# Microbiological Sampling Plans

### Microbiological sampling plans: a tool to explore ICMSF recommendations (some explanation)

##### Two-class sampling plans (2-class enrichments and 2-class counts)

The sheet has several components containing graphs, input/output fields, and calculation results. For the normal user only the graphs and input/output fields shown in lines 1 to 37 and columns A to U should be of interest.

Input fields are shown in yellow. In the following text information that can be entered in such fields is written in italics.

**Graphs** for two-class sampling plans (from left to right):

**Plot 1** – Common type of operating characteristic (OC) curve relating probability of accepting a lot to the proportion defective it contains.

Acceptance probabilities are calculated for given number of sampling units that are examined, *n*, given microbiological limit specified by the sampling plan, *m*, and given maximum number of sampling units that are allowed to exceed the limit, *c*.

**Plot 2** – Normal frequency distribution assumed for Log-transformed colony count numbers per gram to be found in sampling units drawn randomly from a lot characterized by given *mean* Log count per gram and given standard deviation *sigma*.

The pink vertical line indicates the microbiological limit, *m*, specified in the sampling plan. Right to *m*the area under the curve corresponds to the proportion defective the lot contains.

**Plot 3** – Using these proportions defective and the given sampling plan specifications for *n* and *c*corresponding probabilities of lot acceptance are calculated. Results are plotted (black) to show the OC curve in relation to Log arithmetic mean counts per gram.

**Input fields** for two-class sampling plans:

**Yellow fields in the center** –

- Lot characteristics:
*mean*Log count per gram and standard deviation*sigma*, shown in Plot 2. - Sampling plan specifications: the microbiological limit,
*m*, (and for enrichments the sample amount), the number of sampling units,*n*, and the number of sampling units that are allowed to exceed the limit,*c*.

To change any of these values go to that field, type in a new value, and press *Enter*. Calculation results, like *P(accept)*, and graphs will be changed accordingly.

- The desired acceptance probability
*P(accept).*

*The desired **P(accept)* can be changed for example to 5%, and by clicking the upper grey box the mean Log count per gram accepted with this new probability *P(accept)* is calculated and graphs are changed accordingly.

**Yellow field on the left **–

To calculate the acceptance probability P(accept) for a specific proportion defective type in this proportion as Pd and press *Enter*. The result will be shown in the next field to the right.

For this calculation only the given values for *n* and *c* are needed (see Plot 1).

**Yellow fields on the right** –

Only sampling plan specifications (*n* and *c*) can be manipulated here, lot characteristics remain unchanged. When new values for *n* and/or *c* are entered the corresponding acceptance probability *P(accept)* can be calculated using the microbiological limit *m* as given. When the grey box on the right side is clicked a new value for *m* is derived that yields the same combination of *mean* Log count per gram and lot acceptance probability *P(accept)* that is achieved by the sampling plan described in the center.

Furthermore the upper percentile of the concentration distribution can be determined (implied acceptance level).

##### Three-class sampling plans (3class)

**Graphs** for three-class sampling plans (from left to right):

**Plot 1** – Operating characteristic (OC) surface showing probabilities of accepting a lot depending on two proportions: the proportion defective in the lot exceeding the microbiological limit *M*, named Pd, and the proportion marginally defective between the two microbiological limits *m* and *M*, named Pm.

Acceptance probabilities are calculated for given number of sampling units that are examined, *n*, given microbiological limits *m* and *M*, and given maximum number of sampling units that are allowed to be marginally defective, *c*, i.e. that are allowed to exceed the limit *m* but not *M*. The number of sampling units that are allowed to exceed *M* is assumed to be zero.

**Plot 2** – Normal frequency distribution assumed for Log-transformed colony count numbers per gram to be found in sampling units drawn randomly from a lot characterized by given *mean* Log count per gram and given standard deviation *sigma*.

The red vertical line indicates the microbiological limit *M* specified in the sampling plan. Right to *M*the area under the curve corresponds to the proportion defective the lot contains, Pd. The pink vertical line indicates the microbiological limit *m* specified in the sampling plan. The area under the curve between *m* and *M* corresponds to the proportion marginally defective in the lot, Pm.

**Plot 3** – Using these proportions defective, these proportions marginally defective, and the given sampling plan specifications for *n* and *c* corresponding probabilities of lot acceptance are calculated. Results are plotted (black) to show the OC curve in relation to Log arithmetic mean counts per gram.

**Input fields** for two-class sampling plans:

**Yellow fields in the center **–

- Lot characteristics:
*mean*Log count per gram and standard deviation*sigma*shown in Plot 2. - Sampling plan specifications: the microbiological limits,
*m*and*M*, the number of sampling units,*n*, and the number of sampling units that are allowed to be marginally defective,*c*, i.e. that are allowed to exceed the limit*m*but not to exceed*M*.

To change any of these values go to that field, type in a new value, and press Enter. Calculation results, like P(accept), and graphs will be changed accordingly.

- The desired acceptance probability
*P(accept)*

The* *desired* **P(accept)* can be changed to for example 5% and by clicking the upper grey box the *mean* Log count per gram accepted with this new probability *P(accept)* is calculated and graphs are changed accordingly.

**Yellow fields on the left** –

To calculate the acceptance probability P(accept) for a specific combination of proportion defective, i.e. exceeding *M*, and proportion marginally defective, i.e. between *m* and *M*, type in these proportion as *Pd* and *Pm*, respectively, and press *Enter*. The result will be shown in the next field to the right.

For this calculation only the given values for *n* and *c* are needed (see Plot 1).

**Yellow fields on the right** –

Only sampling plan specifications (*n* and *m*) can be manipulated here, lot characteristics remain unchanged. When new values for *n* and/or *c* are entered the corresponding acceptance probability *P(accept)* can be calculated using the microbiological limits *m* and *M* as given. When the grey box on the right side is clicked a new value for *m* is derived that yields the same combination of *mean*Log count per gram and lot acceptance probability *P(accept)* that is achieved by the sampling plan described in the center. The microbiological limit *M* is left unchanged (as long as *M*>*m*).

Furthermore the upper percentile of the concentration distribution can be determined (implied acceptance level).