The Class Survey on Study Hours
A class of 40 students records how many hours they studied on a given weeknight. The results are grouped: 0 hours (5 students), 1 hour (12 students), 2 hours (14 students), 3 hours (6 students), 4 hours (3 students). A student computes the mean and reports it as 1.7 hours. Another claims the median is 2 hours. A third says, 'Because most students studied 2 hours, the mean must also be 2.' The teacher asks the class to check each claim and to consider what happens to the mean if the three students who studied 4 hours had instead studied 0 hours.
1. Compute the mean and median study time from the grouped data. Check the first two claims.
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Total students: $5 + 12 + 14 + 6 + 3 = 40$. Good.
Total hours: $0(5) + 1(12) + 2(14) + 3(6) + 4(3) = 0 + 12 + 28 + 18 + 12 = 70$.
Mean $= \dfrac{70}{40} = 1.75$ hours.
Median: the 20th and 21st values in order. Cumulative counts: 5 (0 h), 17 (1 h), 31 (2 h). Both the 20th and 21st values fall in the '2 hours' group, so the median is 2 hours.
Mean = 1.75 hours (the 1.7 claim is slightly off). Median = 2 hours (correct).
2. Evaluate the third student's reasoning: 'Most students studied 2 hours, so the mean must be 2.'
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The mode is 2 hours (14 students), but the mean is 1.75 hours. The mean is pulled down by the 5 students who studied 0 hours and the 12 who studied 1 hour - together 17 students below 2 hours, versus 9 students above 2 hours.
The reasoning is wrong. The most common value (mode) is not generally equal to the mean. The mean balances total hours across all students, so low values pull it below the mode.
3. Recompute the mean if the three students who studied 4 hours had instead studied 0 hours. Describe the effect on the mean and explain why.
Read the worked answer
New total hours: change each of the three 4-hour values to 0, removing $3 \times 4 = 12$ hours.
New total: $70 - 12 = 58$ hours over 40 students.
New mean: $\dfrac{58}{40} = 1.45$ hours.
The mean drops from 1.75 to 1.45, a decrease of 0.30 hours.
Explanation: the mean is sensitive to extreme values. Removing the highest values (4 hours) and replacing them with the lowest (0 hours) reduces the total while the count stays 40, so the mean falls. The median, by contrast, would stay at 2 hours because the middle values do not change.