Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the 21st, 26th, 61st, and 71st percentiles. If needed, round your answers to two decimal digits.
To compute percentiles, we need to sort the data values in ascending order.
The sorted data set is as follows: 15, 20, 25, 25, 27, 28, 30, 34.
To find percentiles, we use the formula:
(percentile / 100) * (N + 1), where N is the number of data values.
For the 21st percentile:
(21 / 100) * (8 + 1) = 1.89
Since 1.89 is not a whole number, we take the value that corresponds to the rank 1, which is the value between the 1st and 2nd data values:
15 + (20 - 15) * 0.89 = 15 + 5 * 0.89 = 15 + 4.45 = 19.45
For the 26th percentile:
(26 / 100) * (8 + 1) = 2.34
Again, we take the value between the 2nd and 3rd data values:
20 + (25 - 20) * 0.34 = 20 + 5 * 0.34 = 20 + 1.7 = 21.7
For the 61st percentile:
(61 / 100) * (8 + 1) = 5.49
This falls between the 5th and 6th data values:
27 + (28 - 27) * 0.49 = 27 + 1 * 0.49 = 27 + 0.49 = 27.49
For the 71st percentile:
(71 / 100) * (8 + 1) = 6.79
This falls between the 6th and 7th data values:
28 + (30 - 28) * 0.79 = 28 + 2 * 0.79 = 28 + 1.58 = 29.58
Therefore, rounding to two decimal places:
21st percentile = 19.45
26th percentile = 21.7
61st percentile = 27.49
71st percentile = 29.58