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 | The standard deviation of the lognormal distribution, <math>{\sigma }</math> , is given by [[Appendix: Weibull References|[18]]]: | ||
::<math>\sigma =\sqrt{\left( {{e}^{2\mu '+\sigma {{'}^{2}}}} \right)-\left( {{e}^{\sigma {{'}^{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}'</math> , in terms of <math>\bar{T}</math> and <math> | 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 '=\sqrt{\ln \left( \frac{{{\sigma }^{2}}}{{{{\bar{T}}}^{2}}}+1 \right)}</math> | ::<math>\sigma '=\sqrt{\ln \left( \frac{{{\sigma }^{2}}}{{{{\bar{T}}}^{2}}}+1 \right)}</math> | ||
Revision as of 16:53, 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]