To compare two grouped data sets, compare the centres first, then the spreads, then write one conclusion that uses the numbers. A smaller interquartile range means the set is more consistent.
This lesson is the last in dispersion of grouped data. It builds on constructing and interpreting an ogive.
What is the sentence frame?
Write three sentences in this order.
- The centres: “Class A has a median of … and Class B has a median of …”.
- The spreads: “The interquartile range of A is … and of B is …”.
- The conclusion: “So class … is more consistent, because …”.
Worked example: two classes with similar means
Class A is the 40-student table from the mean and ogive lessons: frequencies 4, 9, 14, 8, 5 in the classes 10 to 19 up to 50 to 59. Class B also has 40 students, with frequencies 2, 8, 20, 8, 2.
Class A. Mean = 1 390 ÷ 40 = 34.75. Median ≈ 34.5, Q1 ≈ 26.2, Q3 ≈ 43.3, so the interquartile range is about 17.1.
Class B. The products are 2 × 14.5 = 29, 8 × 24.5 = 196, 20 × 34.5 = 690, 8 × 44.5 = 356, 2 × 54.5 = 109. Σfx = 1 380, so the mean is 1 380 ÷ 40 = 34.5.
The cumulative frequencies of B are 2, 10, 30, 38, 40. Q1 is at 10, the end of the class 20 to 29, so Q1 = 29.5.
The median at 20 gives 29.5 + (10 ÷ 20) × 10 = 34.5. Q3 is at 30, the end of 30 to 39, so Q3 = 39.5, and the interquartile range is 10.
| Mean | Median | Interquartile range | |
|---|---|---|---|
| Class A | 34.75 | 34.5 | 17.1 |
| Class B | 34.5 | 34.5 | 10.0 |
Conclusion. The two classes have almost the same centre, because both medians are 34.5. The interquartile range of B (10.0) is smaller than that of A (17.1), so the marks in Class B are more consistent.
The wording slip
A common slip is to write “Class B is better”. Better at what? The data shows only that B is more consistent, because the medians are equal.
Another slip is to compare only the means, which differ by just 0.25, and skip the spread. The means alone cannot tell these classes apart.
When the centres differ
If Class B had a higher median and a smaller interquartile range, you could say it scored higher and more consistently. If the centre and the spread point in different directions, write both results and do not force a single winner.
Use the graph evidence comparison lab to practise writing conclusions from invented graphs.
Check yourself
Class P has a median of 24.7 and an interquartile range of 14.4. Class Q has a median of 24.5 and an interquartile range of 8.0. Write a conclusion.
Answer
The medians are almost equal (24.7 and 24.5), so the centres are similar.
The interquartile range of Q (8.0) is smaller than that of P (14.4), so the marks in Class Q are more consistent.
What to study next
Test the whole chapter with the dispersion of grouped data practice set. Return to the chapter overview to revisit any lesson.
For a teacher to read your conclusions with you, see online one-to-one Mathematics tuition.