Percentiles: learn both course methods
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A percentile describes a position in an ordered dataset. The 50th percentile is the median. A 90th percentile is a cutoff near the upper end, not “90% marks”.
What a percentile means
For the lecture’s hand-rule definition, a 100p percentile is a cutoff with at least 100p% of the observations at or below it and at least 100(1−p)% at or above it. Here p is a proportion: use 0.25 for the 25th percentile. Ties mean the percentages on either side need not be exactly 25% and 75%.
Method A: the lecture’s hand-calculation rule
First sort the observations. For an interior percentile, 0 < p < 1, calculate np. If np is not a whole number, round it UP and take the observation at that position. If np is a whole number, average the observations at positions np and np+1. At p = 0 use the minimum; at p = 1 use the maximum, rather than looking for a nonexistent next observation.
Use the sorted marks 35,38,47,58,61,66,68,68,70,79, with n = 10.
| Percentile | p | np | Hand-rule result |
|---|---|---|---|
| 10th | 0.10 | 1 | (35+38)/2 = 36.5 |
| 25th | 0.25 | 2.5 | 3rd value = 47 |
| 50th | 0.50 | 5 | (61+66)/2 = 63.5 |
| 75th | 0.75 | 7.5 | 8th value = 68 |
| 90th | 0.90 | 9 | (70+79)/2 = 74.5 |
| 100th | 1 | 10 | Maximum = 79 |
Method B: spreadsheet linear interpolation
Split the rank into whole part i and fractional part d. Start at ordered value x[i], then move fraction d of the gap to x[i+1]. If rank is an integer, just use that ordered value; at the maximum rank use the maximum.
For the 25th percentile, rank = 1 + 9×0.25 = 3.25. Position 3 has 47 and position 4 has 58. Move one quarter of the 11-mark gap: 47 + 0.25×11 = 49.75.
For the 90th percentile, rank = 1 + 9×0.9 = 9.1. Positions 9 and 10 have 70 and 79. The result is 70 + 0.1×9 = 70.9. It differs from the hand-rule 74.5 because the definitions differ.
Using a spreadsheet
With the ten raw marks in A1:A10, enter =PERCENTILE(A1:A10,0.25). This interpolation-based function gives 49.75. For 90th percentile, use 0.9 rather than 90. Use the method requested in the question; a spreadsheet result does not automatically check a calculation done with a different definition.
Do not mix percentile definitions within one calculation. State the rule you used. The two methods are both explicitly presented in the supplied lecture.
Try it yourself
For sorted 10,20,30,40,50, find the 30th percentile using the hand rule.
Show explained answer
n×p = 5×0.3 = 1.5; round up to position 2, giving 20.
Try it yourself
For the same values, find it using interpolation.
Show explained answer
Rank = 1 + 4×0.3 = 2.2. Result = 20 + 0.2×(30−20) = 22.
Try it yourself
Does the 90th percentile mean a student answered 90% correctly?
Show explained answer
No. A percentile is a position relative to other observations. Percentage marks measure a fraction of the available marks.
Source: Lecture 3.5. Extra examples and diagrams are teaching material prepared for these notes.