## 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.

## Z1.4 Split Sampling

Q: I have two questions about Z1.4-2008: Sampling Procedures and Tables for Inspection by Attributes.

1. Does the plan allow one to “split” sampling plans among multiple items, or is only one item per plan intended?

2. The plan states a 95% confidence level, which means the findings of the sampling will statistically show that the findings (or number of defects) will be consistent with the findings of the entire inspected lot. So, if we split the sampling, how can you determine what happens to the confidence level?

A: Thank you for submitting your question to ASQ’s Ask the Experts Program. Answers to your inquiries follow.

1. In attempting to answer any given question, one needs to understand the question with respect to its gist and terms used.

Z1.4 uses the term “unit” to represent an individual “product” entity (unit here can represent a discrete fairly simple product, such as a bolt or nut), or it can represent a complex product (such as a computer, or a large piece of machinery, or even a square meter of cloth or other material, a length of wire or other material, etc.).

It is assumed here that the use of the term “item” in the question refers to a “unit.” It might, however, refer to a quality characteristic, and the explanation given here will attempt to explain either case.

Now, units can have a single principal quality characteristic or they can have many different quality characteristics.

Z1.4 allows for some of these quality characteristics to be of greater importance (severity for example, with respect to quality and/or economic effects) than others, whereby separate sampling is applied to each group with different sampling parameters (such as sample size, acceptance number, lot size). Hence, units with a single quality characteristic can be checked by sampling via Z1.4 and units with multiple quality characteristics can be checked by sampling via Z1.4.

In each case, the chosen Acceptable Quality Limit (AQL) and what it stands for applies to whatever is included in the inspection made on each unit. It is also assumed that this separate handling of units and quality characteristics is what the question means with respect to the term “split.”

Furthermore, it should also be understood that sampling inspection can be conducted with respect to two distinctly different statistics. One is the number of nonconforming units found in the sample. These are sometimes referred to as “defectives.” The second is the number (sum) of nonconformities found on all units in the sample, where any given single unit can have multiple nonconformities. These are often referred to as “defects.”

A “nonconforming unit” is defined as a unit with one or more nonconformities (defects) — but counted only as one “defective” unit. A “nonconformity” is any departure for any quality characteristic being considered in the inspection of each unit. In Z1.4, one can use either statistic as desired. The choice is largely dependent on the nature of product units and the reason for doing the sampling inspection — whether it is to control or oversee defective units or to control or oversee defects.

In the tables of Z1.4, note the top line above the range of AQLs: “Acceptance Quality Limits (AQLs), Percent Nonconforming Items and Nonconformities per 100 Items”. It should also be pointed out that Z1.4 is intended to be a sampling scheme or system, not just a selection of a given sampling plan. Please review the standard and any number of excellent books available on sampling inspection covering Z1.4, ISO 2859, and etc.

2. If one examines the Z1.4 standard from cover to cover, one will not encounter the term “confidence level.” Z1.4 contains no confidence intervals (or levels) related to any of its features.

Furthermore, the 95% figure is a very general figure associated with the expected “probability of acceptance” at the designated (selected) AQL. This is NOT a confidence level! In fact, the AQL is NOT a statistic!

Setting an AQL is generally an agreement/negotiation process between the customer and supplier. It is more of an index. Essentially, it refers to a level of nonconformity that is generally “acceptable” — a value of 0 being desired of course — but otherwise, a compromise figure.

And it is not by any means a constant, as can be seen by examining the Operating Characteristic (OC) Curves for the various code letters A through R using the same AQL in every table.

For example, for an AQL of 2.5% with the code letter C plan, incoming quality p must be 1.03% for Pa to be 95%, and Pa at 2.5% is less than 90%; for the code letter F plan, p must be 1.80% for Pa to be 95% and Pa at 2.5% is between 90% and 95%, etc.

If confidence intervals at chosen levels are desired for any given sampling plan, one most resort to the theory and methodologies of statistical inference with the available information provided by the sample statistics.

Kenneth Stephens
ASQ Fellow
ASQ Quality Press Author

For more on this topic, please visit ASQ’s website.

## Z1.4:2008, Using Acceptance Quality Limit (AQL)

Q: I have a question about how to use ANSI/ASQ Z1.4-2008 Sampling Procedures and Tables for Inspection by Attributes.

I am looking to achieve a 99.5% production yield.  How do I calculate that using the Acceptance Quality Limit (AQL) in this standard?  Is it as simple as taking (100-AQL) to calculate the expected yield?

A: The ANSI Z1.4-2008 standard is not intended for calculating production yield or expected production yield.  The AQL is the maximum percent non-conforming that can be considered acceptable as a process average.  Typically we set this as the percent defective that would be accepted at a 95% confidence.  If you want to sample such that you have 95% confidence that the average production yield is 99.5%, you can find a sampling plan with an AQL of 0.5%.  Also, please understand that the tables in the standard are not exact value for AQL.  Using the binomial distribution (or hypergeometric for sampling with no replacement) you can calculate the exact probability.

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

For more on this topic, please visit ASQ’s website.