## Determining Statistically Significant Sample Size

Question

I am developing an internal audit process within our supply chain to determine packaging and Finalizing SOP’s are being followed. I need to determine what will be the sample size needed to accurately represent the population. We are currently shipping out 650k cartons a day. How do I determine how many audits I need a day for statistical significance?

Statistical sampling theory shows that for large populations, the sample size is not a function of the population size, assuming all units in the population have an equal probability of being selected for the sample.  To ensure a representative sample, stratified random sampling is employed to represent in the audit sample. This method requires that each category (or stratum) is specified, and that none of them overlap (i.e., items to be audited must fall in only one category).  For example, you can break the packaging records in groups of 25,000 (26 stratum for 650,000 records), sampling 1/26th of the sample from each stratum.

To determine the sample size, we employ the binomial distribution where a records is either confirming or nonconforming.

The basic formula for the binomial confidence interval is

For a given sample size (n) with a given number of defects (x), the probability of the sample coming from a population with probability (p) is given by the value alpha (a).  The above equation can be solved for probability (p) at a given a level or can be solved for a at a given population probability (p).

In other words, you specify the percent defective in the population you can accept.  The only when to ensure 0% defective is 100% sampling. You solve the equation for n by setting 1-alpha (1-a) equal to a high probability (i.e. 95%).  If you desire to accept zero (0) defects in the sample then set x equal to zero. In this case, the equation reduces to ln(1- a)/ln(1-p).

Hope this helps with the question.

Thanks

Steven

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

## Question

Regarding part of your answer to a post found here, you state:

“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.”

Can you explain for the non-statistical folks like me people how that math works? Specifically, I am wondering what the minimum sample size would be for an AQL of 0.25, when using Special Inspection level S2? Would it be a minimum of 50, no mater what the lot size is?

Acceptance sampling procedures were developed during the early 1920s at Western Electric Company and later formalized at Bell Telephone Laboratories where terms like producer’s risk and consumer’s risk were established.  Later, during World War II, sampling plans such as MIL-STD-105 were developed by Harold F. Dodge and others working with the Army Quartermaster Corps (Dodge, 1967).

Two special features were employed in order to gain agreement with the large body of military suppliers.  One was the use of the acceptable quality limit (AQL) as opposed to the RQL in presenting the plans.  The goal at the time was to focus on rewarding suppliers for production whose quality levels were considered good.  RQLs were recognized but not often brought to the surface during discussions. Also, at that time, the term “AQL” was deliberately vague or inexact.  It was a close approximation, not an exact probability statement.

The other feature was the practice of increasing sample sizes with increased lot sizes.  As noted in Section 3, in most situations, the lot size does not factor in plan construction (based on the binomial).  For many, however, this lacks intuitive appeal.  Therefore, in the development of MIL-STD-105 and its derivatives a deliberate increase in sample sizes for higher lot sizes was introduced, with corresponding increases in acceptance numbers for similar AQLs.  Clearly, this practice resulted in over-sampling and consequent increased inspection costs.  Government operatives believed that the increased sampling cost was of small consequence relative to the power to persuade.

For the binomial distribution you solve for the AQL that gives a high probability of passing.  Usually this probability is set at 95%.  For example if you have a sample size of 80 units with an accept/reject of 1, an AQL of 0.65% would have a 90% probability of passing the sampling plan.

You can use Excel to solve this with the function

=BINOMDIST(1,80,0.0065,1)

Hope this helps,

Steven Walfish

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

## Sampling Plan Review?

Question

When following ANSI/ASQ Z1.4-2003 (R2018), if a product has been placed in a “reduced” sampling plan based on the previous 10 lots results, is it a requirement to convert back to a “normal” sampling plan on an annual basis, or should that decision remain based on supplier performance? I have been told that we should revert to normal sampling each year, but I do not see that in the AQL inspection manual.

The standard does not require annual (or periodic) review of the sampling plan.  The switching rules are time invariant, and reflects just the normal flow of lots, which can span more than a year.  Unless the supplier requires a change in the inspection level, the standard is silent on resetting to the normal level annually.

Steven Walfish

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

## Z 1.4 AQL Levels

Question

I need help understanding the AQL values in the tables of ASQ Z1.4. They are defined in paragraph 4.5 as percentages or ratios, but there are some values that are less than 1 and greater than 100. How should these values be interpreted?  Since this standard is for attribute data, is there a standard for variable data?

A percentage can be from 0 to more than 100% depending on what the ratio represents.  First we need to define AQL.  Section 4.2 states “The AQL is the quality level that is the worst tolerable process average when a continuing series of lots is submitted for acceptance sampling.”  Therefore, an AQL of 0.65% means that on average we can accept 65 defects per 10,000 units in a lot.  The sampling plans with percentages greater than 100% are carried over from the MIL-STD-105 and are considered to be antiquated and not used any longer.

