Reading Challenge: Combined Mean, Totals and a Misleading Chart
Two P5 classes took part in a four-week reading challenge. Class A has 28 pupils and recorded a mean of 145 minutes of reading per pupil per week. Class B has 30 pupils and recorded a mean of 138 minutes per pupil per week. The school newsletter prints a bar chart whose vertical axis starts at 130 minutes rather than at 0, and in that chart Class A's bar is nearly twice as tall as Class B's. The headline reads: "Class A reads almost twice as much as Class B!" A parent has written in to complain that the chart is misleading, while a teacher argues that Class B actually read more in total and therefore deserves the trophy. The organiser needs one combined figure for the whole year group and must decide which class should be awarded the trophy.
1. Calculate the combined mean for both classes and judge whether the claim of "about 141.5 minutes" is accurate.
Read the worked answer
Work with totals, never by averaging the two means.
Class A total = 28 × 145 = 4,060 minutes. Class B total = 30 × 138 = 4,140 minutes. Combined total = 8,200 minutes over 28 + 30 = 58 pupils.
8,200 ÷ 58 = 141.379… ≈ 141.4 minutes.
The claimed 141.5 is exactly (145 + 138) ÷ 2 - the simple average of the two class means, which is only correct when both classes have the same number of pupils. Because Class B has 2 more pupils, the combined mean is pulled slightly towards 138, so 141.5 overstates the true figure. The method matters more than the small difference.
2. Which class read more in total, and does that settle who should get the trophy?
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Totals: Class A = 4,060 minutes; Class B = 4,140 minutes.
Class B read 80 minutes more in total, even though its mean is lower.
The reason: if Class B had only 28 pupils at its mean, its total would be 28 × 138 = 3,864, which is 196 minutes less than Class A. Its 2 extra pupils add 2 × 138 = 276 minutes, turning a 196-minute deficit into an 80-minute lead.
So both statements are true at once: Class A has the higher mean (per pupil) and Class B has the larger total. The trophy rule must be decided first - a "per pupil" trophy goes to A, a "total minutes" trophy goes to B. Reporting one number without saying which measure it is would mislead.
3. Explain why the newsletter's chart makes Class A look almost twice as large, and state the true comparison.
Read the worked answer
The bar heights are measured from a baseline of 130, not 0.
Class A bar height = 145 − 130 = 15 units. Class B bar height = 138 − 130 = 8 units.
Ratio shown = 15 : 8 ≈ 1.9, hence "almost twice as much".
The real comparison is 145 : 138 ≈ 1.05 - a difference of 7 minutes, about 5%. Truncating the axis magnifies small differences and is a classic way a graph misleads even when every number on it is correct. A fair bar chart starts at 0, or uses a dot plot or a table instead.