The ordered values are \(10,14,18,18,22,26,30,34,38,42,46,50,54,58,62,70,90\). Exclude the overall median when forming the two halves, and take the median of each half as its quartile.
Find \(\displaystyle Q_{1}\). Explain your reasoning. [2]
Find \(\displaystyle Q_{3}\). Explain your reasoning. [2]
Find the interquartile range. Explain your reasoning. [2]
Solution
Hint
Use the explicitly stated median-of-halves convention. Take the mean of the fourth and fifth lower-half values.
Worked solution
(a)
\(\displaystyle \frac{18+ 22}{2}= 20\)
Answer: \(20\).
Answer: \(20\)
(b)
\(\displaystyle \frac{54+ 58}{2}= 56\)
Answer: \(56\).
Answer: \(56\)
(c)
\(Q_3-Q_1=36\)
Answer: \(36\).
Answer: \(36\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
1 mark: Take the mean of the fourth and fifth lower-half values.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not confuse an outlier fence with an observed extreme.
(b) · 2 marks
1 mark: Take the median of the upper eight observations.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Use \(\displaystyle Q_{3}-Q_{1}\), not \(\displaystyle Q_{3}\)−median, for the IQR.
(c) · 2 marks
1 mark: Subtract the first quartile from the third.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not confuse an outlier fence with an observed extreme.