Class Test Scores and a Suspicious Mean
A class of 24 students sits a mathematics test out of 50. The teacher calculates the mean as 31.5 and the median as 33. A student who was absent sits the test later and scores 18. The teacher adds this score and recomputes: the new mean is 30.96 and the new median is 32. A parent queries the drop in the median, arguing that adding one low score should not move the median by a whole mark. The teacher also notices that exactly one score of 47 appears twice in the list, and wonders whether removing one of those duplicates would change the median.
1. Verify the new mean after adding the score of 18, and explain why the median changed as it did.
Read the worked answer
Original total = 24 × 31.5 = 756.
New total = 756 + 18 = 774 over 25 students.
New mean = 774 ÷ 25 = 30.96. ✓ This matches the teacher's figure.
Median reasoning: with 24 scores the median is the average of the 12th and 13th values, giving 33. With 25 scores the median is the 13th value alone. Inserting 18 at the bottom shifts every position up by one, so the old 12th value becomes the 13th value and is now the median. That value is 32, so the median becomes 32. The drop is genuine, not an arithmetic error: the median is a positional measure and a single insertion can move it by a whole mark.
2. The teacher removes one of the two 47s. Determine the effect on the median of the 25-score list.
Read the worked answer
The 25-score list has median equal to its 13th value, which is 32.
A score of 47 lies above the median, so it occupies one of the top positions (positions 14 to 25 at most). Removing one value from above the median leaves the lower 13 values unchanged in order.
After removal there are 24 scores, and the median is the average of the 12th and 13th values. The 12th and 13th values of the reduced list are the 12th and 13th values of the original list, because nothing below them was removed.
So the new median = average of the original 12th and 13th values. Since the original 13th value is 32 and the original 12th value is at most 32 (values are ordered), the median becomes the average of those two, which is 32 or slightly less - it cannot rise. Removing a high duplicate lowers or holds the median; it never raises it.