## DPMO

Question

My question concerns the process performance metric DPMO (defects per million opportunities). I want to use this to quantify a particular supplier’s performance. My question is, is the number of defects referred to in the calculation the number of defects produced by the supplier (in which case it would involve data I don’t have access to), or is it the number of defects experienced by the customer (which is us)? I of course can count the number of defects we receive from the supplier, but if this metric is supposed to be based on the number of defects produced by an organization, I would have no way of knowing how many defects are produced by the supplier’s process, but contained within the supplier’s facility. My hope is to be able to characterize the supplier’s process performance in terms of sigma level.

The DPMO metric is not usually considered a point estimate of the true percent defective in the lot (either at the supplier or customer site).  It is a relative performance metric used to equate the observed percent defective from a sample to defective units per million opportunities.  If a supplier culls out all the defective units before shipping to you (i.e. perfect inspection system), your internal DPMO would be 0, even if the supplier DPMO is high. If your goal is to characterize the supplier’s process performance in terms of sigma level, you would need their data, as the data you collect internally is just an estimate for the average outgoing quality from the supplier and not their process performance.

Steven Walfish

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

## Z1.4 Sample Size

Question

I am trying to determine the sampling size using my ANSI/ASQ Z1.4 table and I wanted to get some clarification. If I am using Table II A and my Sample Size Code letter is D, what would be my sample size? If it falls on an arrow does it mean that I have to change to the next sample size based on where the arrow points?

From Charlie Cianfrani:

If you are using Z1.4, your sample size is selected based on your lot size.  You would pick the AQL you need based on the risk you are willing to take for the process average of percent defective.  It is important to understand what you are doing when using sampling plans, what they are and the protection you are trying to ensure. Thus, the important step is to determine the AQL. Then you select the sample size to provide the level of protection you are striving to ensure. It is more important to understand the theory behind the tables than to mechanically use the tables.

From Fred Schenkelberg:

Use the sample size where the arrow points. In the 2008 and 2013 versions it explains this in section 9.4, “When no sampling plan is available for a given combination of AQL and code letter, the tables direct the user to a different letter. The sample size to be used is given by the new code letter, not by the original letter.”

From Steven Walfish:

The standard sample size for Code Letter D from IIA is a sample size of 8.  But depending on your AQL, a sample size of 8 would be inappropriate, so the standard has arrows to delineate alternative sample sizes to reach the target AQL.  So, you sample size and accept/reject values are changed.  For example, at an AQL of 0.25, you would move down to a sample size of 50, with an accept/reject of 0/1.  If the lot size is less than 50, you would need to do 100% inspection.  In other words, there is no sampling plan that can give an AQL of 0.25 without a minimum sample size of 50.

From James Werner:

Yes.  When using Z1.4 two items need to be known, lot size and the AQL (Acceptance Quality Limit).  You use Table I – Sample size code letters to determine the Sample size code letter based on the Lot or batch size.  In the question below that was determined to be “D”.  Next step is to use Table II-A to find the sample size related to the sample size code letter – D and the AQL.  On Table II-A go across the table’s row for letter D until it intersect the given AQL column heading.  If an arrow is in that intersection point, follow the arrow then go back to the sample size code letter column to find the actual sample size (if a up/down arrow is in there then you choose).

Example 1.  Code letter is D (as in the question below).  Let’s say the AQL is 0.25.  Starting at code letter D, move across that row until you intersect at the AQL 0.25 column.  There’s a down arrow this row/column intersection.  Follow the arrow downward until the “Ac Re” reads ” 0 1″.  Staying on this row go back to the Sample size code letter column and find Code Letter H and Sample size = 50.  This means for the lot size with code letter D and with an AQL of 0.25 the sample size = 50 and accept the entire lot if no nonconformances were found else reject the entire lot if 1 or more nonconformance were found in the sample.

Example 2.  Let’s say the Sample size code letter was determine from Table I to be “F”.  Looking at Table II-A; If the AQL = 0.65, then the sample size would be 20 and the lot would be accepted zero nonconformance.  But if the AQL = 0.15 then the sample size would be 80.

## Six Sigma Statistical Meaning

Question

I need to understand the statement, “Adding a 1.5 sigma shift in the mean results …….”
I’m used to the bell curve and + /- three sigma.
How does the extra +/- three sigma fit in, and what is this about moving the mean?
Does ASQ have a good book that includes this detail in with basic statistics?

The idea of 6-sigma leading to a process with 3.4 parts per million defective is not a totally statistical statement.  Using the normal distribution, we know that a process that is centered on its mean will have 0.135% of the distribution outside 3 standard deviations on each tail.  That same process would have 0.00000010% outside of 6 sigma, which does not lead to the aforementioned 3.4 million parts per million outside.  Dr. Mikal Harry in 1992 published a book (see chapter 6) entitled Six Sigma Producibility Analysis and Process Characterization, written by Mikel J. Harry and J. Ronald Lawson. In it is one of the only tables showing the standard normal distribution table out to a z value of 6.  Here is where he stated that processes can shift by 1.5 sigma leading to only having 4.5 sigma limits and the 3.4 parts per million outside the “6-sigma” limits.  I would suggest you look at ASQ’s Six Sigma Forum Division that will help to better explain the rationale for the shift.

