Quartiles divide a sample into four nearly equal pieces. In general, a sample of size n can be broken into k nearly equal pieces by using the cutpoints (i/k)(n + 1) for i = 1,..., k – 1. Consider the following ordered sample:

Question

2 18 23 41 44 46 49 61 62 74 76 79 82 89 92 95

a. Tertiles divide a sample into thirds. Find the tertiles of this sample.
b. Quintiles divide a sample into fifths. Find the quintiles of this sample.

Answer 1

(a) The sample size is n = 16. The tertiles have cutpoints (1?3)(17) = 5.67 and (2?3)(17) = 11.33. The first tertile is therefore the average of the sample values in positions 5 and 6, which is (44 + 46)?2 = 45. The second tertile is the average of the sample values in positions 11 and 12, which is (76 + 79)?2 = 77.5.

(b)The sample size is n = 16. The quintiles have cutpoints(i?5)(17) for i = 1,2,3,4. The quintiles are therefore the averages of the sample values in positions 3 and 4, in positions 6 and 7, in positions 10 and 11, and in positions 13 and 14. The quintiles are therefore (23 + 41)?2 = 32, (46 + 49)?2 = 47.5, (74 + 76)?2 = 75, and (82 + 89)?2 = 85.5.