# 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.