Combining two sections' test results
The examination committee of a large higher secondary school is comparing two sections in a 50-mark business mathematics test. Section A has 40 students with mean 62 and standard deviation 8. Section B has 25 students with mean 68 and standard deviation 12. The principal wants a single combined mean for the school newsletter and also asks which section is more consistent. A parent-teacher representative argues that Section B must be both better and more consistent because its mean is six marks higher. The committee has the section-wise means and standard deviations but not the individual scripts. It plans to use the coefficient of variation and the combined standard deviation formula, and it is aware that the two sections wrote the same test under similar conditions.
1. Compute the combined mean for the two sections.
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The combined mean is a weighted average using the section sizes:
Combined mean = (40 times 62 + 25 times 68) / (40 + 25)
= (2,480 + 1,700) / 65
= 4,180 / 65
= 64.31 approximately.
So the combined mean is about 64.31 marks. It lies between 62 and 68, and closer to 62 because Section A has more students.
2. Compute the combined standard deviation and the coefficient of variation for each section. Which section is more consistent?
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Deviations of section means from the combined mean: dA = 62 - 64.31 = -2.31; dB = 68 - 64.31 = 3.69.
Combined variance = [nA(sA^2 + dA^2) + nB(sB^2 + dB^2)] / (nA + nB)
= [40(64 + 5.34) + 25(144 + 13.62)] / 65
= [40(69.34) + 25(157.62)] / 65
= [2,773.6 + 3,940.5] / 65
= 6,714.1 / 65
= 103.29 approximately.
Combined standard deviation = square root of 103.29 = 10.16 approximately.
CV for A = 8 / 62 = 0.1290, or 12.90%.
CV for B = 12 / 68 = 0.1765, or 17.65%.
CV combined = 10.16 / 64.31 = 0.1580, or 15.80%.
Section A is more consistent because it has the smaller coefficient of variation, even though Section B has the higher mean.
3. Is the representative's claim fully supported? Explain what the statistics do and do not show, and state one check the committee should make before publishing a 'more consistent' conclusion.
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The claim is only partly supported. Section B does have the higher mean, so it may be described as stronger on average. But its standard deviation and coefficient of variation are larger, so it is less consistent in relative terms. A higher mean does not imply lower dispersion.
The coefficient of variation compares dispersion relative to the mean, but it does not reveal the shape of the distribution. The combined standard deviation uses the section means and within-section standard deviations; it does not show whether either section has outliers, skewness, or a few very low scores pulling the mean down. The committee should inspect the scripts, or at least a frequency table, for outliers and marking consistency before publishing a conclusion about consistency.