Best Proximate Value

My Oxford English Dictionary describes proximate as meaning,

Closely neighbouring, immediately adjacent, next, nearest (in space, serial order, quality, etc.); close, intimate

In this article, a proximate value will mean a value p which we shall use as a stand-in for an unknown value x.  The unknown variable x is known to have the property that

a \leq x \leq b

Choosing some value p as a proximate value for x entails incurring an absolute error of p-x and a relative error 0f \displaystyle\frac{p-x}{x}.  If we intend to choose a proximate value that minimizes absolute error, we are solving

\displaystyle\min_p\left(\max_x\left|p-x \right| \right).

If we are intending to minimize relative error, we are solving

\displaystyle \min_p\left(\max_x\frac{\left|p-x \right| }{x}\right)

That is to say, we are looking for the proximate value that minimizes the damage of being given the worst possible value of x for that proximate value.  The solution to the first problem is the arithmetic mean of the end points of the interval,


and to the second is the harmonic mean of the end points,


This is of some practical relevance as I often notice that scientists choose the arithmetic mean by default when choosing a proximate value.  At least in cases where the relative error is of greatest concern, this may be an error.

I try to keep posts relatively short so I will go into the proofs in a second installment.

This article is based on a paper by George Pólya, On the Harmonic Mean of Two Numbers.

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