Scale and Shape Parameters for Lognormal Distributions

[sjacobs]

I have strength data that is lognormal. I am trying to plot the lognormal distribution and am confused on the shape and scale parameters. I am using the formulas given in the subject index on the main page of weibull.com. Under lognormal, there are links that give formulas for the lognormal distributions (pdf, mean, stdev, etc.). My problem is that the scale parameter, which is the mean of the natural logs of the times to failure, is defined at the top of the page as the simply the natural log of the times to failure. Simple enough. However, on the link that says 'lognormal statistics properties' there is a definition of the mean of the natural log of the times to failure (exactly the same parameter as above) given in terms of the mean and the stdev of the original data (before the nat. log is taken). Now, since the parameters are the same one would expect them to be equivalent. I get similar but different answers when I use both methods to calulate them. The distribution visually fits my data better using the 2nd method (the equation). But which one is correct? I am having exactly the same problem with the shape parameter (or the sigma term). Both are simply defined as one thing and then given as a formula later but do not seem to give me the same results. Thanks in advance to anyone who attempts to read this long message.

[adam]

It is a bit confusing, but remember that the mean of the original data is actually the expected value of the random variable, or, the mean of the distribution (lognormal in this case). The log-mean is the mean value of the transformed data (i.e., log-times), since the Normal distribution is fitted to these log-times. When you transform back to the actual scale you get the Lognormal distribution which is no longer the same as the normal, and has its own properties, such as the expected value, standard deviation, etc.
The lognormal is a skewed distribution and by simply taking e^log-mean does not give you the expected value (or the mean value). To get the expected value for the lognormal you need to use the equation given in the link you mentioned.

I hope this makes sense. If not let me know and I'll take another try.

[Scott Jacobs]

Thanks. It all makes sense. I just figured out that I was making a simple mistake with my numbers.

Can you anwser one more question? I have a textbook that goes over the lognormal dist. (Auth. Karl Bury, "Statisical distrib. in eng.). It explains the lognormal dist. and describes the shape and scale factors but does not give equations for them. However, the equations given in this site define them as the mean and stdev of the natural logs of the original data. Makes sense but why didn't Mr. Bury give an equation or define them in this way? Are the equations given here standard or are there different ways of getting these factors?

I really appreciate your help. It is very important that I get this right.

[adam]

There are different ways of reporting the parameters of the lognormal distribution (and of other distributions as well). The most common one is the way that are presented in this site.