I spent a good amount of time agonizing over this same question. I ended up reaching out to a math professor. He said that quantiles, and specifically percentiles, can be very tricky as there is no standard definition.

The CFA curriculum defines the percentile “y” as the value AT or BELOW “y” percent of the distribution lies. Let’s call this “definition one.”

There’s an alternative definition that says the percentile “z” corresponds to the smallest value within the distribution that is greater than “z” percent of the distribution. Let’s call this “definition two.”

Here is some numerical data to help demonstrate what’s going on here:

Rank – Value

1 – 3

2 – 5

3 – 7

4 – 8

5 – 9

6 – 11

7 – 13

8 – 15

Let’s calculate the 25th percentile.

By the first definition (the smallest value greater than or equal to 25% of the scores), the 25th percentile would be 5. That’s because rank positions 1 and 2 represent 25% of the total number of data points.

By the second definition (the smallest value greater than 25% of the scores), the 25th percentile woulud be 7.

There’s a third procedure for finding the percentile and it happens to be the one used by the CFA. The formula is familiar: L = (n + 1) * y/100

This procedure is the WEIGHTED AVERAGE of the first two definitions. (Follow the link below if you need help convincing yourself of that point.)

(The source for all of this:

http://onlinestatbook.com/2/introductio ... tiles.html)

What’s interesting and a bit frustrating is that the DEFINITION provided by the CFA curriculum does not match the PROCEDURE (or formula) they provide.