Calculates the rank order for finding distribution-free tolerance bounds for large samples. This function should only be used for computing B-Basis for samples larger than 28 or A-Basis for samples larger than 298. This function is used by basis_nonpara_large_sample().

nonpara_binomial_rank(n, p, conf)



the sample size


the desired content for the tolerance bound


the confidence level for the desired tolerance bound


The rank corresponding with the desired tolerance bound


This function uses the sum of binomial terms to determine the rank of the ordered statistic that corresponds with the desired tolerance limit. This approach does not assume any particular distribution. This approach is described by Guenther (1969) and by CMH-17-1G.

The results of this function have been verified against the tables in CMH-17-1G and agreement was found for all sample sizes published in CMH-17-1G for both A- and B-Basis, as well as the sample sizes n+1 and n-1, where n is the sample size published in CMH-17-1G.

The tables in CMH-17-1G purportedly list the smallest sample sizes for which a particular rank can be used. That is, for a sample size one less than the n published in the table, the next lowest rank would be used. In some cases, the results of this function disagree by a rank of one for sample sizes one less than the n published in the table. This indicates a disagreement in that sample size at which the rank should change. This is likely due to numerical differences in this function and the procedure used to generate the tables. However, the disagreement is limited to sample 6500 for A-Basis; no discrepancies have been identified for B-Basis. Since these sample sizes are uncommon for composite materials testing, and the difference between subsequent order statistics will be very small for samples this large, this difference will have no practical effect on computed tolerance bounds.


W. Guenther, “Determination of Sample Size for Distribution-Free Tolerance Limits,” Jan. 1969. Available online:

“Composite Materials Handbook, Volume 1. Polymer Matrix Composites Guideline for Characterization of Structural Materials,” SAE International, CMH-17-1G, Mar. 2012.


nonpara_binomial_rank(n = 1693, p = 0.99, conf = 0.95)
#> [1] 11
## [1] 11

# The above example indicates that for a sample of 1693 observations,
# the A-Basis is best approximated as the 11th ordered observation.
# In the example below, the same ordered observation would also be used
# for a sample of size 1702.

nonpara_binomial_rank(n = 1702, p = 0.99, conf = 0.95)
#> [1] 11
## [1] 11