### Go/no-go decisions, so-called Bernoulli trials

Attribute acceptance sampling makes ‘go/no-no’ decisions about entities. The basis for these decisions are observations on specific items which can be by variable (i.e. quantitatively), but more generally are by any attribute of the entities (which includes even qualitative and ordinal observations).

#### Specific sampling

In specific sampling, the entity is a single item (e.g. one packet)

Incorrect decisions about specific entity conformance in go/no-go trials can arise for test results (by variable or attribute) close to the specification limit on the attribute of interest owing to measurement uncertainty.

There are, basically, three zones where:

- entities are clearly conforming – test value inside region of permissible values,
*R*_{PV} - entities clearly non-conforming – test value outside
*R*_{PV} - entities which might be conforming (or not) – test value within measurement uncertainty of specification limit

If every item in a batch is sampled specifically then an estimate of how many items are non-conforming can be made, allowing for certain risks of incorrect decisions associated with measurement uncertainty on each entity.

#### Limited sampling in global assessment

Only in exceptional cases (e.g. matters of high individual risk) is one interested in specific compliance – in most cases one is content with knowing if lots of items on the average satisfy requirements. In such global sampling, the entity is then typically a batch (e.g. of packets).

However, if one does not have the time or resources to sample all items **globally**, there will be sampling uncertainties. This leads to a distribution*, g _{attrribute}*, of the probability of sampling a certain fraction non-conforming items,

*d*/

*N*,where the maximum likelihood is at the mean fraction non-conforming, , for sample size

_{sample}*N*

_{sample}, and the width of the sampling distribution is in general determined by sampling uncertainties.

In cases where **statistical** sampling uncertainties dominate globally, sampling probabilities follow the Binomial distribution. This is *irrespective* of whether specific assessment is made by variable or more generally by attribute (even when observations are purely qualitative).

The **distribution **of the probability of sampling a certain fraction non-conforming can be plotted against the fraction non-conforming entities:

###### Binomial distribution of fraction non-conforming (statistical sampling of mass of pre-packaged goods)

### Conformity assessment based on samples of finite size. Risks of incorrect decisions when assessing fraction non-conforming entities

- e.g. number of entities in/out of specification <=> compared with specification limit on fraction non-conforming
- estimates need to be made of consequences of lack of representativeness since sampling uncertainty will lead to risks of incorrect decision-making in the vicinity of the specification limit

Over and above the risks of incorrect decisions in specific go/no-go trials, there are additional risks – for both customer and supplier – of incorrect decisions when testing globally against a specification limit for fraction non-conforming entities (*U _{SL,p}* in above figure) arising from sampling uncertainty associated with limited sampling.

Note that, even in cases where the measurements underlying the go/no-go trials are qualitative, the distribution of fraction non-conforming entities due to limited sampling is itself quantitative.

Supplier and customer risks with limited sampling can be estimated in terms of the ‘tail’ in the sampling distribution, *g _{attrribute}*, beyond the specification limit

*U*on fraction non-conforming entities

_{SL,p}*p = d*/

*N*.

_{sample,}Example: **Customer attribute risk**

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### References

Bashkansky E and Gadrich T 2010 ”Some metrological aspects of ordinal measurements”, *Accred Qual Assur* **15**, pp. 331 – 6 doi: 10.1007/s00769-009-0620-x

L R Pendrill 2008, “Operating ‘cost’ characteristics in sampling by variable and attribute” *Accred. Qual. Assur*., **13**, pp. 619-31, DOI: 10.1007/s00769-008-0438-y

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