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,

\displaystyle\frac{a+b}{2}

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

\displaystyle\frac{2ab}{a+b}.

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.

One response to “Best Proximate Value

  1. Pingback: Carnival of Mathematics #60 « ∑idiot's Blog

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