Characteristics of the Lognormal Distribution


Reliability HotWire	Issue 47, January 2005
Reliability Basics
Characteristics of the Lognormal Distribution The lognormal distribution is commonly used to model the lives of units whose failure modes are of a fatigue-stress nature. Since this includes most, if not all, mechanical systems, the lognormal distribution can have widespread application. Consequently, the lognormal distribution is a good companion to the Weibull distribution when attempting to model these types of units. As may be surmised by the name, the lognormal distribution has certain similarities to the normal distribution. A random variable is lognormally distributed if the logarithm of the random variable is normally distributed. Because of this, there are many mathematical similarities between the two distributions. For example, the mathematical reasoning for the construction of the probability plotting scales and the bias of parameter estimators is very similar for these two distributions. Lognormal pdf The lognormal pdf is given by: where: = ln(T), the T values are the times-to-failure = mean of the natural logarithms of the times-to-failure = standard deviation of the natural logarithms of the times-to-failure The lognormal distribution is a distribution skewed to the right. The pdf starts at zero, increases to its mode, and decreases thereafter. The degree of skewness increases as increases, for a given . For the same , the pdf's skewness increases as increases. For values significantly greater than 1, the pdf rises very sharply in the beginning (i.e. for very small values of T near zero) and essentially follows the ordinate axis, peaks out early, and then decreases sharply like an exponential pdf or a Weibull pdf with 0 < β <1. The parameter (or the mean life, or the MTTF), in terms of the logarithm of the s, is also the scale parameter, and not the location parameter as in the case of the normal pdf. The parameter , or the standard deviation of the s in terms of their logarithm or of their , is also the shape parameter and not the scale parameter, as in the normal pdf, and assumes only positive values. Lognormal Distribution Parameters in Weibull++ In Weibull++, the parameters returned for the lognormal distribution are always logarithmic. That is, the parameter represents the mean of the natural logarithms of the times-to-failure, while represents the standard deviation of these data point logarithms. Specifically, the returned value is the square root of the variance of the natural logarithms of the data points. Even though the software denotes these values as mean and standard deviation, the user is reminded that these are given as the parameters of the distribution and are thus the mean and standard deviation of the natural logarithms of the data. The mean value of the times-to-failure, not used as a parameter, and the standard deviation can be obtained through the QCP or the Function Wizard.
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