SOP: Adjusting the Process Mean to Specification Limits

GIFSQ — Global Institute for Food Safety & Quality

Principle

To achieve complete freedom from rejection on drained weight, the mean fill must sit at least three standard deviations above the lower control limit. The area table under the normal curve converts a distance in standard deviations (sigma) into the expected percentage of cans falling below the minimum — so the fill target can be set from the minimum requirement plus the measured spread of the process.

Procedure

Compute z = (average fill − minimum required fill)/σ

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Read the area for z from the normal-curve area table

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Percentage of cans above the minimum = 50 + area

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Percentage of cans below the minimum = 100 − (50 + area)

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Set the target mean = minimum required fill + 3σ

Calculation

z = (x̄ − L_min)/σ

% above minimum = 50 + A

% below minimum = 100 − (50 + A)

target mean = L_min + 3σ

Where,

• x̄ = average fill (g)
• L_min = minimum required fill (g)
• σ = process standard deviation (g)
• z = number of standard deviations the average sits above the minimum
• A = area under the normal curve between the mean and z, in per cent, read from the normal-curve table

Worked example: minimum required fill 540 g, average fill 546.42 g, σ = 4.31 g.

Target mean = 540 + 3 × 4.31 = 552.93 g, rounded to 553 g.

z = (546.42 − 540)/4.31 = 1.49 sigma above the minimum.

From the normal-curve table, 1.49 sigma covers 43.19% of the distribution. Adding this to the 50% of the distribution above the mean, 93.19% of the cans may be expected to be above the minimum fill requirement, and 100 − 93.19 = 6.81% to be underfilled.

Note (required flag 1): the book uses σ = 4.31 in this example while the preceding revision gave σ = 3.096 — carried as printed, flagged.

Note (required flag 2): the book prints the underfilled percentage as “6.18%”, but 100 − 93.19 = 6.81 — carried as printed, flagged.

Precautions

• These limits must not be mistaken for specification requirements — they position the process mean, not the permissible range for individual cans.
• Keep the mean line of the filling-weight chart at least three standard deviations above the lower control limit; only 6.42 g separated the old mean (546.42 g) from the 540 g minimum.

References

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

Enrick, N.L., Quality Control and Reliability, Industrial Press, New York, 1977 — cited in the book.

Besterfield, D.H., Quality Control — A Practical Approach, Prentice-Hall, New Jersey, 1979 — cited in the book.

Kramer, A. & B.A. Twigg, Fundamentals of Quality Control for the Food Industry, Vol. 1, AVI Publishing Co., Westport, 1970 — cited in the book.