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FAQs  Frequently Asked Questions
Acceptance Sampling Questions
What are the origins of
the "square root of n plus
one" sampling rule?
The best that I can determine is that the rule probably had its origin in the USDA in
the 192030's. But no confirming documents exists. There are actually three versions
of the rule:
 Take the square root of the lot size to get the sample size. Accept on zero
defects. This version of the rules determines the sampling plan to use.
 Take the square root of the number of cartons, open that number of cartons and select
the required number of samples from them. In this case the sampling plan including
the number of samples is already determined. For example, if the sample size is 20
and 50 cartons exist, SQRT(50)+1=8 cartons must be opened. Select 3 samples from 4 cartons
and 2 from the remaining 4. The square root of n plus one rule is used to obtain a
representative sample.
 Take the square root of the number of drums, sample from this number of drums and
composite the samples together to run a single test. For example, if 50 drums are
received, take samples from SQRT(50)+1=8 drums, composite them together and measure the
characteristic of interest. The rule is used again to obtain a representative
sample.
Much of the discussion seems to confuse these distinct uses. Rule 1 should never be
used. Sampling plans should be selected based on operating characteristics such as
AQL and LTPD using tables of sampling plans like those given in my book Guide to Acceptance Sampling or one of the many standards such as ANSI Z1.4. The operating characteristic does not
depend on the lot size as explained in the article "The
Effect of Lot Size". Therefore, such plans can be selected independent on
lot size. One reference that might be of interest is:
Keith Borland (1950), "The Fallacy of the Square Root Sampling Rule,"
Journal of the American Pharmaceutical Association, 39, No. 7, p373377.
This reference describes why the square root of n plus one rule should not be used to
select a sampling plan.
Rule 2 and 3 represent a reasonable compromise in many cases balancing the cost of
testing with the precision of the results. However, there are certainly situations
rule 2 and 3 should not be used. For example, printing defects where the process could
produce 100 consecutive bad units all packed in a single carton and then correct itself.
Despite the lack of justification and documentation, this rule is commonly used.
For example:
GUIDE TO INSPECTIONS OF
MANUFACTURERS OF MISCELLANEOUS FOOD PRODUCTS  VOLUME 1
"For microscopic filth, excess shell, etc., sample the square root of the number
of bags in the lot. Collect a minimum of six and a maximum of eighteen subs each
consisting of 900 grams (2 lbs) taken 340 grams (2/3 lb) from each of the three bags.
Collect the subs in duplicate for the 702(b) portion."
"For retail size containers, sample the square root of the number of containers in
the lot with a minimum of six and a maximum of 18  900 gram (2 lb) subs."
"Bulk containers  collect 1 pint in duplicate from each container in lot. Sample
55 gal drums on a square root basis, collecting 1 pint from a minimum of 6 and maximum of
24 in duplicate."
Investigators Operations
Manual  FDA May 1996
Subchapter 420, Section 427.2 on random sampling states: "Sample size is usually
described in your assignment, IOM Sample Schedule, Compliance Program, or the applicable
schedules. If none of these furnish the sample size, 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. If there are less than 12 containers, all
should be sampled. Discuss sample size and 702(b) requirements with your supervisor. See
IOM 422.1. "
All reference to this rule has been deleted from the 2005
version.
I would love to hear from anyone who has further references, information or examples.
Back to Acceptance Sampling Questions
How do statistical sampling plans/concepts
apply to destructive testing?
Two issues occur with destructive testing, First is the cost of testing. With
destructive testing, the cost of testing is at least as great as the cost of producing the
unit. However, there are cases involving nondestructive testing where the cost of
testing exceeds the cost of the unit. As the cost of testing increases, economics dictate
that the sample sizes be reduced lessening the protection.
The second issue is the fact that rejected lots can be rectified by 100% inspection. For
destructive testing, the choice is between release and discarding (or recycling or
downgrading). However, again there are cases involving nondestructive testing where the
cost of performing a 100% inspection is more expensive than the potential value of the
lot, so that 100% inspection is not a viable option.
