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determines its spread. For any log countVin a log-normal distribution, a
certain proportion of counts will lie aboveVdetermined by:

ðV
mÞ=sð¼KÞð 11 : 7 Þ

where Kis known as the standardized normal deviate.For example,
when:

K¼0(V¼m) then 50% of values will lie aboveV;

K¼1(V¼mþs) then 16% of values lie aboveV;

K¼1.65 (V¼mþ1.65s) then 5% of values lie aboveV.

Rearranging, we get:

mþKs¼V ð 11 : 8 Þ

IfVis chosen to represent a log count related to a safety or quality limit
andKdetermines the acceptable proportion of samples in excess ofV
then a lot would be acceptable if:

mþKsV ð 11 : 9 Þ

and unacceptable if:

mþKs 4 V ð 11 : 10 Þ

In practiceVis likely to be very close to the logarithm ofM, used in
three-class attribute plans.
Since we do not knowmors, we must use estimates derived from our
testing,x,themeanlogcount,ands, the sample standard deviation.Kis
replaced with a value k 1 , derived from standard Tables, which makes
allowance for our imprecision in estimatingKand chosen to give a desired
lowest probability for rejection of a lot having an unacceptable propor-
tion of counts greater thanV. This gives us the condition for rejection:

xþk 1 s 4 V ð 11 : 11 Þ

Somek 1 values are presented in Table 11.4. If we decrease the desired
minimum probability of rejection for a given proportion exceedingV,i.e.
decrease the stringency of the plan, thenk 1 decreases. Application of a
variables plan for control purposes will give a lower producer’s risk than
the equivalent attributes scheme.
It is possible to apply the variables plan as a guideline to Good
Manufacturing Practice (GMP). In this case the criterion is:

xþk 2 sov ð 11 : 12 Þ

wherek 2 is derived from Tables and gives a certain minimum probability
of acceptance provided less than a certain proportion exceedsv, a lower
limit value characteristic of product produced under conditions of Good
Manufacturing Practice (Table 11.4). The valuevwill be very similar to

408 Controlling the Microbiological Quality of Foods