Range, quartiles and interquartile range | Year 11

01 / LEARN

Understand the idea.

Locate the central half of the observations

The range measures the full span: maximum minus minimum. The interquartile range, $Q_{3} - Q_{1}$ , measures the span of the central half, so it is less affected by unusually extreme endpoints.

Use the chapter's median-of-halves convention: order the values, find the overall median, and take the median of each half. When the sample size is odd, exclude the overall median from both halves. Other software conventions can give different quartiles.

IQR = Q_{3} - Q_{1}

Read a box plot as a summary, not individual observations

A five-number summary lists minimum, $Q_{1}$ , median, $Q_{3}$ and maximum in order. A min–max box plot places the box from $Q_{1}$ to $Q_{3}$ , the median line inside it and whiskers extending to the two extremes.

The box describes the central half's spread, while the whiskers show the outer spans. A long section indicates more distance between its summary values, not necessarily more observations. The plot does not reveal every individual value or determine the mean.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

An odd sample

Find the five-number summary and IQR for 1,3,4,6,8,9,12.

Worked solution

1
Locate the overall median.
$median = x_{4} = 6$
Seven ordered observations have their middle at position 4.
2
Find the quartiles after excluding the median.
$\begin{aligned} (1, 3, 4) & \Rightarrow Q_{1} = 3 \\ (8, 9, 12) & \Rightarrow Q_{3} = 9 \end{aligned}$
Each half has three observations, and the middle of each half is its quartile.
3
Assemble the five-number summary.
$(1, 3, 6, 9, 12)$
The summary runs from the minimum through both quartiles and median to the maximum.
4
Calculate the central span.
$IQR = 9 - 3 = 6$
Interquartile range subtracts the lower quartile from the upper quartile.

Answer

(1,3,6,9,12); IQR 6.

EXAMPLE 02

An even sample

Find Q1, median and Q3 for 2,4,5,7,8,10,12,14.

Worked solution

1
Find the overall median.
$median = \frac{7 + 8}{2} = 7.5$
Eight observations put the middle between the fourth and fifth values.
2
Find the lower-half median.
$Q_{1} = \frac{4 + 5}{2} = 4.5$
The lower four values are 2, 4, 5 and 7; their central pair is 4 and 5.
3
Find the upper-half median.
$Q_{3} = \frac{10 + 12}{2} = 11$
The upper four values are 8, 10, 12 and 14; their central pair is 10 and 12.

Answer

Q1=4.5, median 7.5, Q3=11.

03 / YOUR TURN

Try it yourself.

Check your understanding

A set has Q1=12 and Q3=21. Find its IQR.

Show answer & reasoning

The central half extends from $12$ to $21$ . Its width is $IQR = Q_{3} - Q_{1} = 21 - 12 = 9$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find the range of 3,4,7,8,20.
2 marks
Find the five-number summary of 2,4,6,8,10,12,14.
2 marks
Find the IQR of 1,2,4,5,7,9,11,15.
2 marks
A box plot has summary $(5, 8, 11, 16, 25)$ . State the median, IQR and range.
2 marks
Replace the largest observation 30 in 1,3,5,7,9,11,30 with 100. Compare range and IQR.
3 marks
Two sets have the same five-number summary. Must their means be equal? Explain.
3 marks

Read the worked solutions

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

Find the range of 3,4,7,8,20.
2 marks
$20 - 3 = 17$ .
17.
Find the five-number summary of 2,4,6,8,10,12,14.
2 marks
The median is 8; half medians are 4 and 12.
$(2, 4, 8, 12, 14)$ .
Find the IQR of 1,2,4,5,7,9,11,15.
2 marks
$Q_{1} = \frac{2 + 4}{2} = 3$ ; $Q_{3} = \frac{9 + 11}{2} = 10$ .
IQR $10 - 3 = 7$ .
7.
A box plot has summary $(5, 8, 11, 16, 25)$ . State the median, IQR and range.
2 marks
IQR $16 - 8 = 8$ ; range $25 - 5 = 20$ .
Median 11; IQR 8; range 20.
Replace the largest observation 30 in 1,3,5,7,9,11,30 with 100. Compare range and IQR.
3 marks
Median 7; $Q_{1} = 3$ and $Q_{3} = 11$ in both samples.
Changing only the extreme leaves the central-half summary unchanged.
Range rises from 29 to 99; IQR stays 8.
Two sets have the same five-number summary. Must their means be equal? Explain.
3 marks
The summary fixes only selected positions and endpoints, not every intermediate observation.
Different intermediate values can change the total and mean.
No.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.

Official curriculum source ↗

In this lesson

Understand the idea.

Locate the central half of the observations

Read a box plot as a summary, not individual observations

See the method in action.

An odd sample

Locate the overall median.

Find the quartiles after excluding the median.

Assemble the five-number summary.

Calculate the central span.

An even sample

Find the overall median.

Find the lower-half median.

Find the upper-half median.

Try it yourself.

Check your understanding

Keep practising