So far in Uncertainty Wednesday we have seen two different limits on observations: the foundational limit that all observation necessarily entails compression of reality and the further compression resulting from the resolution of measurement. Today we will look at measurement error as a limit on observation. Thanks to a tweet from a reader, I have the perfect opening example for today.

The following is from a show call 99% Invisible, which had an episode called “On Average” that has the following in it:

The modern use of averages was pioneered by a Belgian mathematician and astronomer named Adolphe Quetelet. In the 1830s, astronomers were some of the only people that regularly calculated averages, since early telescopes were extremely imprecise. To obtain more accurate data for say, tracking the orbits of planets, astronomers would take multiple measurements (all of which were slightly different) add them together, then divide by the number of observations to get a better approximation of the true value. Quetelet was the first to take this tool of astronomers and apply it to people.

To put this in the framework we have been using, the stars out there are the reality. We use telescopes to get observations of that reality. The telescopes are our measurement instruments. And all measurement instruments introduce some degree of measurement error.

So what exactly is measurement error? The critical sentence above is “astronomers would take multiple measurements (all of which were slightly different)” Reality here is unchanged. It is not the position of the stars that has changed. It is something in the instrument we are using (or in the way we are reading results from that instrument) that gives us a different result.

Where does measurement error come from? Well every instrument is a series of mechanical, optical, and/or electronic components. Each of those components is not some idealized immutable mathematical concept but rather a physical object that is subject to changes such as thermal expansion (parts are bigger when they are warmer), tolerance (who tightly two mechanical parts fit together), electromagnetic interference (from sources inside and outside the instrument) and so on.

Physical changes in the measurement instrument are not the only possible source of error. There are also recording errors in which the error is introduced when the output from the instrument is recorded – that again happened a lot in early science when scientists would hand write their observations into notebooks. Or user error in which an instrument is set up the wrong way or used outside its applicable range. For instance if you have a thermometer that has a maximum temperature then any temperature above that will be recorded as the maximum.

Whether or not the technique described in the quoted paragraph about “averaging” observations is legitimate depends on the nature of the error. This is something we will cover in the future. In the meantime I would encourage you to read the rest of the linked piece on averages and see if you can discern the important and problematic misstep of reasoning that occurs as Quetelet starts to apply “averages” to the measurement of people.

We are also beginning to see that the various limitations on observations are all related to each other. This will become even clearer when we look at cost as a limit on observations next time. Here is the quick preview of that: increasing measurement resolution and reducing measurement error can often be achieved by building a more costly instrument.