SOP: Fitting the Poisson Distribution to Rot Counts and Testing Goodness of Fit

GIFSQ — Global Institute for Food Safety & Quality

Principle

When the sample size of a lot is very large and the chance of an event occurring is very small — for example, rot counts or insect fragments in spite of good manufacturing practice — the binomial distribution tends towards the Poisson distribution. Fitting the observed counts to the Poisson distribution and testing the goodness of fit with the chi-square test shows whether the samples were drawn at random.

Materials

• The observed counts per sample (e.g. rot counts per gram) for N samples
• The book’s cumulative Poisson probability table for finding expected frequencies

Procedure

Count the number of samples showing 0, 1, 2, … occurrences of the event and record the observed frequencies

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Compute the mean count per sample m using the expression below

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Compute the probability of each count x from the Poisson expression below and multiply by the number of samples N to obtain the expected frequency for each count

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Compare the observed and expected frequencies; where the expected frequency of a class is less than 5, combine neighbouring classes

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Compute the chi-square value from the observed and expected frequencies using the expression below

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Refer the chi-square value to the chi-square table with the appropriate degrees of freedom and judge whether the observed data follow the Poisson distribution

Calculation

Mean count:

m = Σfx / N

Poisson probability:

P(x) = e⁻ᵐ × mˣ / x!

Mean and standard deviation of the Poisson distribution:

μ = m

σ = √m

Chi-square goodness of fit:

χ² = Σ[(O − E)² / E]

Where,

• m = mean count per sample
• f = number of samples showing a given count
• x = observed count
• N = number of samples
• e = 2.71828
• O = observed frequency
• E = expected frequency

Worked example

Rot counts per gram in 50 samples of a product gave a mean of:

m = (sum of f × x)/50 = 1.02

The expected frequencies computed from P(x) = e⁻¹·⁰² × 1.02ˣ/x! tally closely with the observed frequencies, suggesting that the samples were drawn at random, so the mean rot count of 1.02 can be accepted with confidence.

To test the hypothesis, the chi-square test is applied. Two constants — the frequency and the mean — have been fitted, giving 1 degree of freedom:

χ² = 0.345

The chi-square value of 0.345 is not significant; the actual significance level lies between p = 0.70 and p = 0.50, which confirms that the observed data reasonably follow a Poisson distribution.

Note: the book’s Table 33-6 carrying the class-by-class expected frequencies is illegible in the available copy; the mean (1.02), the chi-square value (0.345) and the conclusion are carried as printed.

Precautions

• When the values of m are small, the distribution slants towards one side or the other of the mean; established tables must then be used.
• When the values of m are large, the Poisson distribution approaches the normal distribution, and in most cases the results lie within twice the standard deviation from the mean.
• Expected frequencies should not be less than 5; combine classes where necessary before applying the chi-square test.

References

Ranganna, S., Handbook of Analysis and Quality Control for Fruit and Vegetable Products, 2nd edition.