Box plots and the semi-interquartile range
Answers are for parents and teachers.
1
The 13 lengths of rope in cm: 18, 22, 25, 27, 30, 31, 33, 35, 38, 40, 44, 47, 52 (already ordered). Find the five numbers, and say where the box and the whiskers are drawn on the scale.
n + 1 = 14: places 3.5, 7, 10.5. Q₁ = (25 + 27) ÷ 2 = 26, Q₂ = 33, Q₃ = (40 + 44) ÷ 2 = 42. Five numbers 18, 26, 33, 42, 52. The box from 26 to 42 with the line at 33; whiskers to 18 and 52.
2
Read the five numbers from the box plot and find the range, the interquartile range and Q.
Five numbers 20, 30, 50, 70, 90. Range 70. Interquartile range 70 − 30 = 40. Q = 20.
3
Waiting times in minutes at two bus stops have the five numbers: stop 1: 4, 8, 11, 15, 22; stop 2: 3, 9, 12, 14, 18. Compare the stops in two sentences (centre and spread). In the picture the lower box plot is stop 1 and the upper one is stop 2. Which stop would you choose?
The medians 11 and 12 are about the same. Stop 1 has the longer box (15 − 8 = 7 against 14 − 9 = 5) and the longer range (18 against 15), so its waiting times are less predictable. Stop 2 is steadier: choose it.
4
Find Q for the 12 values in the plot (key 3|1 = 31).
key 3|1 = 31
| Stem | Leaves |
|---|---|
| 3 | 1 3 6 8 |
| 4 | 0 4 5 7 |
| 5 | 0 2 5 9 |
n + 1 = 13: places 3.25 and 9.75. Q₁ = 36 + 0.25 × 2 = 36.5; Q₃ = 50 + 0.75 × 2 = 51.5. Q = (51.5 − 36.5) ÷ 2 = 7.5.
5
The reaction times of 40 pupils are in the classes 0–, 5–, 10–, 15–, 20–25 with frequencies 5, 11, 14, 7, 3. Find Q.
Cumulative 0, 5, 16, 30, 37, 40; orders 10 and 30. Q₁ = 5 + (10 − 5) ÷ 11 × 5 = 7.27; Q₃ = 10 + (30 − 16) ÷ 14 × 5 = 15. Q = (15 − 7.27) ÷ 2 = 3.86.