xmean2

xmean2 is a statistical concept used to denote the second moment of a data set, specifically the mean of the squared values. For a sample x1, x2, ..., xn, xmean2 is defined as (1/n) ∑i=1..n xi^2. It represents the second raw moment about zero and is related to but distinct from the root-mean-square (RMS), which is sqrt(xmean2).

Relation to other statistics: xmean2 captures the magnitude of values but does not by itself describe central tendency. The variance is E[X^2] − (E[X])^2, so xmean2 equals the expected value of X^2, while the variance also accounts for the mean. xmean2 is always non-negative and scales with the square of the units of xi.

Computation and interpretation: computing xmean2 is straightforward—average the squares of the data. In many software environments, xmean2 can be obtained as mean(x.^2) or equivalent. A higher xmean2 indicates greater overall magnitude of observations, while a low xmean2 suggests smaller values on average.

Applications and variants: xmean2 appears in energy calculations, signal processing, and data diagnostics where the magnitude of values matters. It can be used alongside the first moment (the mean) to assess dispersion of energy or to form composite metrics such as the RMS. A related concept is the centered second moment, the variance, which uses deviations from the mean rather than raw values.

Example: for data [1, 2, 3, 4], xmean2 = (1^2 + 2^2 + 3^2 + 4^2)/4 = 7.5; RMS is sqrt(7.5) ≈ 2.74.