Mathematics Year 7: Median and range: where the centre is, how spread out the data are
Mathematics, Year 7 (Portugal): The median says where the centre of the data is, the range says how spread out they are; learn to compute both, to see how an outlier moves them, and to pick the measure that fits the data.
Two questions, two kinds of measure
Summary measures answer two different questions. Measures of location say where the centre of the data is: you know the mean and the mode, and now there is the median. Measures of dispersion say how spread out the data are: the range. The median is the middle value of the data in order; with an even number of data, it is the mean of the two middle ones. The range is the greatest value minus the smallest.
Same centre, different spread
Two groups can have the same centre and still be very different. Five pupils read, in minutes: A: 20, 20, 20, 20, 20; B: 5, 15, 20, 25, 35. In both, the median is 20 and the mean is 20, but the range of A is 0 and that of B is 30 (35 − 5). Only the dispersion shows that in B some read very little and some read a lot. The range also depends on context: the ages in a Year 7 class vary less than those of a whole school.
Nine journeys, then eight
Nine pupils took 12, 60, 9, 15, 8, 11, 20, 10 and 14 minutes to reach school. Step 1: order them: 8, 9, 10, 11, 12, 14, 15, 20, 60. Step 2: 9 values, the middle one is the 5th: median 12 min. Step 3: range 60 − 8 = 52 min; mean 159 ÷ 9 ≈ 17.7. Remove the 60 (a very long journey, an outlier): 8 values, median (11 + 12) ÷ 2 = 11.5 min, mean 99 ÷ 8 ≈ 12.4, range 20 − 8 = 12. The outlier barely moved the median, and moved the mean and the range a lot.
A table is not the list
18 pupils, number of siblings: 0 siblings, 3 pupils; 1, 8 pupils; 2, 5 pupils; 3, 2 pupils. Error 1: the median of the frequency column (3, 8, 5, 2): 4, and nobody has 4 siblings. Error 2: the median of 0, 1, 2, 3: 1.5, ignoring how many pupils each value has. The table is the list in short: three 0s, eight 1s, five 2s, two 3s. With 18 data, the median is the mean of the 9th and 10th, both 1 (places 1 to 11 are 0s and 1s). Median: 1. If data come only in classes, you only find the class that contains the median, not its value.
Which measure fits the data?
Which measure to choose? Categories only have a mode. With numbers and no outliers, the mean and the median say almost the same. With outliers or very uneven data, the median describes the centre better, which is why people quote the “median price” or “median wage”. With data in classes, the mode becomes the modal class and you only locate the class of the median. Always give a measure of location together with a measure of dispersion: where the data are and how spread out.
Test what you learned
Two groups have the same median reading time, 20 minutes. Group A has range 0 and group B has range 30. What can you say?
- a) B read more, because its range is bigger
- b) A's data are more spread out than B's
- c) Both are centred on 20 minutes, but B's data are much more spread out
- d) The two groups have exactly the same data
Correct answer: c) Both are centred on 20 minutes, but B's data are much more spread out — Right: the median is a measure of location and the range a measure of dispersion. Same location, different dispersion.
Journey times (min): 14, 9, 22, 11, 9, 30, 16, 12. What are the median and the range?
- a) Median 13, range 21
- b) Median 10, range 21
- c) Median 13, range 30
- d) Median 12, range 21
Correct answer: a) Median 13, range 21 — Ordered: 9, 9, 11, 12, 14, 16, 22, 30. Eight values, so the median is (12 + 14) ÷ 2 = 13, and the range is 30 − 9 = 21.
18 pupils, number of siblings: 0 siblings, 3 pupils; 1, 8 pupils; 2, 5 pupils; 3, 2 pupils. What is the median number of siblings?
- a) 4
- b) 1.5
- c) 8
- d) 1
Correct answer: d) 1 — 18 data, so the median is the mean of the 9th and 10th. The three 0s and the eight 1s fill places 1 to 11, so both are 1.
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