Food Protection Trends - July/August 2017 - 274

actual measurements or observations of CLs of times and/
or temperatures need to be recorded. For the variable
sampling plan, the actual core temperatures observed
must be recorded for each fish in the sample to estimate
the mean (µ) and standard deviation (σ) for all the fish in
the precook batch.
Normal distribution parameter estimators
A computer can be used by the fish preparation personnel
to estimate µ and σ from the sample mean
, as well as
the sample standard deviation(s), thereby allowing for the
determination of acceptance of the batch.
The use of the range/d2 to estimate the standard deviation,
σ, was first suggested by Tippett (43) in 1925. An excellent
explanation of the mathematical derivation and the statistical
validity is detailed by Luko (31). A list of d2 factors for n = 2
thru 100 was published in Grant and Leavenworth (21), and
selected values are listed in Table 2 (7).
The median
can be used as an estimator of the mean
in a normally-distributed population, with n > 25, which
has been shown to be adequate by Hozo et al. (25). With
these two pieces of knowledge, median
and range/d2,
decisions can be made (e.g., determining whether the fish
have been cooked sufficiently), using statistical methods
when those techniques are needed. If the estimates are done
manually, i.e., without using a calculator, the median and
range/d2 can be used to estimate µ and σ (2). The median
and range/d2 is offered so that a factory without a computer
could still use this technique.
The range of core temperatures will impact how close the
lowest measured temperature will be to the value calculated
to be three standard deviations (range/d2) below the mean
or median. For a fixed sample size, the standard deviation
will increase with the size of the range. It is beneficial to the
factory operators to minimize the range of the precooking
core temperatures. The goal is to make the actual minimum
core temperature as close as possible to the calculated lower
limit of the core temperature.
Tests for normality
To use the variable sampling plan, processors will need
to verify that a normal distribution provides a good model for the distribution of the actual core temperatures
being measured. There are good discussions of testing
for normality or near-normality in Geary (20) and Hart
and Hart (24). Two statistical tests for normality that
work well for grouped data are the Ryan-Joiner test (38)
and the Shapiro-Wilk test (40); these are similar tests
(6). The data can also be easily analyzed and plotted
in Minitab © or another statistical commercial software.
The Shapiro-Wilk test is available online (10). The Ryan-Joiner test is available in Minitab© and can be easily
calculated with a computer spreadsheet with a statistical
package installed. The spreadsheet application (to be

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Food Protection Trends July/August

discussed in a future paper) uses the Ryan-Joiner method
for ascertaining normality.
METHODS AND MATERIALS
Sample size determination
The Minitab© statistical software system was used to
develop the sample sizes for attribute and variable sampling
plans, depending on confidence limits and reliability or
percent acceptable. The sample size, with sampling by
attribute, can change depending on the size of the population
being sampled. If the approximate number of pieces in the
lot is known, the attribute plan will have fewer samples
than it would have if an infinite lot size was assumed. A
detailed example of how the sampling sizes were developed
for a variable acceptance plan using the sample median and
standard deviation is presented in Appendix B.
Testing core temperature data for normality
To test if the core temperature data are normally
distributed, two completely different data sets were
collected from the same tuna processing factory and
tested for normality. The first data set was collected from
four different size groups of fish, but with different core
temperature distributions (ranges) that were standardized
and combined into one data set for analysis by use of a
Minitab normality plot. The second data set was provided
by one of the authors (30), who collected precooking exit
core temperatures from 24 fish, each from 296 precooking
cycles, covering nine different fish size groups. Each 24fish sample had a histogram charted for inspection and
was tested for normality with a Minitab normality plot.
The Ryan-Joiner statistic was calculated and recorded for
all the data sets.
Testing attribute data for normality
No normality tests were required or conducted: the
test used is a simple pass/fail test of whether or not the
temperature passes the minimum target temperature.
Modeling how the lower limit of the tolerance interval
changes based on a fixed minimum core temperature
and varying ranges of core temperatures
The range in core temperatures will impact the
lower end of the tolerance limit or percent acceptable.
This approach was modeled by fixing the lowest core
temperature and varying the range. The minimum
measured core temperature of 60°C was chosen and the
temperature ranges were varied in increments of 5°C, so
Max/Min of 60°C/60°C, Max/ Min of 65°/60°C, Max/
Min of 70°/60°C and so forth were modeled. Using that
information, we could estimate a median (midpoint of
an ordered data values) and estimate the batch standard
deviation, using range/d2. We used an n of 60 with a d2
value of 4.638 from Table 2.

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