Every Number Has a Range

A number without an uncertainty is not a measurement. It is a number. Without a stated range there is no way to tell whether a difference from some other result is meaningful or noise, which is most of what anyone wants to know.

Behind the single word error sit two entirely different problems, and confusing them causes more trouble than almost any other methodological mistake.

Random error is scatter. Read the same quantity repeatedly and the values vary a little in both directions. This is tractable: averaging reduces it, and the improvement goes as the square root of the number of readings, so more data genuinely helps. This is the error that statistics textbooks are about, and it is the one people have in mind when they say a study needs a larger sample.

Systematic error is bias, present in every reading in the same direction. Averaging does nothing to it whatsoever. A miscalibrated instrument produces beautifully repeatable numbers that are all wrong by the same amount, and the tight scatter can actively increase confidence in a wrong result. Nothing internal to the measurement reveals it. That is the whole difficulty: the data looks excellent.

Which is why the serious work in precision measurement goes into cross-checking against methods built on different principles, rather than into taking more readings. Two instruments of the same design share their systematic errors and their agreement confirms the design. Instruments relying on unrelated physics are unlikely to be wrong in the same direction by the same amount.

The 2011 result that appeared to show neutrinos travelling faster than light is the useful worked example. The discrepancy was traced to a loose fibre-optic connector and an oscillator running slightly off. What deserves remembering is how the collaboration handled it: they published the anomaly explicitly asking the community to find the mistake, rather than claiming a discovery. That is the correct response to a result that contradicts a well-tested framework, and it is also a clean illustration of a systematic effect no amount of internal consistency checking could expose.

The failure mode most people meet in daily life is polling. A stated margin of error covers sampling variation only: how much a random sample of that size would be expected to vary. It says nothing about who declined to answer. Non-response is systematic, because people who refuse differ from those who agree in ways correlated with what is being measured, and it has been the larger source of error in most recent polling misses. Weighting helps and cannot fully correct for a difference you cannot observe.

There is a smaller habit worth acquiring, which is reading significant figures as a claim. The digits written down assert how well the quantity is known. A length of about three metres converted into feet does not become known to a thousandth of a foot, however many digits the calculator supplies. Unnecessary precision is not harmless: it is a claim the original measurement cannot support, and it travels onwards into everything built on it.