|Why be approximate?|
Seeking excess precision:
A thoughtful approach to precision is needed. We seek saticficing, `good enough' precision.
We get simplicity and power by selectively reducing precision.
When you ask someone "What time is it?", you rarely hear "one thirty three". But why? Perhaps they haven't spent the time needed to keep their watch accurate to the minute. And perhaps they judge that "one thirty" is all the precision you really care about.
Yet when introduced to the volume of a ball, one hears it is 4/3 pi r3, rather than it is "one half the volume of the box the ball came in" (1/2 d3). As with "one thirty", this simpler approximation is only off by 5%. But how often does one hear the simpler version? And how often do you remember the hairier one?
When a value is not known precisely, the amount of uncertainty is usually called "error". This has given the whole business of uncertainty analysis a bad name, because in common usage "error" implies sloppiness, very likely caused by sinfulness. (She saw the error of her ways.)
In the language of technology, this section deals with the analysis of errors - how to judge their magnitude, how to describe them in conventional ways, and how to take them into account in calculating numerical values based on a number of individual measurements. As we shall emphasize, error represents uncertainty and has nothing to do with mistakes or sloppiness. Indeed, reducing the amount of error in a measurement beyond the immediate need is usually a mistake and a sign of poor judgement.
We categorize the reporting and handling of uncertainty in four stages, each involving a particular degree of precision. The first is concerned only with order of magnitude of a number. Consider this to be a zeroth approximation. Next are the conventions regulating the use of significant figures, a first approximation of limited usefulness. The second approximation deals with the maximum and minimum range of measured quantities. The rules of manipulation of such error limits are simple, and this system should be the one most often used by students of introductory science. Finally, there is a third approximation to error citation and analysis, involving rules derived from probability and statistics. This system is frequently misunderstood and misapplied. Its use is justified only when the primitive data fulfill certain requirements of quantity, distribgution, and probability.
Comments encouraged. - Mitchell N Charity <firstname.lastname@example.org>
 Paraphrase of Swartz, Used Math, p1.