Minimize squared relative error

Suppose you have a list of positive data points y₁, y₂, …, y_n 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].

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

[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

One thought on “Minimize squared relative error”

21 June 2025 at 10:17

Similarly, minimizing the sum of _absolute_ distances gives you the median (if the number of points is even then anything between the middle two does equally well). You can think of all these things as max-likelihood estimators.

Minimize squared relative error

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One thought on “Minimize squared relative error”

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