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

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

## Switch from ANSI/ASQ Z1.9 to ANSI/ASQ Z1.4?

Question

Hi,

We are using ANSI Z1.9 for a dimension test of packaging components. As dimension is under variable, can we switch to ANSI Z1.4? The reason for this is to align with our supplier who is using ANSI Z1.4.

Can you please advise if this switching is acceptable. If yes, what should be taken into consideration like AQL, etc.?

The ANSI/ASQ Z1.4 standard is for incoming inspection of attribute characteristics.  As your measurement is a variable measurement, it is appropriate to use ANSI/ASQ Z1.9.  Both plans are indexed by AQL, but have different sample size requirements based on the level of protection you are looking to maintain.  I assume your real question is can you switch from a variable plan (Z1.9) to an attribute plan (Z1.4) for your inspection to align with your supplier’s use of Z1.4.   Though I do not believe harmonizing with the supplier’s use of Z1.4 for your acceptance testing is necessary, it is possible to use Z1.4 by redefining the variable measurements as either good or no-good.  Choosing to move to Z1.4 from Z1.9 will increase your sample size for the same level of protection and same lot size.  For example, a lot size of 5000 would have a sample size of 75 in Z1.4 and 200 for Z1.4 for a General Inspection Level II plan.  Both plans give approximately the same AQL and LTPD, though the Z1.4 will require 2.67x more samples.

Steven Walfish
Chair Z1, U.S. TAG to ISO/TC 69
ASQ CQE
Staff Statistician, BD

## 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.

## Z1.4:2008 Inspection Levels

Q: I am reading ANSI/ASQ Z1.4-2008: Sampling procedures and tables for inspection by attributes, and there is a small section regarding inspection level (clause 9.2). Can I get further explanation of how one would justify that less discrimination is needed?

For example, my lot size is 720 which means, under general inspection level II, the sample size would be 80 (code J). However, we run a variety of tests, including microbial and heavy metal testing. These tests are very costly. We would like to justify that we can abide by level I or even lower if possible. Do you have any advice?

The product is a liquid dietary supplement.

A: Justification of a specific inspection level is the responsibility of the “responsible party.” Rationale for using one of the special levels (S-1, S-2, S-3, S-4) could be based on the cost or time to perform a test. Less discrimination means that the actual Acceptable Quality Level (AQL) on the table underestimates the true AQL, as the sample size has been reduced from the table-suggested sample size (i.e. Table II-A has sample level G of 32 for a lot size of 151 to 280, while General Inspection level I would require Letter E or 13 samples for the same lot size).

Justification of a sampling plan is based on risk and a sampling plan can be justified based on the cost of the test, assuming you are willing to take larger sampling risks. If you use one of the special sampling plans based on the cost of the test, it is helpful to calculate the actual AQL and Limiting Quality (LQ) using the following formulas.

You solve the equation for AQL and LQ for a given sample size (n) and defects allowed (x):

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

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

## 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

## ANOVA for Tailgate Samples

Q: I have a question that is related to comparison studies done on incoming inspections.

My organization has a process for which it receives a “tailgate” sample from a supplier and then compares that data with three samples of the next three shipments to “qualify” them. The reason behind this comparison is to determine if the production process of the vendor has changed significantly from the “tailgate” sample, or if they picked the best of the best for the “tailgate.”

It seems a student’s t-test for comparing two means might be a simple and quick evaluation, but I believe an ANOVA might in order for the various characteristics measured (there are multiple).

Can an expert provide some statistician advice to help me move forward in determining an effective solution?

A: Assuming the data is continuous,  ANOVA (or MANOVA for multiple responses) should be employed. Since the tailgate sample is a control, Dunnett’s multiple comparison test should be used if the p-value from ANOVA is less than 0.05.  If the data is discrete (pass/fail), then comparing the lots would require the use of a chi-square test.

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

## Is C=0 in Z1.4?

Q: I have ANSI/ASQ Z1.4-2008 Sampling Procedures and Tables for Inspection by Attributes. I looked through it rapidly, and I still can’t find the C=0 plan directly, so I am a little confused. I thought C=0 is included in Z1.4. Is the C=0 plan spirit/concept contained in Z1.4 or does C=0 need to be calculated from the several tables in Z1.4? (if yes, which tables?).

A: Z1.4:2008 is a general sampling plan for attributes.  It is tabled by AQL with varying accept reject numbers.  The standard gives a framework for attribute inspection plans. Though Z1.4 does have some plans where C=0, they are NOT optimal to minimize the Type II error. For C=0 plans specifically, I would recommend purchasing Zero Acceptance Number Sampling Plans, Fifth Edition.  The value of the Z1.4 standard is the switching rules used for incoming inspection.

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.

## 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.