Form Factor

The form factor, in the sense of a ratio of means, is the root-mean-square divided by the mean absolute value (Aarts 2021, p. 87). For a real random variable with , it is

(1)

where denotes the expectation value. The Cauchy-Schwarz inequality gives , with equality iff is constant almost surely. The form factor is unchanged by multiplication of by a nonzero real constant. It is undefined for an identically zero input.

For real data values , ..., that are not all zero,

(2)

and

(3)

The lower bound is attained when all are equal, and the upper bound when exactly one value is nonzero.

For a zero-mean normal distribution with positive standard deviation, . For a nonzero sinusoid averaged continuously over a full period, (Aarts 2021, pp. 88 and 91). For centered data, the reciprocal is the mean deviation divided by the population standard deviation, using the same averaging convention for both quantities (Geary 1947, p. 219).