## Z1.4: Selecting the Sample Size

Q: I work for a pharmaceutical company that manufactures soft gel capsules. What is the proper way to select a sample size when using ANSI/ASQ Z1.4-2008: Sampling Procedures and Tables for Inspection by Attributes?

I’ll further illustrate my question with an example.  If one were to have a batch size of 20,000 units, according to General Inspection Level II, Normal, the corresponding letter code is “M.” In the master table for Acceptable Quality Levels (AQLs), the sample size would be 315 units.  If my AQL is 0.010 (with an acceptance/rejection number of 0/1 based on the table), does my sample size change to 1250 units? Or does it remain at 315 units?

A: The simple answer is 1250, not 315 suggested for sample size letter M.  General Inspection Level II, Normal, shows that for a lot size of 20,000, a sample size code level of M corresponds to a sample size of 315.  For an AQL of 0.01, the arrow points to a sample size of 1250 (sample size letter code Q) to have the required AQL of 0.01.

The calculation of AQL is not dependent on lot size.  In other words, a sample size of 315 gives a minimum AQL of 0.04, so a larger sample is required to estimate an AQL of 0.01.

Q2: Could you please add another layer to your response? The reason I’m seeking additional clarification is that the first step in determining the sample size is to find the letter code and the corresponding sample size. To me, it feels like the first step should be to determine the AQL.

A2: Let me expand with a more technical explanation.  Attribute sampling is based on the hypergeometric distribution and is estimated using the binomial distribution (which assumes an infinite population size).

The basic formula for the binomial is:

AQL and LQ for a given sample size (n) and defects allowed (x):

If n=30, x=0; AQL=0.17%; LQ=7.4%:

If you are using Z1.4, your sample size is selected based on your lot size.  Then, you would pick the AQL you need based on the risk you are willing to take for the process average of percent defective.  If you decide to not use Z1.4, but instead use the binomial directly, then you are correct that you would decide on the AQL and lot tolerance proportion defective (LTPD) first, then calculate a sample size for c=0, c=1, c=2, and etc.

Steven Walfish
Secretary, U.S. TAG to ISO/TC 69
ASQ CQE
Principal Statistician, BD
http://statisticaloutsourcingservices.com

Related Content:

Acceptance Sampling With Rectification When Inspection Errors Are Present, Journal of Quality Technology

Zero Defect Sampling, World Conference on Quality and Improvement, open access

Explore ASQ’s website for more case studies, articles, benchmarking reports, and other content about zero defect sampling.

## Sampling Plan for Pharmaceuticals

Q: We are a U.S. dietary supplements manufacturer operating under c-GMP conditions set by the U.S. Food & Drug Administration (FDA).

As such, we perform analyses of incoming raw materials (finished product ingredients), intermediate products (during manufacturing), and finished products. Analyses include identity testing (incoming raw materials), and other types of analysis (e.g. microbiological, heavy metals, some quantitative assays on specific compounds). These tests would be the attributes we wish to assess.

Basically, we are refining our sampling procedures and need to ascertain an acceptable number of samples to be taken for the various testing purposes outlined above.

The World Health Organization’s (WHO) Technical Report Series No. 929,  Annex 4, “WHO Guidelines for sampling of pharmaceutical products and related materials” references ANSI/ISO/ASQ 2859-1:1999 Sampling procedures for inspection of attributes – Part 1: Sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection in reference to the selection of a statistically-valid number of samples for testing purposes.

I note from your website that there are a number of other sampling standards available. I am seeking some guidance as to the most appropriate standard(s) for our particular purposes.

Any assistance you can offer would be much appreciated.

A: Though many of the sampling plans are similar, many standards organizations have published different interpretations of sampling schemes.  Since WHO recommends using ISO 2859-1 as the guidance document, I suggest selecting that plan.

There are similar documents that could be used as an alternative, if necessary:

2. BS 6001-1:1999/ISO 2859-1:1999+A1:2011 Sampling procedures for inspection by attributes. Sampling schemes indexed by acceptance quality limit (AQL) for lot-by-lot inspection

3. MIL-STD-105E – Sampling Procedures and Tables for Inspection by Attributes*

4. JIS Z9015-0-1999 Sampling procedures for inspection by attributes — Part 0 Introduction to the JIS Z 9015 attribute sampling system

A few points to consider:

• Usually for FDA-regulated products, a c=0 sampling plan is appropriate. See H1331 Zero Acceptance Number Sampling Plans, Fifth Edition, by Nicholas L. Squeglia
• Based on risk, an Acceptable Quality Level (AQL) should be selected
• Your sample size is usually set to be proportional to lot size.  If you are doing testing on bulk raw materials, the sample size will be set based on the variability of the lot as well as the variability of the method.

Steven Walfish
Secretary, U.S. TAG to ISO/TC 69
ASQ CQE
Principal Statistician, BD
http://statisticaloutsourcingservices.com/

Note:

*Military standard, cancelled and superceded by MIL-STD-1916, “DoD Preferred Methods for Acceptance of Product”, or ANSI/ASQ Z1.4:2008, according to Notice of Cancellation