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.
