Measurement Uncertainty

When we measure the length of the table with the folding rule, we first have to look at the length of the folding rule we are measuring with. A folding rule has a specified uncertainty of ±1 mm in its length. So folding rules are between l = 1.999 m and l = 2.001 m long. At first, we do not know how long the folding rule actually is. Nor do we know whether it is too long or too short, which is why we state the interval with “±”.

If we trust the manufacturer that the length of the folding rule lies within its specification, we can only say with certainty that it is at least l = 1.999 m and at most l = 2.001 m long. If we measure the length of the table with this folding rule of unknown length, the measurement result is just as uncertain as the measurement uncertainty – here the length uncertainty – of the measuring instrument.

Of course, we can use a precision measuring instrument to measure how long a folding rule is. But the precision instrument itself is not arbitrarily accurate either. Suppose the precision instrument itself has an uncertainty of ±100 µm; then we know the length of the folding rule afterwards with an uncertainty of ±100 µm. Our folding rule in the example is l = 2.0005 m long. We always state the measurement uncertainty with measurement results. That is why we would give the length of this folding rule as l = 2.0005 m ± 100 µm. Everyone can then understand how certain we are about the statement. The true length of the folding rule could therefore also be 2.0004 m or 2.0006 m.

Because we now know that the folding rule is 0.5 mm too long, we correct this known measurement deviation in every result we determine. So we measure a table and read off, for example, l = 2 m. Then we know that the table is actually 2.0005 m long. And we know that, after this correction, measurement results with the folding rule only have an uncertainty of ±100 µm.

For high-quality measurement technology, the measurement uncertainty of a measurement system is stated in the data sheet.

To prove in a folding-rule factory that the folding rules have a maximum length uncertainty of ±1 mm, a reference system with a measurement uncertainty of at most ±100 µm is necessary. In practice, to prove a measurement uncertainty, a measuring instrument must be about a factor of 10 more accurate than the instrument whose accuracy is to be verified. Of course, you can also use more accurate reference systems; then the proof of your measurement system's uncertainty is even more reliable. The next chapter deals with the reference systems for verifying measurement results.