Uncertainties. It's something you can't escape in physics. No matter how hard you try, no matter how good your instruments are, there will always be some "uncertainty" in your measurements.

Since we can't get rid of uncertainties entirely, the next best thing we can do is to quantify them - that is, give others an idea of how big our uncertainties are. You may already have seen uncertainties in some measurement results:

measurement result = best estimate ± uncertainty.

What we will be examining is this "plus-or-minus" uncertainty -- how big should it be? We will also be asking ourselves how to obtain a best estimate.

Notation. Suppose we are measuring some quantity A. Then the "best estimate plus-or-minus uncertainty" will be denoted symbolically by

A ± DA.

Comment 1. The symbol DA when used to denote an uncertainty denotes a positive quantity. The symbol DA has also another use, namely, as notation for the change in the quantity A during a given time interval. Such changes, in general, may be either positive or negative. Thus, the symbol DA, when used to denote an uncertainty, has a different significance than when used to denote a change in a quantity.

Comment 2. There are different ways of quantifying an uncertainty. The symbol DA is used in all cases and does not make it clear how the uncertainty was determined. Thus, a statement like "DA = 5 cm" will contain different information, depending on how the 5-cm uncertainty has been determined. It needs to be made clear in the context, how exactly DA has been determined.

Uncertainties vs. Errors. "Uncertainties" are also called errors -- however, the term "uncertainty" describes better what one is dealing with in measurements. An uncertainty in a measurement result is, after all, not like an error or a mistake in a calculation. The latter should and can be eliminated, the former is an aspect of nature that cannot be eliminated, although one should try to reduce uncertainties.

The term "error" occurs in certain time-honored combinations like "absolute error" and "relative error". It would be better to refer to these as "absolute uncertainty" and "relative uncertainty". You will find the terms "uncertainty" and "error" used interchangeably in the literature.