The idea, step by step
Quartiles divide ordered positions
The median separates an ordered data set into lower and upper halves. The lower quartile, Q1, is the median of the lower half; the upper quartile, Q3, is the median of the upper half. This lesson uses the median-of-halves convention and leaves out the overall median when the number of values is odd.
Different software can use different quartile conventions. A valid report states the convention and uses it consistently rather than treating a small software difference as an arithmetic error.
Five numbers describe location and spread
Write the minimum, Q1, median, Q3 and maximum in that order. Check that they cannot decrease. The first and last values show the full recorded span, while Q1 to Q3 describes the central half of the observations.
Quartiles are positions in ordered data, not four separate measurements chosen at random. If values repeat, the intervals between quartiles can have different widths even though the position groups have similar numbers of observations.
A common trap
Q1 is one-quarter of the largest measurement.
Instead: Q1 is determined by ordered positions. Divide the data positions, not the maximum value.
See the thinking
An even-sized sample
Find the five-number summary for 2, 4, 5, 6, 7, 8, 10, 12.
- There are 8 values; median=(6+7)÷2=6.5.
- Lower half 2, 4, 5, 6 has median 4.5.
- Upper half 7, 8, 10, 12 has median 9.
An odd-sized sample
Find the five-number summary for 3, 4, 6, 8, 9, 11, 15.
- The middle fourth value is 8.
- Exclude 8: the lower half 3, 4, 6 has median 4.
- The upper half 9, 11, 15 has median 11.
Check your understanding
Using median-of-halves, find Q1 for 1, 2, 4, 5, 7, 9, 10.
Show answer
2
Exclude the median 5; the lower half 1, 2, 4 has middle value 2.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Order 9, 4, 7, 2, 6, 8, 3, 5 and find their median.2 marks
- Find Q1 and Q3 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 Q1=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: (5+6)÷2=5.5.
Order 2, 3, 4, 5, 6, 7, 8, 9; median 5.5. - Find Q1 and Q3 for 2, 3, 4, 5, 6, 7, 8, 9.2 marks
Lower half 2, 3, 4, 5 gives (3+4)÷2.
Upper half 6, 7, 8, 9 gives (7+8)÷2.
Q1=3.5; Q3=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=(3+3)÷2=3.
Lower half median 2; upper half median(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.
Q1=12 exceeds the median 9, so the summary is invalid. - Software reports Q1=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.
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 ↗