# Statistically Valid Sampling Plans

Dr. Wayne A. Taylor

Numerous 483’s issued by the FDA have cited sampling plans as not being “statistically valid”.  So what is required of a sampling plan to be statistically valid?  Selecting statistically valid sampling plans is a two-part process.  First, one must determine an appropriate objective for the inspection.  Second, one must select a sampling plan capable of meeting this objective.  Let us look at a couple of examples.  In all these examples suppose that one is inspecting for major defects and that an acceptable quality level (AQL) of 1.0% defective has been established.

First, suppose that a sampling plan is being selected to inspect the first lot produced.  The objective of this sampling plan is to demonstrate that the process average is below the AQL.  The single sampling plan with sample size n=300 and accept number a=0 is statistically valid for this purpose.  Passing this sampling plan allows one to state that, with 95% confidence, the process average is below 1% defective.

Having passed the first lot, future lots are to be inspected to ensure the average outgoing quality is less than the AQL and that no released lot is worse than 4% defective.  For this purpose, the single sampling plan n=125 and a=2 is statistically valid.  This sampling plan has an LTPD of 4.20% defective.  This makes the release of lots worse than 4% unlikely.  Further, this sampling plan has an AOQL of 1.10% ensuring that the worse the customer receives as a long-term average is 1.10%.

Now suppose that the process has run for six months with a process average of 0.1% defective and no major problems.  While the process has a good history, there may still be concern that something could go wrong in the future.  As a result, a small number of samples should continue to be inspected from each lot.  It is decided that the sampling plan should ensure that a major process failure resulting in 20% defective or more should be detected on the first lot.  The single sampling plan n=13 and a=0, with an LTPD of 16.2% defective, is statistically valid for this purpose.

Simply using sampling plans from a recognized source such as Mil-Std-105E does not ensure that the sampling plans are statistically valid.  One of the sampling plans in Mil-Std-105E is to take 2 samples and accept on 30 defects.  This sampling plan would not be valid if used to inspect for critical defects.  All sampling plans, regardless of their source are valid for certain applications but not valid for others.  Different sampling plans are statistically valid at different times during a process’s life.  Therefore, all sampling plans should be periodically reviewed.  To ensure a sampling plan is valid, first, the objective of the inspection should be clearly stated.  Then the protection provided by the sampling plan such as the LTPD and AOQL should be documented to demonstrate that the sampling plan meets the objective.  If you don’t know the protection provided by your sampling plans or are unclear as to the purposes of your inspections, you are at risk.

Appeared in FDC Control, Food Drug & Cosmetic Division ASQC, No. 102, March 1994, p. 5