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3 Responses
This is one of those spots where perfectly sensible people use two different conventions. If you exclude the median when n is odd, you split 2,4,5,7,9 as [2,4] | [7,9], so Q1 is the median of 2 and 4 (that’s 3) and Q3 is the median of 7 and 9 (that’s 8), giving IQR = 8 − 3 = 5. If you include the median in both halves, you split as [2,4,5] | [5,7,9], so Q1 is the median of the lower half (4) and Q3 is the median of the upper half (7), giving IQR = 7 − 4 = 3. Your notes are using the “include the median” rule; your calculation used the other common rule. Neither is wrong-just be consistent with whichever your course expects.
For your data, if you’re following the notes’ convention, Q1 = 4, Q3 = 7, so IQR = 3. And yes, the range is 9 − 2 = 7. I used to overthink this too; now I just pick the rule my teacher is using and stick with it.
Both answers are defensible; they come from two common conventions. If you exclude the median when n is odd, you split as [2,4] | [5] | [7,9], giving Q1 = (2+4)/2 = 3, Q3 = (7+9)/2 = 8, so IQR = 5. If you include the median in both halves (Tukey’s “hinges”), you split as [2,4,5] and [5,7,9], so Q1 = 4 and Q3 = 7, giving IQR = 3. Your notes are using the inclusive/Tukey method. The range 9 − 2 = 7 is correct in any case.
A sensible rule is: pick a convention and be consistent with your course or calculator. Many school texts use the inclusive method when n is odd (giving IQR = 3 here), while some calculators default to the exclusive method (IQR = 5). For background on the different quartile definitions, see the discussion and comparisons at Wikipedia: https://en.wikipedia.org/wiki/Quartile#Different_methods
Which convention does your class or exam specify? If you try the same data on your calculator or software, which Q1 and Q3 does it report?
You’re running into a common quirk: there are two standard conventions for quartiles when n is odd. For the data 2, 4, 5, 7, 9, the median is 5. If you use the “exclusive-median” rule (drop the median when forming the halves), you split as [2, 4] | 5 | [7, 9], giving Q1 = median(2, 4) = 3 and Q3 = median(7, 9) = 8, so IQR = 8 − 3 = 5. If you use the “inclusive-median” rule (include the median in both halves), you split as [2, 4, 5] and [5, 7, 9], so Q1 = median(2, 4, 5) = 4 and Q3 = median(5, 7, 9) = 7, giving IQR = 7 − 4 = 3. Your notes’ answer (IQR = 3) matches the inclusive rule, which many school texts prefer for hand calculations. The range is indeed 9 − 2 = 7. To stay consistent with your course, use the rule your notes adopt: for odd n, include the median in both halves; for even n, just split into equal halves. Which convention does your class or calculator default to, and would you like a quick checklist for even-sized samples too?