Percentile why n+1

Why we take n+1 in percentile. I didn’t got that part

Probably taking an example with fewer elements might help with the intuition part.

Suppose there are 3 elements, and you want to find the location of 50th percentile. If you use only n (instead of n+1), location will be 0.5*3 = 1.5th location. That implies only 1st location is less than or equal to 50th percentile, which is incorrect, since 2nd location should also have been included (2nd location is in middle and hence part of 50th percentile).

Not that, percentile formula p(n+1)/100 calculates the location and location count starts with 1 (not 0).

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