one quarter of the cases, not an equal length on the axis. A long section shows more spread, not more people. Box plots do not reveal sample size or every individual value.

A common trap

The longest box-plot section contains the most people.

Instead: Each quarter represents about 25%; length shows numerical spread.

02 / WORKED EXAMPLES

See the thinking

Example 1

Find five numbers

Summarise 2, 4, 5, 7, 8, 10, 12, 16.

Median $(7 + 8) / 2 = 7.5$ .
Q1 $(4 + 5) / 2 = 4.5$ ; Q3 $(10 + 12) / 2 = 11$ .
Minimum 2; maximum 16; IQR 6.5.

(2, 4.5, 7.5, 11, 16); IQR 6.5.

Example 2

Interpret the middle half

Q1 is 12 min and Q3 is 20 min. What does that show?

Q1 and Q3 bound the middle half.
$20 - 12 = 8$ .

The middle half spans 12–20 min; IQR 8 min, not all observations.

Check your understanding

Q1=9, Q3=17. Find IQR.

Show answer

$17 - 9 = 8$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Summarise 1, 3, 4, 6, 8, 10, 11, 15.
2 marks
Find IQR for that set.
2 marks
Exclude the median to find Q1, median, Q3 for 2, 4, 7, 9, 11, 14, 18.
2 marks
A summary is (3, 8, 12, 18, 27). Find range and IQR.
2 marks
Can box plots alone show which group contains more people?
2 marks
What does a long upper whisker suggest?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Summarise 1, 3, 4, 6, 8, 10, 11, 15.
2 marks
Median 7; lower/upper half medians 3.5 and 10.5.
(1, 3.5, 7, 10.5, 15).
Find IQR for that set.
2 marks
$10.5 - 3.5 = 7$ .
7.
Exclude the median to find Q1, median, Q3 for 2, 4, 7, 9, 11, 14, 18.
2 marks
Exclude 9; use medians of 2, 4, 7 and 11, 14, 18.
4, 9, 14.
A summary is (3, 8, 12, 18, 27). Find range and IQR.
2 marks
$27 - 3 = 24$ ; $18 - 8 = 10$ .
Range 24; IQR 10.
Can box plots alone show which group contains more people?
2 marks
They show summary positions and proportions, not counts.
No, unless sample sizes are given separately.
What does a long upper whisker suggest?
2 marks
It does not prove why or reveal individual values.
Higher observations extend across a wide interval.

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Mean/median and variability

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: statistical investigation, univariate data, sampling, displays, summaries and interpretation.

Official curriculum source ↗

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