I have attempted to make the case that it is not two distinct situations, destructive
versus nondestructive testing, but really a continuum from inexpensive testing to very
expensive testing. The impact of this is that defects types are frequently
segregated not only by severity (critical, major, minor) but also by cost of testing
(visual, functional) and inspected using different sampling plans. For example major
visuals might be assigned an AQL of 0.25% while major functionals might be assigned an AQL
of 1.0%. Since major visuals and major functionals are of the same consequence but major
visuals are easier to rectify, the AQL is set lower so that one is quicker to reject
them.
More on selecting sampling plans based on economics can be found in Chapter 4 of my book Guide to Acceptance Sampling. One concept, I mention in the
article Classifying Defects and Selecting AQLs posted on
the web site is that of break even quality. The formulas I give there are for the case of
nondestructive testing only. Chapter 4 gives break even qualities for the case when 100%
inspection is not possible.
Back to Acceptance Sampling Questions
How does acceptance sampling apply to ISO 9000
and GMPs?
ISO 9000 requires "The supplier shall establish and maintain documented procedures
for the inspection and testing activities set out in the quality plan ...". The Good
Manufacturing Practices (GMPs) regulations issued by the FDA has a similar requirement in
Subpart H, Section 820.80. The book The FDA and Worldwide Quality Systems Guidebook
for Medical Devices by Kim Trautman compares the two sets of requirements and provides
some guidance.(pages 125137).
My own advice is:
 The emphasis should be on defect prevention including SPC, not inspection.
 Prevention does not eliminate the need for inspection. Most processes produce some
defects and have the potential of failing. Acceptance sampling is a required part of
the quality system as indicated
by ISO 9001 and the GMPs.
 Both SPC and acceptance sampling require the routine inspection of units from the
process. The trick is to use the same data for both purposes. The article "The Importance of Trending Attribute Data" talks
about combining SPC and acceptance sampling for attribute (pass/fail) data.
Acceptance control charts can be used in the case of variables (actual measurements) data.
 Deciding which sampling plan to use is generally based on a risk assessment. The
articles on selecting valid sampling plans, "Statistically
Valid Sampling Plans" and "Selecting
Statistically Valid Sampling Plans", describe the rational I generally use.
 A good tool for performing a risk assessment is an FMEA. This can be used to establish
the entire control plan, of which acceptance sampling is one part. The issue is what
failures could reasonable occur, are severe enough to be of concern, and for which no
other means of detection are in place. In this case, acceptance sampling can be used.
However, more preferable is implementation of a mistake proofing device which either
prevents the defect from being produced or ensures it does not pass by undetected. In this
sense, acceptance sampling is the method of last resort. FMEA's and their
relationship to acceptance sampling is described in "Methods
and Tools for Process Validation".
Back to Acceptance Sampling Questions
What are the differences between MILSTD105E
and ANSI/ASQZ1.4?
ANSI/ASQ Z1.4 (1993) is nearly identical to MilStd105E. There are no
changes in the tables of sampling plans. The only change in the
switching rules is that ANSI Z1.4 makes the use of the limit numbers for
switching to reduced optional. In addition, ANSI/ASQZ1.4 contains additional OC curves called scheme OC
curves that describe the protection provided by the switching procedure during periods of
constant quality. Numerous changes where also made to the explanatory text but which
do not affect any procedures.
Back to Acceptance Sampling Questions
Why was
MILSTD105E and MILSTD414 cancelled?
ANSI/ASC Z1.4 (1993) is nearly identical to MILSTD105E. Likewise ANSI/ASQC
Z1.9 (1993) is very similar to MILSTD414. Both were cancelled to reduce costs through the elimination of duplication. Many other standards were
cancelled as well where nearly equivalent civilian standards existed.
Back to Acceptance Sampling Questions
