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.

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

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

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