S3, S4Cumulative frequency and box plots

Answer: Box plot with smallest value 10, lower quartile 12, median 18, upper quartile 23, largest value 26 (minutes).

• B1 for a box drawn from the lower quartile (12) to the upper quartile (23)
• B1 for the median (18) marked with a line inside the box
• B1 for both whiskers drawn, out to the smallest value (10) and the largest value (26)

Worked solution: Draw the box from 12 to 23 with a line at the median, 18. Draw whiskers from the box out to the smallest value, 10, and the largest value, 26.

(a) Answer: A larger range means the times are more spread out, so Group A is less consistent.

• C1 for explaining that a larger range (or interquartile range) means more spread, so less consistent

Worked solution: Consistency means small spread. Group A's range (150 ms) is larger than Group B's (90 ms), so Group A is less consistent.

(b) Answer: Group A has the lower median (240 ms compared with 250 ms), so Group A generally reacted faster. Group B has the smaller interquartile range (25 ms compared with 45 ms), so Group B's times were more consistent.

• C1 for comparing the medians in context: Group A's median (240) is lower, so Group A generally reacted more quickly
• C1 for comparing a measure of spread in context: Group B's IQR (25) is smaller than Group A's (45) (or range 90 < 150), so Group B's times were more consistent

Worked solution: Medians: A 240 ms, B 250 ms, so on average Group A reacted faster.
IQRs: A \(260 - 215 = 45\) ms, B \(255 - 230 = 25\) ms, so Group B's times were more consistent.

(a) Answer: Cumulative frequencies 8, 28, 52, 72, 80. Points plotted at (20, 0), (30, 8), (40, 28), (50, 52), (60, 72), (70, 80) and joined with a smooth curve (or straight line segments).

• M1 for the cumulative frequencies 8, 28, 52, 72, 80 (may be seen in the table or implied by the points)
• M1 for at least 4 points plotted at the upper class boundaries, (30, 8) ... (70, 80), each within ½ small square
• A1 for a fully correct graph: all 5 points correct and joined with a smooth curve or line segments, starting from (20, 0)

Worked solution: Cumulative frequencies: 8, 28, 52, 72, 80.
Plot each upper class boundary against its cumulative frequency: (20, 0), (30, 8), (40, 28), (50, 52), (60, 72), (70, 80). Join the points with a smooth curve, starting from 20 on the horizontal axis.

(b) Answer: 45 m (accept 43 to 47)

• B1 for an answer in the range 43 to 47 m (ft a correctly read value from their graph)

Worked solution: Read across from a cumulative frequency of \(\frac{80}{2} = 40\) to the graph, then down: median \(\approx 45\) m.

(c) Answer: 18 m (accept 14 to 20)

• M1 for reading the graph at cumulative frequencies 20 and 60 (lower quartile 34 to 39, upper quartile 51 to 56)
• A1 for an answer in the range 14 to 20 m

Worked solution: Lower quartile: read across from \(\frac{80}{4} = 20\), giving about 36 m.
Upper quartile: read across from \(\frac{240}{4} = 60\), giving about 54 m.
IQR \(\approx 54 - 36 = 18\) m.

(d) Answer: 64 (accept 62 to 68)

• M1 for reading the graph at 34 m (cumulative frequency 12 to 18)
• A1 for an answer in the range 62 to 68

Worked solution: At 34 m the cumulative frequency is about 16, so about \(80 - 16 = 64\) throws have a distance of more than 34 m.