Handling Data 6
3 marks · NormalThe histograms below each show the distribution of the finishing times (in minutes) for two groups, Group P and Group Q, in a 10 km race. Both groups contain exactly 60 runners. Both axes are labelled the same and the scales match.
| Time (min) | Frequency density (P) | Frequency density (Q) |
|---|---|---|
| 30 < t ≤ 40 | 1.8 | 0.7 |
| 40 < t ≤ 50 | 1.2 | 2.0 |
| 50 < t ≤ 60 | 0.7 | 1.3 |
| 60 < t ≤ 70 | 0.3 | 0.4 |
Which group had a greater proportion of fast runners? Give a reason for your answer.
Which group had finishing times more spread out? Explain why using the histogram.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a) Group P had a greater proportion of fast runners, because the frequency density is higher in the 30 < t ≤ 40 class for P than for Q.
(b) Group Q had finishing times more spread out; the higher frequency density in 40 < t ≤ 50 and a noticeable density in the 60 < t ≤ 70 group indicate a wider spread compared to Group P, which has most frequencies concentrated in lower time brackets.
Common mistakes: Just stating frequencies instead of proportions; Not relating spread to histogram bar widths or shape
Show mark scheme and worked solution
Marking points
B1 (a) · 1 mark · States Group P had a greater proportion of fast runners, with valid reference to histogram (e.g., highest frequency density in lowest class for P).
B2 (b) · 1 mark · Identifies Group Q as having a more spread out distribution (oe), or equivalent reasoning.
E1 (b) · 1 mark · Explains reasoning by comparing frequency densities and discussing distribution shape, not just reading frequencies.
Worked solution
a) Group P had a greater proportion of fast runners because its frequency density is highest in the lowest time group (30 < t ≤ 40), meaning more runners finished quickly.
b) Group Q had finishing times more spread out because while it has the highest density in 40 < t ≤ 50, it also maintains relatively high densities in both 50 < t ≤ 60 and 60 < t ≤ 70 ranges. The bars are higher in several different bins, so the times are more widely distributed than in Group P, where most finishing times are in the lower intervals.