The idea, step by step
Locate the central half of the observations
The range is maximum minus minimum and uses only the endpoints. Quartiles divide the ordered data into four parts. In this chapter, find the median, then find the median of the lower and upper halves; when n is odd, exclude the overall median from both halves. Other software conventions can differ, so state and apply the chosen convention consistently.
Read a box plot as a summary, not individual observations
The five-number summary is minimum, Q1, median, Q3 and maximum. An ordinary min–max box plot places its box from Q1 to Q3, a median line inside, and whiskers to the extremes. The IQR describes the central half and resists extreme values better than range. A longer half-box suggests more spread in that portion, but the plot alone does not reveal every value or the mean.
A common trap
Including the overall middle observation in both halves without stating a different quartile convention.
Instead: Use the specified median-of-halves convention consistently, and report the convention if software produces different quartiles.
See the thinking
An odd sample
Find the five-number summary and IQR for 1,3,4,6,8,9,12.
- Median is 6; exclude it.
- Lower half 1,3,4 has median 3. Upper half 8,9,12 has median 9.
- IQR 9−3=6; range 12−1=11.
An even sample
Find Q1, median and Q3 for 2,4,5,7,8,10,12,14.
- Median=(7+8)/2=7.5.
- Lower-half median=(4+5)/2=4.5; upper-half median=(10+12)/2=11.
Check your understanding
A set has Q1=12 and Q3=21. Find its IQR.
Show answer
9.
Subtract the lower quartile from the upper:21−12=9.
Your turn
6 questions · 14 marks · Work at your own pace.
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
Q1=(2+4)/2=3; Q3=(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; Q1=3 and Q3=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.
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 ↗