Two Sections, One Average: Choosing the Right Summary
At a higher secondary school in Nagaon, the statistics teacher compares two Class XI sections. Section A has 40 students with mean marks 62, standard deviation 8 and median 60 in the half-yearly test. Section B has 35 students with mean 68, standard deviation 6 and median 68.5. A parent-teacher meeting asks which section performed better overall and which is more consistent. The teacher also notices that Section A contains three students who scored above 95, while most students scored between 45 and 70. The principal wants a single summary statement for the school magazine and asks whether the mean or the median should be quoted for each section. The teacher has the raw marks but wants to demonstrate the reasoning using only the given summary values. No student is to be identified.
1. Compute the combined mean mark of the two sections. Explain why the simple average of 62 and 68 would be wrong.
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The combined mean is a weighted mean because the sections have different sizes:
[ \bar{x}_{combined}=\frac{40(62)+35(68)}{40+35} =\frac{2480+2380}{75}=\frac{4860}{75}=64.8 ]
The simple average ((62+68)/2=65) gives Section B's 35 students the same weight as Section A's 40 students, so it over-weights the smaller section. The correct combined mean, 64.8, lies between 62 and 68 and is closer to 62 because Section A has more students. The difference is small here, but the principle matters: a mean of means is valid only when the group sizes are equal.
2. Which section is more consistent? Use an appropriate relative measure and justify your choice.
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Section B is more consistent. Its standard deviation is smaller in absolute terms (6 versus 8), but the means differ, so the coefficient of variation is the fairer comparison:
[ CV_A=\frac{8}{62}\times100\approx12.90% ] [ CV_B=\frac{6}{68}\times100\approx8.82% ]
Section B has less variability relative to its own mean. If the question asked only about absolute spread in marks, Section A's SD of 8 would be larger; but for consistency when averages differ, CV is the appropriate relative measure. The conclusion: Section B is both higher on average and relatively more consistent.
3. Which measure of central tendency should be quoted for each section in the magazine? Give the reasoning from the summary values.
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For Section A, quote the median (60). The mean (62) is greater than the median (60), and the teacher knows there are three very high scores above 95. That pattern suggests a right-skewed distribution: the high scores pull the mean upward, so the mean overstates the typical student's mark. The median is resistant to those outliers and better represents a typical Section A student.
For Section B, mean 68 and median 68.5 are very close, so the distribution is nearly symmetric. Either measure is reasonable; report the mean and, if space allows, the median as a check. The magazine should not quote Section A's mean 62 without noting the skew, because readers would wrongly infer that 62 is typical.