According to Z1.4, section 9.1: "A sampling plan indicates the number of units of product from each lot or batch which are to be inspected (sample size or series of sample sizes) and the criteria for determining the acceptability of the lot or batch (acceptance and rejection numbers)." Z1.4 is a statistically valid and statistically based attribute sampling plan with the following characteristics:
Now consider the attribute sampling plan based on Sqrt (N) + 1, where Ac = 0 and Re = 1. It is also statistically valid and statistically based in that it has these statistical characteristics:
Consider again the raw material with N = 1000. The sample size is n = Sqrt (1000) + 1 = 32.6 or n = 33. Using Ac = 0, and Re = 1, software calculations show (3) that the AQL for P a = 0.95 is 0.153%. For all practical purposes, the two plans are the same, given the slight difference in the sample size.
Sample sizes for Sqrt(N) + 1 are very close to the sample sizes in Z1.4 General Level I. For small lot sizes, Sqrt (N) + 1 samples are actually larger than Z1.4 General Level I (see Figure 1).
Both Z1.4 General Level I and Sqrt(N) + 1 are universally used in the industry for attribute sampling and inspection. Both are, as demonstrated, statistically based and statistically valid. We can have as much confidence in Sqrt(N) + 1 as we do of Z1.4 General Level I. Any comments or publications to the contrary are incorrect.
Sqrt(N) + 1 sampling is recommended in the FDA document, "CBER 03/01/92 Draft Points to Consider in the Manufacture of In Vitro Monoclonal Antibody." According to FDA's docket no. 91N-0466, section V3, " ... another protocol for testing representative lots (e.g., square root n + 1/yr, where 'n' equals the number of lots of product produced per year) may also be found to be satisfactory" (4).
Sqrt(N) sampling is recommended in FDA's Investigations Operations Manual, which states, " ... a general rule is to collect samples from the square root of the number of cases or shipping containers but not less than 12 or more than 36 subs in duplicate" (5).
Sqrt(N) + 1 is a statistically correct and valid sampling plan and can be used with the same care and caution as Z1.4 General Level I would be used.
1. H. Saranadasa, "The Square Root of N Plus One Sampling Rule: How Much Confidence Do We Have?" Pharm. Technol. 27 (5), 50 (2003).
2. American Society for Quality (ASQ), ANSI/ASQ Z1.4-2008 (Milwaukee, WI, 2008).
3. W. Taylor, Guide to Acceptance Sampling (Taylor Enterprises, Inc, Lake Villa, IL, 1992).
4. FDA, CBER, "Draft Points to Consider in the Manufacture of In Vitro Monoclonal Antibodies," Docket 91N-0466, Mar. 1992.
5. FDA, Investigations Operations Manual, Subchapter 4.3: Collection Technique, section 220.127.116.11 Random Sampling, http://www.fda.gov/ICECI/Inspections/IOM/ucm122525.htm#18.104.22.168, accessed July 31, 2009.