Suppose you have a list of positive data points y1, y2, …, yn and you wanted to find a value α that minimizes the squared distances to each of the y‘s.

sum_{i=1}^n (y_i - alpha)^2

Then the solution is to take α to be the mean of the y‘s:

alpha = frac{1}{n} sum_{i=1}^n y_i

This result is well known [1]. The following variation is not well known.

Suppose now that you want to choose α to minimize the squared relative distances to each of the y‘s. That is, you want to minimize the following.

sum_{i=1}^n left( frac{y_i - alpha}{alpha} right)^2

The value of alpha this expression is the contraharmonic mean of the y‘s [2].

alpha = frac{sum_{i=1}^n y_i^2}{sum_{i=1}^n y_i}

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[1] Aristotle says in the Nichomachean Ethics “The mean is in a sense an extreme.” This is literally true: the mean minimizes the sum of the squared errors.

[2] E. F. Beckenbach. A Class of Mean Value Functions. The American Mathematical Monthly. Vol. 57, No. 1 (Jan., 1950), pp. 1–6

The post Minimize squared relative error first appeared on John D. Cook.

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