March 29, 2018

The variance of a set of numbers, xnn = 1, …, N, with a known mean value of μ, is given by Equation 3:

(1)   \begin{equation*} \sigma^2=\frac{1}{N}\sum_{n=1}^{N}(x_{n}-\mu)^2 \end{equation*}

Equation 3

If the mean value has been estimated from a set of numbers, then the variance is given by Equation 4.

(2)   \begin{equation*} \sigma^2=\frac{1}{N-1}\sum_{n=1}^{N}(x_{n}-\bar{x})^2 \end{equation*}

Equation 4

The value in the denominator before the summation is the number of independent values used, or degrees-of-freedom, in the summation. X has been computed from the same set of numbers as used in Equation 4, so the value of the last term in the summation is predetermined. Therefore, the number of independent values in the summation is N – 1.

For a vibration signal with a mean value of zero, the variance is equal to the mean-square value. In general:

(3)   \begin{equation*} \sigma^2=\overline{x^2}-(\bar{x})^2 \end{equation*}

Equation 5

When summing independent random variables, the variances add (even with non-zero means).