# Quadratic Mean

The **quadratic mean** $f_{\text{QM}}$ of a list of values $a_1, \ldots, a_k$ is the mean with the property that the square of the mean is equal to the average of the squares of the values.

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## Definition

Suppose one is given data weighted by their squares. In that case, one might consider a mean with the property such that $k$ times the square of the mean equals the sum of the squares of the values:

$k f_{\text{QM}}^2 = a_1^2 + \cdots + a_k^2.$

This leads to the quadratic mean defined as

$f_{\text{QM}} = \sqrt{\frac{a_1^2 + \cdots + a_k^2}{k}}.$

## Properties

By the QM-AM-GM-HM inequality, the harmonic mean does not exceed either the arithmetic mean or geometric mean.