The ANSI standard for variable data sampling plans is ANSI/ASQ Z1.9.  It is based on probability of being outside the acceptance region.

Steven Walfish

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

## Z 1.4 Inspection Levels

Question

I am using a reduced switching rule and I don’t understand the meaning of the numbers in the first box. Total noncomforming less than limit number? What’s my limit number?Does production stability mean capability? Would I use 1.33?  The table has an arrow to reduced, so would I move to the next box?

The ANSI/ASQ Z1.4 standard has three inspection levels: normal, reduced and tightened inspection.  Initially you start at normal inspection, and can move to either tightened or reduced inspection depending on how lots are dispositioned.  Based on Figure 1 of the standard, the determination to move amongst the levels can be ascertained.  When you get to the reduced inspection level (Table II-C), you need to read the footnote (†).  It states “If the acceptance number has been exceeded, but the rejection number has not been reached, accept the lot, but reinstate normal inspection.”

A stable process or production is less about a capability index, and more about the control chart of the data showing a stable process.  In other words, the process is stable over time.

Steven Walfish

## Confidence Levels

Question

I would like to confirm if ASQ Z1.4-2008 attribute tables are calculated based on 95% confidence level? I am using Table II-A, on page 11.

ANSI/ASQ Z1.4 tables are not technically calculated based on a 95% confidence level.  The technical definition of AQL is the quality level that is the worst tolerable process average when a continuing series of lots is submitted for acceptance sampling.  Some interpret it to mean if a lot has AQL percent defective or less, a lot would have a high probability of being accepted based on the sampling plan.  The standard does not specify the probability of acceptance explicitly.  The operating characteristic curve (OC Curve and the tables define the AQL as the percent defective that has a 95% probability of acceptance.  So though it is not a 95% confidence level, it is a 95% probability of acceptance.

Steven Walfish

## Zero Acceptance Number Sampling Plans

Question

Regarding Nicholas Squeglia’s Zero Acceptance Number Sampling Plans, in the 4th edition for lot size 151-280 (1% AQL), a sample size of 20 is provided.  However, in the 5th edition, for the same lot size 151-280 and AQL of 1%, the sample size is 29. Which is correct – a sample size of 20 or 29?

In the 5th edition of Nicolas Squeglia’s book, he mentions on page xii the rationale of the change in sample sizes.  From  the 5th edition, “in the early 2000’s, a large aerospace manufacturer was given permission by ASQ to reproduce the c=0 sampling table.  They modified the table by changing several sample sizes, and for convenience it was therefore originally decided to carry those modifications into the fifth edition.”

Table 1a is the original tables (4th edition and previous) which has the sample size of 29.  Use this table unless otherwise specified by contract.

Table 1b is the modified table which has a sample size of 20.

Thanks

Steven Walfish

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

## Sampling Foils, Films, and Labels

Question

My question is about sampling aluminium foils, films used in packaging and sticker labels received in rolls which are wound around a core. I can decide to chose the number of rolls to sample from using the tables given in Z1.4, but how should I decide on the amount of stickers and aluminium foil and film to be sampled? I ask this question since it is practically impossible to sample from within a wound roll.

The ANSI Z1.4 and Z1.9 standards might be applicable when all units do not have the same probability of being selected.  Since you cannot sample units closer to the core, and defects would never be detected unless they occur at the end of the roll, I would recommend a different strategy, either using a vision system (100% inspection) or in process inspection.

If you want to use the standard, the sample size should be based on the number of samples, not the number of rolls.  For example, a roll with 5000 labels would be an N=5000 not N=1.

Steven Walfish

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

## ANSI Z1.4 Reduced Inspection

Question

If you have Ac=0 and Re=2 what do you do for 1? I have not used the reduced sampling before, so am curious what should be done in this instance.

If you review the footnotes for Table II-C of ANSI Z1.4, you will see that there is a note (†) that states: If the acceptance number has been exceeded , but the rejection number has not been reached, accept the lot, but reinstate normal inspection (see 10.1.4).  So in your case, with a single reject, you would accept and reinstate normal inspection.

Steven Walfish

## AQL Clarifications

Question

I am confused about the values used for AQLs. For example in Table II-A the AQL values range from 0.010 to 1000. Where do these values come from and what do they mean?

The table states, “AQLs, in Percent Nonconforming Items and Nonconformities per 100 Items .” At first I thought the values were percentages, but how can you have more than 100, as in 100%, as the values go up to 1000? Also how can there be more than 100 nonconformities per 100 items, unless one part can have multiple nonconformities?

Just looking for clarification on the AQL numbers, what they mean, and how to interpret them.