Find the median, the quartiles and the interquartile range of a data set, and use the interquartile range to compare how spread out two sets are.
The survey is filed and the park is roped off. Wren Lane runs a two-lap race round it every summer, and this year eleven people finished. Their times in minutes, in order, were 21, 23, 24, 26, 27, 29, 30, 32, 35, 41 and 58. These are the real times, with no bands and nothing to estimate. The median you already know: position (11 + 1) / 2 = 6, so the sixth time, 29 minutes. Every summary here is found by counting to a position, which only means something once the list is in order.
The median cut the list in half. Two more cuts give you quarters, and they are found the same way, by position. The lower quartile is at position (n + 1) / 4 and the upper quartile is at position 3(n + 1) / 4. For the eleven runners that is (11 + 1) / 4 = 3 and 3 × 12 / 4 = 9. Count along: the third time is 24 and the ninth is 35. So the lower quartile is 24 minutes and the upper quartile is 35 minutes. When a position lands halfway between two places, take the halfway point of the two times.
Those three cuts split the runners into four groups holding the same number of people, not the same spread of time. About a quarter of the field was home by 24 minutes. About a quarter took longer than 35. The half in the middle finished somewhere between 24 and 35. That middle half is the part of a race most people care about, because it is where the ordinary runner is, away from the one who sprinted and the one who stopped to fix a shoelace.
The interquartile range is the distance between the two outer cuts: upper quartile minus lower quartile. For this race that is 35 - 24 = 11 minutes. It measures how spread out the middle half of the field was. Set it beside the range, which is the whole spread from first to last: 58 - 21 = 37 minutes. Two honest numbers, three times apart. Turn the subtraction round and the upper quartile is the lower quartile plus the interquartile range. Two races with the same median but interquartile ranges of 5 and 20 minutes had very different middles: one bunched, one strung out.
That gap is the point. The 58 belongs to a runner who pushed a buggy round both laps. Suppose she had walked it in 90 minutes instead. The range would leap from 37 to 90 - 21 = 69, nearly doubling, on the strength of one person. The lower quartile stays 24 and the upper quartile stays 35, so the interquartile range is still 11. The cuts are made by position, and a bigger last number is still just the last number. One odd entry cannot drag a quartile.
The lower quartile sits at position (n + 1) / 4 and the upper quartile at 3(n + 1) / 4 of an ordered list, with the median's position halfway between those two positions. The interquartile range is upper quartile minus lower quartile, and it measures the spread of the middle half. Because the cuts are made by counting positions, one extreme value moves the range a long way and leaves the interquartile range where it was.
Race day. Eleven finishers, and this time the clock gave exact numbers instead of tick boxes: 21, 23, 24, 26, 27, 29, 30, 32, 35, 41 and 58 minutes, already in order. Sam wants a line for the newsletter about how spread out the field was, and reaches for the range.
The range is 58 - 21 = 37 minutes. What is awkward about putting that in the newsletter?
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