EC-8.2 Calibration, Traceability and Propagating Error

What this is and why it exists

A reading with no uncertainty attached is not a measurement. It is a number.

This topic is what turns one into the other. It is also the reason later work can ask whether two results actually differ, rather than noticing that they are not identical and stopping there.

It settles the most common piece of misplaced confidence in engineering work, which is copying six figures off a display when the instrument guarantees three.

The vocabulary

• Calibration — comparing an instrument against a better one and recording the difference.
• Traceability — an unbroken record of comparisons leading back to a recognised standard.
• Systematic error — an error that shifts every reading the same way.
• Random error — an error that scatters readings around the true value.
• Uncertainty — the range within which the true value is believed to lie.
• Absolute uncertainty — an uncertainty expressed in the units of the quantity.
• Relative uncertainty — an uncertainty expressed as a fraction of the reading.
• Drift — a slow change in an instrument's response over time.

The mental model

Take calibration first, because the word is used loosely. Calibration compares an instrument against a better one and writes down the difference. It does not repair the instrument and it does not make it correct.

A calibrated instrument can still be the wrong instrument for your measurement. A perfectly calibrated meter with a one-megohm input is still going to load a ten-megohm source and read low.

Traceability is the chain behind that comparison. Your meter was checked against a laboratory reference, which was checked against a national one, which participates in international comparisons. Each link adds a little uncertainty, which is why the chain is kept short and every step of it is recorded.

Now the split that does the work. Random error moves readings around and shrinks when you average. Systematic error moves all of them the same way and averaging does nothing at all.

Taking a hundred readings and averaging improves the random part by a factor of ten. The improvement goes as the square root of the count. A meter reading two per cent high stays two per cent high across all hundred. This is why the two must be found separately and reported separately, and why more readings is not a general answer.

Propagating uncertainty through a calculation follows two rules worth memorising. When quantities are added or subtracted, the absolute uncertainties combine. When they are multiplied or divided, the relative uncertainties combine. Combining means adding in quadrature when the sources are independent, which is the square root of the sum of the squares.

The practical consequence is more useful than the arithmetic. One input almost always dominates. Suppose a resistance is known to one per cent and a voltage to five per cent. The power computed from them is uncertain by a little over five per cent. Improving the resistance measurement buys nothing.

That is where measurement effort should go: find the dominant term and improve that one.

Significant figures follow from the uncertainty and from nothing else. The uncertainty decides how many figures may be written, not the display. If the uncertainty is three millivolts, then a result of 1.234567 volts should be written as 1.235 volts, plus or minus 0.003.

Writing the extra digits is not harmless. It tells a reader that you know something you do not, and the next person to use your number will carry that false precision forward.

Finally, a certificate is conditional. It holds at a stated temperature, after a stated warm-up time, for a stated interval before the next calibration. Outside any of those the figures are an estimate.

Drift is the slow version of the same problem. An instrument that agreed with itself last year may not now, and the only way to know is to check it against something better. This is why a laboratory keeps one reference that is used for nothing except checking the others.

What you should now be able to explain or do

• State what a calibration certificate promises and under what conditions it holds.
• Describe the traceability chain behind a measurement and why each link adds uncertainty.
• Separate a systematic error from a random one, and say what averaging does to each.
• Propagate uncertainty through a sum and through a product, and identify the dominant term.
• Choose the number of figures to report from the uncertainty rather than from the display.
• Explain why drift and warm-up conditions make a certificate conditional.

Check yourself