Terms of Probability Statistics: Part 1 of 4 of Understanding Random Vibration Signals

A random test performed with Crystal Instruments' Spider-81

A random test performed with Crystal Instruments' Spider-81

Verifying the robustness of products (or their packaging) by subjecting them to shaker-induced vibration is an accepted method of “improving the breed”. While shock bumps and sine sweeps are intuitively obvious, random shakes with their jumps and hissing are anything but. Even the language of a random test is confusing at encounter. Let’s try to improve upon that first introduction to random signals!

For starters, a random time-history is simply a signal that cannot be precisely described by a simple equation in time; it can only be described in terms of probability statistics. Two of the most important of these statistical measures are the mean and the variance. The mean, μ, is the central or average value of a time history, x(t). It is the DC component of the signal.


The variance, σ2, is the averaged (unsigned) indication of the signal’s AC content, its instantaneous departure from the mean value, defined by:


The square root of the variance, σ, is termed the standard deviation.

These functions are closely related to a third time-domain value the mean-square:

The square root of the mean-square is the familiar root mean-square (RMS) value, commonly used to characterize AC voltage and current, as well as the acceleration intensity of a random shake test. Because these statistics are so frequently measured from signals with a zero-valued mean (no DC), the differentiation between standard deviation and RMS and between variance and mean-square has become unfortunately blurred in modern discussions.