# References on how to calculate quantile of a population from a small sample

I'm a novice in statistics. I need to study a problem that basically consists in checking if the 5% quantile of a population (that can be assumed normally distributed) is above or below a given value from a very small sample (15 values). I don't know the mean nor the standard deviation of the population, so both will have to be estimated from the sample. Presumibly I will need to define a confidence interval for that testing.

I need help on how to solve such a problem and references on the background theory I need to study to approach such a problem. Please be specific if you can.

• This problem calls for a tolerance limit (a phrase you could search on).
– whuber
Nov 10, 2015 at 19:11
• @whuber Yeah, tolerance limits are what I was looking for. Thank you, you saved me from digging into thousands of pages of statistics. For future readers of this post, I read the introductory part of this book "Statistical tolerance regions, Theory, Applications, and Computation" and it was very helpful. Hope this can help someone else in future.
– xanz
Nov 11, 2015 at 3:46
• @xanz if you found answer to your question you can always post it as an answer to your question -- it'd be nice, since it'd let others learn from it.
– Tim
Nov 30, 2016 at 21:30

An excellent Excel file has been compiled to allow for the use of calculating Hazen Percentiles. The file not only contains information regarding this particular method (which is useful for percentile with small sample sizes), but it also includes several other percentile calculations.

From the file:

Calculation procedure
Say you have n data, Xi, such that i = 1, 2, … , n. The Hazen procedure used is:

1 Rank the n data from lowest to highest; call these ranked data Yi: i = 1, 2, …, n.

2 Compute the percentile fraction (i.e., proportion) as p = P/100

3 Check if there are enough data to make the calculation, i.e., if
n >= 1/[2(1-p)] and n >= 1/(2p)
[the first limit applies for an upper percentile (p > 1/2), and vice versa]

4 If there are enough data then calculate the Hazen rank (usually non-integer)
rHazen = 1/2 + pn

5 Interpolate between integer ranks (i.e., ranked data) adjacent to the Hazen rank using
Hazen Pth percentile = (1-rf)Yri + rfYri+1
where ri = the integer part of rHazen and rf = fractional part of rHazen
[note that the formula still works if there is just enough data, i.e., for
equalities, instead of inequalities, in the equations in item 3 above]

Note that there is no one correct way to calculate percentiles. The Hazen method is a
"middle-of-the-road" option. Other estimators use the same procedure but with a
different minimum number of data and a different r formula (in Steps 3 & 4), as follows:

rBlom = 3/8 + p(n+1/4), requiring n >= (3+2p)/[8(1-p)] and n >= (5-2p)/(8p)
rTukey = 1/3 + p(n+1/3), requiring n >= (1+p)/[3(1-p)] and n >= (2-p)/(3p)
rWeibull = p(n+1), requiring n >= p/(1-p) and n >= (1-p)/p
rExcel = 1 + p(n-1), requiring n >= 1 (which is dashed odd!)
(The Excel option always give the lowest percentile; Weibull always gives the highest.)

For more details see chapter 8 of McBride, G.B. (2005). Ysing Statistical Methods for
Water Quality Management: Issues, Problemns and Solutions,Wiley, New York.