Template:Lognormal distribution standard deviation: Difference between revisions

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===The Standard Deviation===
===The Standard Deviation===
The standard deviation of the lognormal distribution,  <math>{{\sigma }_{T}}</math> , is given by [18]:  
The standard deviation of the lognormal distribution,  <math>{{\sigma }}</math> , is given by [[Appendix: Weibull References|18]]]:  


::<math>{{\sigma }_{T}}=\sqrt{\left( {{e}^{2{\mu }'+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}</math>
::<math>{{\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>{{\sigma }_{{{T}'}}}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma }_{T}}</math>  is given by:  
The standard deviation of the natural logarithms of the times-to-failure,  <math>{{\sigma'}</math> , in terms of  <math>\bar{T}</math>  and  <math>{{\sigma}</math>  is given by:  


::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>
::<math>\sigma'=\sqrt{\ln \left( \frac{\sigma^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}</math>

Revision as of 16:48, 13 February 2012

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]
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