# Log-Normal Distribution

March 29, 2018

Back to: Random Testing

Many physical phenomena exhibit a *Log-Normal* distribution. When the response amplitude of a system results from the *multiplication* of a number of independent parameters, the distribution of the response amplitude will tend to be Log-Normal. This is because the log of the amplitude is the sum of the logs of the multiplicative terms and the *central limit theorem* implies that the log of the amplitude will have a *Normal* distribution. For example, if *x* = A·B/C, then log(*x*) = log(A) + log(B) – log(C), with A, B, C positive values. Figure 3.23 shows the Log-Normal distribution plotted on both a linear and log *x* scale.

In random vibrations, this applies to the distribution of the vibration magnitude over an ensemble of random conditions. This is illustrated in Figure 3.24 with a graph of the histogram of the RMS acceleration levels (A_{RMS}) measured in the vertical and transverse directions on the coupling housing of 40 nominally identical diesel engines running under nominally identical conditions. The measured data are plotted on a decibel scale [Decibel Level = 20 Log (A_{RMS}/1 μG), dB] and compared to a Normal distribution of the Log levels with a mean of 114.2 dB and *σ* = 3.5 dB.