Template:Lognormal distribution standard deviation

The Standard Deviation

The standard deviation of the lognormal distribution, [math]\displaystyle{ {{\sigma }} }[/math] , is given by 18]:

[math]\displaystyle{ {{\sigma }}=\sqrt{\left( {{e}^{2{\mu }'+\sigma'{2}}} \right)\left( {{e}^{\sigma'{2}}}-1 \right)} }[/math]

The standard deviation of the natural logarithms of the times-to-failure, [math]\displaystyle{ {{\sigma'} }[/math] , in terms of [math]\displaystyle{ \bar{T} }[/math] and [math]\displaystyle{ {{\sigma} }[/math] is given by:

[math]\displaystyle{ \sigma'=\sqrt{\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)} }[/math]