A Measurement with a Biography

A week later the part comes back from the laboratory. Attached is a report: diameter 9.974 mm, standard uncertainty 0.004 mm. Dana opens her logbook. Her five-reading average is 10.0 mm, with a spread-based uncertainty of about 0.07 mm. The laboratory number looks more convincing: more digits after the point, and a smaller uncertainty.

“Copy it over,” says Dana. “Theirs is more precise.”

“How do they know?” asks Timur.

The question sounds like nitpicking. The laboratory has better equipment, and its people measure every day. Dana sent the part there precisely to get a more reliable result. Now it lies before her.

Before reading on, decide: would you replace the logbook entry with the report’s number?

A number with more digits

Two more zeroes can be appended to the entry “10.0”. That gives “10.000”, though we learn nothing new about the part. The digit count tells how a result is written. Judging the measurement itself takes other information.

In the last chapter we evaluated the uncertainty of an average from the spread of five readings. The laboratory report states an uncertainty too, but it claims to cover every component that matters for this measurement. Setting it against a spread-only estimate and declaring victory in precision is premature.

Timur suggests reading the whole report. What exactly was measured? How did they arrive at 9.974? What stands behind the 0.004 estimate? These are the same questions about grounds we asked in the prologue. Only now the answer rests on the work of people far from us.

What exactly was measured

The first question begs a quick answer: the diameter, of course. But we already noticed that the word may not be enough. If the part’s cross-section is slightly oval, the size depends on the direction of measurement. If the part tapers toward the end, the place matters too. Two people can carefully measure one part and get different numbers, because they measured different sizes.

So the measurand needs a description detailed enough: which size of the part, in which section and direction, we are after. Then we can tell whether the report’s number refers to what Dana measured, and whether it will help settle the hole question.

Conditions matter too. When heated, a steel part expands. Geometric sizes are customarily stated at the standard reference temperature of 20 °C. Measuring may be done at another temperature, as long as its influence is accounted for. So the report must distinguish the temperature they measured at from the one the result was referred to. How much the difference matters depends on the conditions and the uncertainty required. The bare entry “9.974 mm” alone cannot tell us that.

How a reading becomes a result

The next question is about the instrument. How is it known the way its readings relate to the part’s size? That takes calibration.

Imagine the instrument measuring the length of a reference standard. Its value is known with a stated uncertainty. It is compared against the instrument’s readings, allowing for the uncertainty of both sets of data. Then, from the established relation, they work out how to get a result from a reading — what correction to apply, for example. That is a fair picture of calibration in two stages.

The instrument itself may stay exactly as it was. Calibration reveals its characteristics so they can be allowed for in measurement. It does not by itself remove a bias of the readings, nor make the scale finer. Calibrate Dana’s caliper, and there will be data for corrections and uncertainty evaluation. The division value will not change from that.

And how is the standard’s value known, the one the instrument was compared against? It too came from a calibration, resting on another standard. From a laboratory measurement we can thus walk back to the original support. For length, such a chain ties a result to the definition of the metre through the speed of light in vacuum and the second. From that definition the unit is realised in practice. Through standards and calibrations its size passes on to the instruments that measure parts.

Every stage has its own uncertainty. Its contribution must be counted into the final result together with the influence of the measurement’s own conditions. All these quantities combine by the rules of uncertainty calculation; plainly adding up the numbers from the certificates will not do here.

A chain you can present

When a result is tied to a stated reference through a documented, unbroken chain of calibrations, that is called metrological traceability. In such a chain every link’s contribution to uncertainty must be accounted for. Traceability belongs to a specific result: the whole path tying it to the reference must be shown.

So the phrase “the instrument is certified” is not enough. We need to know what the certificate attests, which standards were used, and how our result connects to them. The document itself may hold the needed information and references. Their content matters more than the paper’s mere existence. We must also pin down what the result is traceable to. A tie to a particular specimen does not yet mean a tie to a unit of the International System of Units — the SI.

Now Timur’s question sharpens. Which documents does the laboratory rest on? What do they confirm? How were the values obtained that its instrument’s readings were compared against? We need not repeat every calibration in person, but we must understand what exactly we entrust to the specialists, and on what grounds.

A familiar image helps here: a family tree. In the Kazakh tradition of jeti ata seven generations of forefathers down the father’s line are remembered by custom. Names are joined by kinship ties: through them a person understands his place in the family’s history. In a measurement’s history we are after ties too. A calibration certificate’s number becomes useful once it is clear which standard and which comparison it links to. Behind every link of such a chain stand calibration records and calculations, by which the tie between measurements can be checked.

Three different questions

Dana wanted to simply copy the number over. Now she faces three questions to work through.

First — is the laboratory result traceable to the claimed reference? That depends on whether the calibration chain is confirmed and its links counted into the uncertainty evaluation. If yes, we know which measurements the result stands on. That does not yet rule out mistakes in handling the part itself.

Second — is this result enough for our task? The part must enter the hole. That takes the hole’s dimensions, the part’s allowed sizes, and the rule by which fitness is decided allowing for uncertainty. Even the small 0.004 mm will not settle the task if the wrong size was measured or the result sits too close to the allowed limit. Traceability helps justify a result; fitness shows whether we can use it for the intended purpose.

Third — do the two measurements agree? The gap between 10.0 and 9.974 is 0.026 mm. It is smaller than even the uncertainty Dana evaluated from the spread of readings. But for a verdict of compatibility that comparison is not enough. We must make sure one and the same quantity was measured, evaluate the uncertainty of the difference allowing for both results, and pick a comparison criterion. Shared sources of uncertainty may move the answer too.

Dana’s evaluation is still incomplete, so declaring the results compatible is premature. The bare difference of two numbers proves no one wrong either. We have yet to find out how large the disagreement is against what these measurements can be expected to show.

Dana keeps her readings in the logbook and attaches the report. Beside them she writes down what information still needs pinning down. That preserves the history of both measurements, and later it will show why a particular result was picked for the job.

A question for the laboratory

It helps to start with one question: which information in the report refers to our part and the conditions of its measurement?

The instrument’s calibration certificate may have been issued a year ago. It holds information obtained back then. For today’s result we also need to know how the instrument was used, which corrections were applied, which part of the detail was measured, and whether the instrument’s needed characteristics still held. All of that ties the old calibration to today’s number.

We started with two entries and wanted to pick the one that looks more precise. Now the laboratory result grows a biography: what was measured, how it was done, and which earlier measurements it rests on. From it we see more clearly what can be trusted and what still needs clarifying. That is how someone else’s work becomes a ground of knowledge we can share.

In the next chapter steel gives way to peas on the table, and the notebook holds a table of possible trait combinations. Each combination has its share. The peas, however, never read the table.