Why the Last Digit Is Not the Measurement

Why this matters

A tech copies what the display shows, because copying what the display shows feels like honesty. It is the opposite. The trailing digits on a readout are a property of the display's step size, not evidence about the world, and once they are written on a ticket they become a claim your shop made in writing about how precisely it knows something.

The cost of that claim shows up much later and in a direction nobody expects. Over-report your digits today and you forfeit the ability to say "that difference is inside our noise" a year from now, because your own record already said you could see finer than that. This card is about what to write down, and it is a records practice before it is a maths one.

Where the last digit actually comes from

Three separate things get confused into one, and the display is the least meaningful of the three.

The display step is how finely the readout is divided on the range you selected. It is a design choice, it is free to make small, and it is the only one of the three you can see without reading a document.

The instrument's uncertainty is how far the indicated value may sit from the true value, from the published accuracy statement plus whatever your accessory and conditions add. It is almost always several display steps wide, often dozens.

The number of digits worth reporting follows from the second, never the first. If the value is uncertain by about a unit, every digit to the right of the units place is decoration.

The habit that produces bad records is not carelessness, it is diligence pointed at the wrong target. A tech who transcribes 74.36 believes he is being precise. He is being accurate about the display and wrong about the measurement.

The rule worth adopting

Round a reported value to the place of the first uncertain digit, and report that digit as the last one. Carry two extra digits through any intermediate arithmetic and round exactly once, at the end. Report the uncertainty alongside the value wherever the record will outlive the visit.

The unit of analysis is the reported value on a customer-facing or archived record, not the reading as it appears in your raw log. Two related traps, and both are arithmetic that manufactures digits from nowhere:

• A computed value cannot be more precise than what went into it. Divide 223 by 3 in a spreadsheet and it returns 74.333333. Pasting that into a report claims six-figure knowledge of a number that came from three readings.
• A unit conversion does not add precision. A conversion factor carrying six digits applied to a two-digit reading produces a six-digit answer that is still a two-digit measurement. Round back after converting, to the same place you were entitled to before it.

Three places you do not round. The raw capture log keeps every digit the instrument gave, because that is the record you go back to when a calibration comes back out of tolerance and you need to re-evaluate old work. Intermediate arithmetic keeps its extra digits so rounding does not accumulate through a chain. And where a contract, a specification or a regulator names a reporting format, that format wins over this rule, and you report as specified.

The case: a number quoted back a year later

A shop commissioned a system and recorded a baseline. The instrument showed 74.36 on its display, and 74.36 went onto the commissioning sheet that the customer kept.

The figures here are illustrative, but the shape is the point. Say the instrument's published accuracy at that value worked out to about plus or minus 0.9 units once the accessory term was included. That puts the true value somewhere between roughly 73.5 and 75.3. The units digit itself is uncertain: the band covers 73, 74 and 75. The defensible report was 74 units, plus or minus 1, taken with instrument ID such-and-such at the recorded operating state. Everything to the right of that 4 was display, not measurement.

A year later a consultant re-measured and got 76.11 on a comparable instrument, carrying the same roughly plus or minus 0.9. He wrote to the customer that the value had risen from 74.36 to 76.11, an increase of 1.75 units, or about 2.4 percent of the original reading, and asked what the shop intended to do about the degradation.

Check the difference against the instruments. Each reading carries about 0.9 units, so the worst-case combination on their difference is about 1.8 units. The observed difference is 1.75 units. The true change sits somewhere between about minus 0.05 and plus 3.55 units, which includes zero. Under the reporting rule a sibling card owns, a difference has to clear roughly three times the combined uncertainty before it is a finding, which here would be about 5.4 units. The measured 1.75 is not a third of that. There is no detectable change, and the correct technical answer is one sentence long.

Except the shop cannot comfortably say it. Its own commissioning record reads 74.36. To argue that a 1.75 unit difference is inside the noise, it now has to explain that the two decimal places on its own document were never real, in front of a customer who has been reading that document as a precise baseline for a year. The technical position is sound and the credibility position is bad, and it is bad entirely because of two characters written in a hurry.

