The precision of methods using the statistics of extremes for the estimation of the maximum size of inclusions in clean steels

The maximum inclusion size in clean steels influences fatigue behaviour and other mechanical properties. Hence, its estimation and the uncertainties associated with the estimation are important issues for steel makers and users. Here, two methods based on the statistics of extremes, one termed the Statistics of Extreme Values (SEV) method and the other the Generalized Pareto Distribution (GPD) method, are used for the estimation. Both methods use data on the size of inclusions revealed on the surface of sampled areas. The influence of the number of sample areas and the way the sample areas are grouped on the estimated result and its confidence limits is determined and compared. For both the SEV and the GPD methods, the estimated largest inclusion size is relatively insensitive to the number of sample areas but, as might be expected, the width of the confidence interval decreases steeply as the number of sample areas increases. A key point is that the SEV method has a narrower confidence interval than the GPD method for a given number of sample areas, because the SEV method makes an extra assumption about the form of the distribution of large inclusions. The particular assumption is difficult to justify on the basis of the data alone, and leads to a potentially over-optimistic estimate of precision. For practical application of the GPD estimation procedure, the number of sample areas needed for estimation depends on the confidence interval required and the volume of steel of interest. It is suggested on the basis of the GPD size distribution that fatigue failure initiation in a component is unlikely to be caused by the single largest inclusion, but rather by more frequently occurring inclusions near the top of the size range. This provides the conceptual basis for a statistically based design approach in which the estimated distribution of inclusion sizes is used in defect tolerance design of steel components and in control of steel production processes.