Quartiles and five-number summaries | Year 10

01 / LEARN

Understand the idea.

Quartiles divide ordered positions

First order the data. Locate the overall median, then find the median of each half: these are $Q_{1}$ and $Q_{3}$ . This lesson uses median-of-halves, excluding the overall median from both halves when the sample size is odd.

For an even-sized half, average its two central values. Keep repeated observations in the list; they still occupy positions. Software may interpolate quartiles differently, so check both the data and the convention before deciding that different results are wrong.

Five numbers describe location and spread

A five-number summary lists the minimum, $Q_{1}$ , median, $Q_{3}$ and maximum in order. It records the extremes and three dividing positions, allowing you to compare the centre with the spread of the middle observations.

Check that the five values never decrease. Quartiles divide ordered positions, not the numerical scale into four equal lengths. Repeated values can give equal quartiles, and uneven gaps do not imply that the positional quarters have unequal importance.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

An even-sized sample

Find the five-number summary for 2, 4, 5, 6, 7, 8, 10, 12.

Worked solution

1
Find the overall median.
$median = \frac{6 + 7}{2} = 6.5$
Eight ordered values place the centre between positions four and five.
2
Find the lower-half median.
$Q_{1} = \frac{4 + 5}{2} = 4.5$
The lower half is $2, 4, 5, 6$ .
3
Find the upper-half median.
$Q_{3} = \frac{8 + 10}{2} = 9$
The upper half is $7, 8, 10, 12$ .
4
Include the two extremes.
$(2, 4.5, 6.5, 9, 12)$
The summary runs from minimum through the three dividing positions to maximum.

Answer

2, 4.5, 6.5, 9, 12.

EXAMPLE 02

An odd-sized sample

Find the five-number summary for 3, 4, 6, 8, 9, 11, 15.

Worked solution

1
Locate the central value.
$median = 8$
The fourth of seven ordered values is central.
2
Exclude the median and inspect the lower half.
$(3, 4, 6) \Rightarrow Q_{1} = 4$
The stated convention leaves the overall median out of both halves.
3
Find the upper quartile.
$(9, 11, 15) \Rightarrow Q_{3} = 11$
The middle upper-half value is the upper quartile.
4
Write the complete summary.
$(3, 4, 8, 11, 15)$
The first and last observations supply the minimum and maximum.

Answer

3, 4, 8, 11, 15.

03 / YOUR TURN

Try it yourself.

Check your understanding

Using median-of-halves, find Q1 for 1, 2, 4, 5, 7, 9, 10.

Show answer & reasoning

Exclude the overall median $5$ . The lower half is $1, 2, 4$ , whose middle value is $Q_{1} = 2$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Order 9, 4, 7, 2, 6, 8, 3, 5 and find their median.
2 marks
Find $Q_{1}$ and $Q_{3}$ for 2, 3, 4, 5, 6, 7, 8, 9.
2 marks
Give the five-number summary for 5, 7, 8, 10, 12, 13, 16.
2 marks
Give the five-number summary for 1, 2, 2, 3, 3, 4, 5, 8.
2 marks
A reported summary is 4, 12, 9, 18, 25. Explain the problem.
2 marks
Software reports $Q_{1} = 4.25$ where median-of-halves gives 4.5. What should you check before claiming an error?
2 marks

Read the worked solutions

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

Order 9, 4, 7, 2, 6, 8, 3, 5 and find their median.
2 marks
There are 8 values.
Average the fourth and fifth values: $\frac{5 + 6}{2} = 5.5$ .
Order 2, 3, 4, 5, 6, 7, 8, 9; median 5.5.
Find $Q_{1}$ and $Q_{3}$ for 2, 3, 4, 5, 6, 7, 8, 9.
2 marks
Lower half 2, 3, 4, 5 gives $\frac{3 + 4}{2}$ .
Upper half 6, 7, 8, 9 gives $\frac{7 + 8}{2}$ .
$Q_{1} = 3.5$ ; $Q_{3} = 7.5$ .
Give the five-number summary for 5, 7, 8, 10, 12, 13, 16.
2 marks
Median 10 is excluded from halves.
Lower-half median 7; upper-half median 13.
5, 7, 10, 13, 16.
Give the five-number summary for 1, 2, 2, 3, 3, 4, 5, 8.
2 marks
$Median = \frac{3 + 3}{2} = 3$ .
Lower half median 2; upper half median $\frac{4 + 5}{2} = 4.5$ .
1, 2, 3, 4.5, 8.
A reported summary is 4, 12, 9, 18, 25. Explain the problem.
2 marks
Five-number summaries must be nondecreasing.
Quartile positions cannot reverse the ordering.
$Q_{1} = 12$ exceeds the median 9, so the summary is invalid.
Software reports $Q_{1} = 4.25$ where median-of-halves gives 4.5. What should you check before claiming an error?
2 marks
Several interpolation rules are used for quartiles.
Compare results only after matching data and convention.
Check the quartile convention and the entered data.

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Curriculum reference & lesson scope

Australian Curriculum v 9 and SA Curriculum Mathematics

Year 10 continuous distributions, bivariate data, two-way categorical tables, bias and investigations, with the approved percentile and standard-deviation content included in the normal chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Quartiles divide ordered positions

Five numbers describe location and spread

See the method in action.

An even-sized sample

Find the overall median.

Find the lower-half median.

Find the upper-half median.

Include the two extremes.

An odd-sized sample

Locate the central value.

Exclude the median and inspect the lower half.

Find the upper quartile.

Write the complete summary.

Try it yourself.

Check your understanding

Keep practising