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. Since x has been computed from the same set of numbers as used in Eq. 4, the value of the last term in the summation in Eq. 4 is predetermined, so 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).