# 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.

#### Contents

## 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.