Five matching measurements can still miss the reference

What the national terminology actually distinguishes

NIST’s Technical Note 1297 appendix treats accuracy as agreement with the value being measured, repeatability as agreement among successive results under the same conditions, and reproducibility as agreement under specified changed conditions. It cautions against using precision as another word for accuracy.

A correction compensates for an estimated systematic error; NIST notes that compensation cannot be complete because that error is not perfectly known. The following numbers are invented by Gretna Junction, not readings from a Southside school, workshop, laboratory or household.

No instrument was tested or calibrated.

The exercise is suitable for a discussion in Blackstone, Victoria or Keysville, but those names identify readers, not a study location. We will write down a reference value as part of a mathematical problem.

That privilege belongs to the problem’s author. It does not mean an observer ordinarily knows the exact value of a real object merely by looking at its label.

Keeping the invented reference visible is what lets every subtraction below be checked.

Two lists that answer different questions

Set the invented reference at 100 millimeters. List A contains five readings: 103, 103, 103, 103 and 103 millimeters.

Their sum is 515 and their average is 103. The largest and smallest entries are identical, so the range is zero.

Each entry nevertheless sits three millimeters above the reference. Nothing in that arithmetic is contradictory: agreement within the list and agreement with the reference are different comparisons.

List B contains 98, 99, 100, 101 and 102 millimeters. Its sum is 500 and its average is 100.

Its range is four millimeters, calculated as 102 minus 98. The average matches the invented reference, but four individual entries do not.

A caption that says only that the average was correct would hide the spread; a caption that says only that the results varied would hide the average. Neither abbreviated description contains the whole list.

A correction in a problem is not a calibration certificate

For List A, subtract three from every entry. All five become 100 and the new average is 100.

This works perfectly because the exercise supplied an exact reference and deliberately gave every original entry the same difference. We designed the answer into the numbers.

The operation demonstrates a subtraction, not a discovery about the behavior of a real ruler or digital gauge.

Now invent a second reference of 200 millimeters and a second list of 205, 205, 205, 205 and 205. Reusing the earlier subtraction gives 202 each time, not 200.

The leftover difference is two millimeters. Two examples do not establish a correction rule for every intermediate value, either.

A line drawn between them would be a proposed mathematical relationship; it would not be another observation. The source of the rule and the source of the readings should remain separate.

More entries do not change a repeated offset

Copy List A into a notebook twenty times. There are now one hundred entries, all 103.

Their sum is 10,300 and their average remains 103. The distance between that average and the invented reference remains three.

The larger notebook has changed the number of written entries, not the value written in each one. It would be misleading to advertise the larger count as proof that the difference disappeared.

The same point can be made without pretending that copying is measurement. Define a hypothetical instrument in a computer exercise that always returns the input plus three.

Feed it 100 once, ten times or a thousand times. Every output is 103 by the rule we supplied.

Averaging those outputs cannot undo the addition. That is a property of this constructed rule, not a general calculation of how quickly real experimental uncertainty decreases.

Changing the setup changes the question

Imagine that a second observer uses another fictional device on the same 100-millimeter reference. Its five outputs are 97, 97, 97, 97 and 97.

Each observer’s list has zero range. The averages differ by six millimeters.

Before combining the lists, write down which device supplied which result. Otherwise a pooled average of 100 would make two deliberately offset devices look like a single unremarkable set of measurements.

The pooled list would contain five copies of 103 and five of 97. Its total is 1,000, divided by ten to give 100.

That average conceals the grouping, although the grouping remains obvious in the original entries. Reporting two labeled groups preserves the question that needs investigation.

Erasing the labels first and displaying only the pooled average makes that question harder to see. This example is invented; no actual disagreement between local observers is alleged.

A decision rule needs its own evidence

Add an explicitly arbitrary classroom rule: accept a written number only when it lies from 99 through 101, including both endpoints. None of List A’s original entries qualifies.

Three entries in List B qualify, while its average also qualifies. The answer therefore changes depending on whether the assignment evaluates individual entries or their average.

State that choice before marking the page, rather than changing it after seeing which list would win.

This invented interval is not a construction tolerance, school grading standard, health threshold or purchasing specification. Nor does it include a real uncertainty calculation.

Its purpose is narrower: a claim that something passed requires a stated rule and an identified result to which the rule was applied. A neat average cannot supply an absent contract.

Readers confronting a real measurement dispute need the applicable method and decision criteria, not a borrowed interval from this article.

Keep the comparison attached to the conclusion

A useful final description of List A would retain its five identical entries, its average and its three-millimeter difference from the invented reference. A useful description of List B would retain its average and its four-millimeter range.

Those sentences are less dramatic than declaring a winner, but another reader can reconstruct both results from them. The lists in this article are fictional arithmetic, not test data.

The exercise ends with a boundary, not a certificate. We have compared numbers whose reference and rules were supplied on the page.

We have not established the condition of a product, identified a defective instrument or approved anyone’s work. For an actual claim, the next question is what establishes the reference and how the result was obtained.

An unexplained string of matching digits answers neither question on its own.