A practical guide to averaging functions

Averaging is ubiquitous in many sciences, engineering, and everyday practice. The notions of the arithmetic, geometric, and harmonic means developed by the ancient Greeks are in widespread use today. When thinking of an average, most people would use arithmetic mean, “the average”, or perhaps its weighted version in order to associate the inputs with the degrees of importance. While this is certainly the simplest and most intuitive averaging function, its use is often not warranted. For example, when averaging the interest rates, it is the geometric and not the arithmetic mean which is the right method. On the other hand, the arithmetic mean can also be biased for a few extreme inputs, and hence can convey false meaning. This is the reason why real estate markets report the median and not the average prices (which could be biased by one or a few outliers), and why judges’ marks in some Olympic sports are trimmed of the smallest and the largest values.