A misleading average journey time
Fifteen students in a Year 11 class record their journey time to school in minutes: 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 15, 18, 20, 22, 45. The local council is consulting on a new bus route and prints a leaflet saying that 'the average journey to school is about 14 minutes'. A parents' group says this figure makes the route look slower than most students' experience. The 45-minute journey belongs to one student who travels from a rural area and changes buses twice. The council wants a single summary figure for the leaflet, but the parents' group wants the figure to reflect a typical student. No student has a journey time of exactly 14 minutes.
1. Calculate the mean and median journey time. Explain why they differ and which is more representative of a typical student.
Read the worked answer
Sum of times = 5 + 7 + 7 + 8 + 9 + 10 + 10 + 12 + 12 + 13 + 15 + 18 + 20 + 22 + 45 = 213 minutes.
Mean = 213 ÷ 15 = 14.2 minutes, which rounds to about 14 minutes.
There are 15 values, so the median is the 8th value in order. The ordered list is: 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 15, 18, 20, 22, 45. The 8th value is 12, so the median is 12 minutes.
The mean is pulled upwards by the single 45-minute journey. The median is not affected much by that one extreme value. For a typical student, the median of 12 minutes is more representative; the mean of 14.2 minutes describes the total travel burden better but is not typical of most students.
2. The 45-minute student moves closer and now takes 15 minutes. Recalculate the mean and median. Does the council's leaflet claim become more or less defensible?
Read the worked answer
New sum = 213 - 45 + 15 = 183 minutes. New mean = 183 ÷ 15 = 12.2 minutes.
Replace 45 by 15 in the ordered list: 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 15, 15, 18, 20, 22. The 8th value is still 12, so the median remains 12 minutes.
The leaflet's claim of 'about 14 minutes' is now less defensible: the mean has fallen to 12.2 minutes, and the median is 12 minutes. The original 14-minute figure was based on an outlier that has now disappeared. If the council wants an honest summary, it should update the leaflet to about 12 minutes or explain that journey times vary widely.