1) If I have the values for the 10th percentile, 50th percentile, and 90th percentile, can I find the value of any individual percentile? If not what assumptions would I need to make to determine this?

2) Going off of question 1, if I know the total sample size, can I find the number of values at any given percentile?

10th percentile - 24
50th percentile - 30
90th percentile - 35
n = 174

And say I want to find what value the 70th percentile would be and how many individuals there are at that particular percentile.


1 Answer 1


I am supposing that this is not a challenging exercise in some course. The context for this would be of interest.

We suppose from your 1) that you have 3 points known for a function, the quantile function that yields quantiles (percentiles) for given cumulative probabilities.

To get other points, you can use any method you like from general interpolation and extrapolation methods to assuming that data come from particular named distributions (e.g. Gaussian). But all methods are highly problematic; even injecting assumptions to make the problem tractable has the risk that the given quantiles contradict the assumptions being made. A simple example is that distance between the 10th and 50th percentiles and that between the 50th and 90th percentiles is equal in any symmetric distribution, so the assumption of any symmetric distribution would contradict your numeric example. Naturally you would be at liberty to assume an asymmetric distribution instead, or work on the basis that we don't usually expect assumptions to hold exactly.

So, the only safe answer to 1) is "all the data".

The second problem 2) is even more intractable, if possible. Without knowing anything else about the data, it is entirely possible that no values are "at" a particular reported percentile, which is true whenever it lies between two different observed values, but is not itself observed; and also entirely possible whenever ties are present in the data that several values are "at" a particular percentile, or indeed that several percentiles are reported to be identical. Concocted examples such as 1,1,1,1,1,1,3 illustrate.

Knowing the sample size doesn't really help.


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