How do you characterize a distribution that follows an S shape

[Tony]

I have a data set that follows an S-shape on the probability plot. I know I have two sub populations. Given that, how do you characterize the overall population as a group? And/or how do you characterize the subpopulations individually?

[Pantelis]

You can use a multimodal distribution such as the mixed Weibull if you want to consider them all as a whole. (see http://www.weibull.com/LifeDataWeb/the_mixed_weibull_distribution.htm )

If you have additional information (i.e. you can identify specific modes/events) for each data point you may want to use a competing failure modes approach. (see http://www.weibull.com/LifeDataWeb/competing_failure_modes.htm )

[Larry Strouse]

I know the exact number of cycles required to fail certain castings in a test environment. I also know the number of cycles to initiate the cracks which ultimately lead to failure. I can plot both in Weibull++. Should I expect the slope of %failure vs. cycles to be the same as the slope of %initiation vs. cycles? If not, why not?

[Pantelis]

The way you phrased -- and the way I understood the question -- the answer is No.

For simplicity assume that once an initiation occurs a failure always follows, and there is a fixed deterministic time after initiation and until failure, say Z. Then if one fits a Weibull distribution to random variables X1, X2 and X3 (time to initiation), should the shape parameter be the same as fitting the distribution to X1+Z, X2+Z and X3+Z (time to failure). The answer to this is no.