Short fill: how light can a 454-gram jar be?
In the United States, an inspector checks packaged goods by weighing a sample of them and following NIST Handbook 133. You run the filling line. Set how much your machine puts in each jar and how steady it is, then hand twelve jars to the inspector.
A careless machine can underfill and pass more often than a careful one. Fill jars to an average of 453 g, a gram short: with a spread of 1 g the lot passes about 1 time in 8; with a spread of 4 g it passes about 9 times in 10.
The figures come from simulating the inspection below. Try them yourself.
No jar in the sample may be more than its maximum allowable variation short. For a label of 454 g that is 19.9 g, so the line is 434.1 g. A jar below it is an “unreasonable minus error,” and in a sample of 12, none is allowed.
Table 2-5 sets 19.9 g for labels of more than 426 g up to 489 g. Table 2-1 allows zero unreasonable errors in samples of 24 or fewer.
If the sample’s average is under the label, the shortfall is compared with a margin: the sample’s standard deviation times 0.635. A shortfall smaller than the margin passes.
0.635 is the “sample correction factor” for 12 packages, used for lots of 12 to 250 (Table 2-1). The margin grows with the spread of the jars, and this is the whole story.
Your line gives away −1.0 g per jar.
Not inspected
Chance a lot passes, by average fill and machine spread
Each cell is 400 simulated inspections of 12 jars, run in your browser when the page loads. The stronger the shade, the likelier the lot passes. The ring is your line, when its settings fall on the map. Left of the 454 g rule, the passing region leans toward the noisy machines at the top, until the spread is so wide that single jars start to fall below 434.1 g.
The inspector weighs 12 jars, not the whole lot. Twelve jars from a lot that is honest on average will often average a little under the label by chance, and more so when the jars vary a lot. The margin keeps an honest packer from failing on an unlucky draw; the handbook says it covers the gap between the sample and the lot “with an approximately 95 % level of confidence.”
The law still asks for more. The handbook’s average requirement is that the average contents of a lot “must at least equal” the label. The margin decides only whether one inspection can show the lot falls short. A noisy machine makes that harder to show, and so it buys a packer room the law does not give.
The handbook’s glossary describes the correction factor as the 95th percentile of Student’s t, one-sided, with n − 1 degrees of freedom, divided by √n. For 12 jars that gives 1.796 / √12 = 0.518.
The table prints 0.635, which is the 97.5th percentile: 2.201 / √12. The same holds at every size we checked (24 jars: 2.069 / √24 = 0.422; 48 jars: 2.012 / √48 = 0.290). The table, which is what inspectors use, is the more forgiving of the two to the packer.
Moisture allowances for products that dry out, tare weighing, the Category B plan used in USDA meat and poultry plants, and lots larger than 250 packages (which use samples of 24 or 48). The simulated machine fills with a normal distribution; real lines drift.
- National Institute of Standards and Technology, NIST Handbook 133: Checking the Net Contents of Packaged Goods, 2025 edition. Section 1.2 (Average Requirement), Section 2.3.7 (Evaluation for Compliance), Appendix A Tables 2-1 and 2-5, and the glossary entries for “sample correction factor” and “sample error limit.” A US government work.
- Student’s t percentiles (11, 23 and 47 degrees of freedom) are standard values, found in any t table.