Had the sheet read "74 units, plus or minus 1," the consultant's own reading of 76 plus or minus 1 would have been visibly inside the same band, and the conversation would likely never have started. The rounding did not just protect the shop after the fact; it would have prevented the dispute.

The failure mode generalizes. Any number you write to more digits than you can defend becomes a baseline against which future noise reads as change. Commissioning sheets, before-and-after service records, annual inspection logs and warranty submissions are all documents whose whole purpose is to be compared to something later, which makes them exactly the documents where over-reported digits do the most damage.

What to write instead

A reported value that survives a year has four parts, and the number is one of them:

• The value, rounded to its first uncertain digit, with its unit.
• The uncertainty, as a plus-or-minus figure or as the instrument and range it was taken on, so someone can reconstruct it.
• The instrument identity, so a later calibration result can be applied to the reading retrospectively.
• The operating state the reading was taken under, because a value without its condition cannot be compared to anything.

That is four fields and about eight extra seconds. The alternative is a record that is precise, tidy, and indefensible.

The one reading where digits are the wrong question entirely

Establishing that something is de-energized is not a numeric measurement. It is a yes-or-no, and treating it as a number is how people get hurt.

A small non-zero reading on an absence-of-voltage test does not round to dead. A few tenths on the display may be induced or capacitively coupled voltage on an otherwise isolated conductor, or it may be a real source through a high-resistance path, and the display cannot tell you which. Resolve it with an instrument that loads the circuit, such as a low-impedance mode or a listed absence-of-voltage tester, rather than by deciding the number is close enough to zero.

That test sits inside the electrical lockout duty at 29 CFR 1910.333(b)(2) in general industry, or 29 CFR 1926.417 in construction, and the proving sequence is live-dead-live at 120.5 of NFPA 70E-2021, in whichever edition your employer's electrical safety program has adopted: confirm the instrument on a known live source, test the conductor, then confirm the instrument on the known source again, so an instrument that died mid-test cannot report dead. 29 CFR 1910.334(c)(2) requires you to inspect the instrument, leads, cables, probes and connectors for external defects before that use, and the instrument's measurement category and voltage rating under IEC 61010-1, binding through its listing mark, must be at or above the circuit. Where a reading must be taken with the circuit live, 29 CFR 1910.333(a)(1) permits it only where de-energizing is infeasible, with the protective equipment 29 CFR 1910.335(a) requires for the exposure.

The connection back to this card is exact. Everywhere else, over-reporting digits costs you credibility. Here, under-reporting a digit by treating it as zero costs something else, and it is the one place where the answer being non-numeric is the whole point.

How to verify you got this right

Pull the last commissioning sheet or baseline record your shop issued. For each number on it, work out the instrument's uncertainty at that value and ask whether the digits written exceed it. Most shops find every number on the page is over-reported by at least one place, and some by three.

Then run the harder test. Pick the oldest baseline you still rely on and ask what you would say if a customer's consultant produced a new reading a few percent off it. If your answer is "that is within our measurement noise," check whether your own record is written in a way that lets you say so. If it is not, the record is not a baseline, it is a hostage.

References

• Manufacturer published accuracy statements, for the uncertainty that sets how many digits a reading is entitled to
• 29 CFR 1910.333(a)(1) energized-work gate; 29 CFR 1910.333(b)(2) and 29 CFR 1926.417 electrical lockout; 29 CFR 1910.334(c)(2) pre-use inspection of instruments and leads; 29 CFR 1910.335(a) protective equipment
• NFPA 70E-2021, 120.5, absence-of-voltage verification, adopted through an employer electrical safety program or contract; IEC 61010-1 measurement categories, binding through the instrument's listing
• See related: The Difference Between Accuracy and Resolution, which owns the three-to-one reporting rule; What a Reading Actually Is; Commissioning Readings to Record as a Baseline