To compare two data sets, make two separate comparisons: one for the average (usually the mean or the median), and one for the spread (usually the range, interquartile range, or standard deviation). State each comparison in context, naming both data sets and what a higher or lower value means for the situation. A comparison of averages alone, without a comparison of spread, will not earn full marks.
A typical question gives two sets of results (for example, the times of two runners over ten races, or the scores of two classes on the same test) and asks: "compare the two data sets." Many students write one sentence about which set has the higher average and stop there. This earns partial credit at best.
The mark scheme is looking for two ideas, not one. The average tells you which set performed better on the whole. The spread tells you which set was more consistent. A set can have the higher average and still be the riskier choice if its spread is much wider, and O-Level questions are written specifically to test whether you notice this.
Step 1
Calculate or read off the average for each set (mean or median)
Step 2
Calculate or read off the spread for each set (range, IQR, or standard deviation)
Step 3
Compare the averages: which set is higher or lower, and what that means
Step 4
Compare the spread: which set is more consistent, and what that means
The question usually tells you which average and which spread to use, either directly or through the data given. If you are given the mean and standard deviation of both sets, compare those. If you are given a box-and-whisker plot with the median and quartiles marked, compare the median and the interquartile range instead. Do not calculate a value you were not given or asked for.
As a general rule: the mean pairs naturally with the standard deviation, and the median pairs naturally with the interquartile range. Keep to whichever pair the data hands you. A box plot never shows the mean, so a box plot question is asking for the median and the interquartile range. A list of raw values, or a frequency table, lets you work out the mean and the standard deviation, so that is the pair to quote. If you are unsure how to find the mean or median from a frequency table in the first place, see the guide on reading averages off a frequency table before attempting a comparison question.
Two students, Amy and Bala, each sit 5 short quizzes out of 10. Amy's scores: 6, 7, 6, 8, 8. Bala's scores: 3, 10, 4, 9, 9.
Solution
(6 + 7 + 6 + 8 + 8) ÷ 5 = 35 ÷ 5 = 7. Bala's mean: (3 + 10 + 4 + 9 + 9) ÷ 5 = 35 ÷ 5 = 7. Both means are equal, so the mean alone cannot separate them.SD = √(Σx²/n − (Σx/n)²). Amy: Σx² = 36 + 49 + 36 + 64 + 64 = 249, so SD = √(249/5 − 7²) = √(49.8 − 49) = √0.8 = 0.894 (3 s.f.).Σx² = 9 + 100 + 16 + 81 + 81 = 287, so SD = √(287/5 − 7²) = √(57.4 − 49) = √8.4 = 2.90 (3 s.f.).The range reaches the same verdict faster here (Amy: 8 − 6 = 2, Bala: 10 − 3 = 7), so quote it if you are only given the highest and lowest values. With every raw value in front of you, the standard deviation is the stronger answer: it uses all five scores, not just the two extremes.
This is the classic exam setup: equal or near-equal averages, very different spreads. When the averages tie, the entire comparison mark rests on the spread, so always calculate it even when the question feels like it is only asking about "who did better."
A box plot for Class A shows: minimum 40, lower quartile 55, median 65, upper quartile 75, maximum 90. A box plot for Class B shows: minimum 30, lower quartile 50, median 68, upper quartile 82, maximum 100. Both are test scores out of 100.
Solution
IQR = upper quartile − lower quartile. Class A: 75 − 55 = 20. Class B: 82 − 50 = 32.The step students get wrong
Writing "Class A has a smaller spread" or "Class A is better" without naming the actual quantity being compared. A marker cannot tell whether "spread" means range, interquartile range, or standard deviation unless you name it and give both values. Always write the full sentence: which measure, both numbers, and what the difference means for the two groups in the context of the question.
Use this sentence shape for both comparisons: "[Set 1]'s [average or spread measure] is [higher/lower/larger/smaller] than [Set 2]'s, which means [what this means in context]." Naming the sets, naming the measure, and stating the numbers is what separates a full-mark answer from a vague one.
Exam tip: if the question gives you units (marks, seconds, metres), include them in your comparison sentence. "Runner A's times are more consistent than Runner B's, since Runner A's standard deviation of 0.4 seconds is smaller than Runner B's 1.2 seconds" earns more than a bare numerical comparison.
Do I always need to compare both the average and the spread?
Yes, unless the question asks for only one specifically. A "compare" question in O-Level E-Maths is testing whether you understand that a full comparison needs a measure of central tendency and a measure of spread. Two sentences, not one, is the safe default.
What if the two averages are different but the spreads are also different?
State both differences separately, in context. Do not try to combine them into one overall judgement such as "Set A is better" unless the question specifically asks you to make a recommendation, and even then, refer back to both the average and the spread to justify it.
How do I know whether to use range, interquartile range, or standard deviation?
Use whichever measure the question gives you data for. Standard deviation needs the mean and every raw value (or a frequency table). Interquartile range needs the quartiles, usually from a box plot or a cumulative frequency curve. Range only needs the highest and lowest value, and is the easiest to compute but the most easily distorted by a single outlier.
Why can two data sets have the same mean but look very different?
The mean only describes the centre of the data, not how the values are distributed around that centre. Two sets can balance out to the same mean while one is tightly clustered and the other swings between very high and very low values. This is exactly why the spread comparison matters as much as the average comparison.
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