k-factors for determining acceptance based on sample mean and an extremum

k_equiv(alpha, n)



the acceptable probability of a type I error


the number of observations in the sample to test


a vector with elements c(k1, k2). k1 is for testing the sample extremum. k2 is for testing the sample mean


The k-factors returned by this function are used for determining whether to accept a new dataset.

This function is used as part of the procedure for determining acceptance limits for a sample mean and sample minimum. These acceptance limits are often used to set acceptance limits for material strength for each lot of material, or each new manufacturing site. When a sample meets the criteria that its mean and its minimum are both greater than these limits, then one may accept the lot of material or the new manufacturing site.

This procedure is used to ensure that the strength of material processed at a second site, or made with a new batch of material are not degraded relative to the data originally used to determine basis values for the material. For more information about the use of this procedure, see CMH-17-1G or PS-ACE 100-2002-006.

According to Vangel (2002), the use of mean and extremum for this purpose is more powerful than the use of mean and standard deviation.

The results of this function match those published by Vangel within 0.05% for \(n > 2\) and \(\alpha \le 0.25\). Those results published by Vangel are identical to those published in CMH-17-1G.

This function uses numerical integration and numerical optimization to find values of the factors \(k_1\) and \(k_2\) based on Vangel's saddle point approximation.

The value \(n\) refers to the number of observations in the sample being compared with the original population (the qualification sample is usually assumed to be equal to the population statistics).

The value of \(alpha\) is the acceptable probability of a type I error. Normally, this is set to 0.05 for material or process equivalency and 0.01 when setting lot acceptance limits. Though, in principle, this parameter can be set to any number between 0 and 1. This function, however, has only been validated in the range of \(1e-5 \le alpha \le 0.5\).


M. G. Vangel. Lot Acceptance and Compliance Testing Using the Sample Mean and an Extremum, Technometrics, vol. 44, no. 3. pp. 242–249. 2002.

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

Federal Aviation Administration, “Material Qualification and Equivalency for Polymer Matrix Composite Material Systems,” PS-ACE 100-2002-006, Sep. 2003.

See also


qual_mean <- 100 qual_sd <- 3.5 k <- k_equiv(0.01, 5) print("Minimum Individual Acceptance Limit:")
#> [1] "Minimum Individual Acceptance Limit:"
print(qual_mean - qual_sd * k[1])
#> [1] 89.24981
print("Minimum Average Acceptance Limit:")
#> [1] "Minimum Average Acceptance Limit:"
print(qual_mean - qual_sd * k[2])
#> [1] 96.00123
## [1] "Minimum Individual Acceptance Limit:" ## [1] 89.24981 ## [1] "Minimum Average Acceptance Limit:" ## [1] 96.00123