Travel Survey: Which Average Goes on the Form?
A transition-year class surveyed 40 students about how long it takes them to get to school in the morning, because the school is applying for a rural transport grant. The results in minutes were: 5 minutes (4 students), 10 minutes (9 students), 15 minutes (12 students), 20 minutes (8 students), 25 minutes (4 students) and 45 minutes (3 students). The three students reporting 45 minutes all travel on the same morning bus from a village 18 km away, which arrives 20 minutes before class starts. The principal wants one figure for the application form and has written 'average journey time: 17 minutes'. A teacher has objected that the survey covered only one class of 40 out of 620 students in the school, and that this particular class happens to contain the bus cohort. The grant application asks the school to describe the transport needs of its students.
1. Calculate the mean, median and mode journey time from the data, showing your working.
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Step 1 – Total student-minutes. Multiply each time by its frequency:
- 5 × 4 = 20
- 10 × 9 = 90
- 15 × 12 = 180
- 20 × 8 = 160
- 25 × 4 = 100
- 45 × 3 = 135 Total = 685 student-minutes. Check frequencies: 4 + 9 + 12 + 8 + 4 + 3 = 40 ✓
Step 2 – Mean. 685 ÷ 40 = 17.125 minutes, i.e. about 17.1 minutes.
Step 3 – Median. With 40 values the median is the average of the 20th and 21st. Cumulative frequencies: 4, 13, 25, 33, 37, 40. Both the 20th and 21st values fall in the '15 minutes' group (positions 14 to 25), so the median is 15 minutes.
Step 4 – Mode. The highest frequency is 12, at 15 minutes. (The data is grouped, so the mode is the modal class, not a single observation.)
Step 5 – Range. 45 − 5 = 40 minutes, showing a strongly spread distribution.
2. Evaluate the principal's figure of 17 minutes. Which figure would you put on the application, and why?
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Step 1 – Where 17 comes from. 17 is the mean, rounded. It is arithmetically correct but it is not a time any student actually reported - nobody travels 17 minutes.
Step 2 – Test the mean's sensitivity. Remove the three 45-minute students: the total becomes 685 − 135 = 550 over 36 students, a mean of 15.3 minutes. Three students out of 40 (7.5%) have shifted the mean by almost 2 minutes. The mean is being pulled by a small, distinctive group.
Step 3 – Compare the measures. Median 15 and mode 15 both describe the typical student: 25 of the 40 travel 15 minutes or less. So 'average journey time: 17 minutes' overstates the typical journey.
Step 4 – But the purpose matters. This is a transport needs application. Reporting only the median hides the very students the grant exists to help - the three travelling 18 km who must catch a bus that arrives 20 minutes early.
Step 5 – Recommended wording. Quote the median (15 minutes) as the typical journey, and state alongside it that the mean is 17.1 minutes and that 3 students travel 45 minutes from a village 18 km away. Never present 17 minutes alone, because it implies every student faces that journey. Choosing a statistic to fit the funder's expectations rather than the data would be misrepresentation.
3. Using the survey data, find the probability that a randomly chosen surveyed student travels 20 minutes or more, and the probability that a student travelling 20 minutes or more travels 45 minutes. Comment on using these figures for the whole school.
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Step 1 – Count the 20+ group. 8 (20 min) + 4 (25 min) + 3 (45 min) = 15 students.
Step 2 – Probability. P(≥20 min) = 15 ÷ 40 = 0.375, i.e. 37.5%.
Step 3 – Conditional probability. Of those 15, three travel 45 minutes: P(45 | ≥20) = 3 ÷ 15 = 0.2, i.e. 20% of the long-journey group.
Step 4 – As a check, the complement. P(<20 min) = 25 ÷ 40 = 0.625, and 0.625 + 0.375 = 1 ✓
Step 5 – Sampling judgement. The survey is 40 out of 620 students, about 6.5%, and it is not a random sample: it is one class, chosen for convenience, and that class contains all three long-distance bus students. Scaling 37.5% to the school would predict about 233 students travelling 20 minutes or more - a figure the data cannot support, because the bus cohort is likely over-represented in this class. Before quoting any probability to the funder, the school should survey a random sample across year groups (60–100 students), or at minimum survey each year group separately, so that the estimate is not driven by one timetable coincidence.