Math · Introductory statistics · Worked example
Find quartiles and the interquartile range
Find the five-number summary and the interquartile range of the sleep data 3, 4, 5, 6, 6, 7, 8, 9.
Minimum, median and maximum
The data are already sorted: the minimum is 3, the median 6 and the maximum 9.
Take the medians of the halves
Split the sorted data into a lower half, 3, 4, 5, 6, and an upper half, 6, 7, 8, 9. Q1 and Q3 are their medians. With an odd count, leave the median out of both halves.
Find the IQR
Compare interpolation
Much statistical software interpolates instead. Q1 sits a quarter of the way through the sorted list, at position 1 + 0.25 × 7 = 2.75: three quarters of the way from the 2nd value, 4, to the 3rd, 5. Q3 sits at position 6.25, a quarter of the way from 7 to 8. That gives 4.75 and 7.25, an IQR of 2.5. Both methods are accepted; say which one you use.
Result
The five-number summary is 3, 4.5, 6, 7.5, 9, so IQR = 3. Interpolated quartiles are 4.75 and 7.25.
Your turn
Using the halves method, find the median and the IQR of 1, 3, 5, 7, 9, 11, 13, leaving the median out of both halves.
Show the answer and explanation
Median 7; Q1 = 3 and Q3 = 11, so IQR = 8.
The lower half is 1, 3, 5, with median 3, and the upper half is 9, 11, 13, with median 11.
Keep exploring
In Statistics, the Summary lists Q1 = 4.5 and Q3 = 7.5, the medians of the two halves.
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Summarize the data in Statistics Check the arithmetic in Math Open worked example on a board Sample standard deviation in Math ReferenceYour existing work stays on this device. Examples open as editable copies.