AMPTIAC MaterialEASE 7 Statistical Analysis of Material Data Part III: On the Application of Statistics to Materials Analysis

Introduction This is the last of three articles dealing with statistics and its applications to the data analysis of materials and components, as used in Metallic Materials and Elements for Aerospace Vehicle Structures (MIL HDBK 5) and the Composite Materials Handbook (MIL HDBK 17) [1,2]. The objective of this series is to discuss some ideas and philosophies underlying the use of statistical procedures that are usually not familiar to the practicing engineer. Statistics courses are often too crowded with methods and too busy explaining the how-to’s, to discuss the why’s and wherefore’s. In the first article of this series, we discussed some ideas dealing with random variables (R.V.), their distributions and their parameters. In the second article, we discussed some problems dealing with estimation and testing of statistical distributions and parameters, when they were unknown but estimated from a random sample. In this last article we apply some of the material discussed in the first two, in the context of statistical analyses included in MIL HDBKs 5 and 17. We hope the series will generate an on-going dialogue, through the AMPTIAC Newsletter and Web Page, where further questions and topics of interest to statistics practitioners in the area of materials data analysis may be discussed. The whole intent of using statistical data analysis in this context stems from the need to establish types A and B tolerances for materials properties (i.e. estimates of the upper/lower first and tenth percentiles of the distribution of a population characteristic of interest). To obtain them we need to analyze samples from some material, which may come from a single source or from multiple sources or batches. Then, we need to establish the property’s underlying statistical distribution, its parameters and finally to estimate the required property tolerances, according to the specific statistical model. How we implement this is the subject of the present article. In the rest of this article, we discuss statistical procedures in [1, 2] using as guide Figure 8.3 in page 8-20 of Reference 2, (denoted as Figure 1). We discuss how, whether the data come from a single batch or whether there are two or more batches, these are tested for potential outliers. We then see how the outliers are removed from the sample, if necessary. If there are two or more batches, we assess whether these can be pooled together (e.g. if they come from the same population). Otherwise, the desired tolerances must be obtained separately, on each individual batch. Then, whether analyzed individually or pooled, the samples need to be tested for Goodness of Fit (GoF) for three statistical distributions: Weibull, Normal and Lognormal. Finally, and once having determined the underlying distribution, we apply the corresponding method of A or B basis tolerance estimation to obtain the corresponding A or B basis allowable. Conversely, we apply nonparametric methods if neither of the above mentioned distributions fit the data. Establishing the Underlying Distribution and Parameters An A or B basis allowable of a material property is an estimation of (γ0), the lower/upper first or tenth percentile of all the population values of the property. This means, with probability 0.95, ninety nine percent (A basis allowable) or ninety percent (B basis allowable) of all population values are smaller/larger than the estimate of percentile, γ0. A and B allowables depend on the specific statistical distribution of the parameters of the population in question. Therefore, the estimation, with high probability and accuracy of both the underlying distribution and the corresponding parameters of the population from which the materials sample was obtained is very important. If there is a serious estimation error in this initial procedure, everything else that we do (since it is based on this) will be wrong. In the first article we saw how F(x), the Cumulative Distribution (CDF) Function and f(x), the probability density function (pdf), are related to each other via: F(x) = ∫×f(t)dt. Hence, two types of GoF tests exist to assess the composite hypothesis (H0) that a completely specified distribution F0(x;θ) fits a data set. One type of test compares the actual (observed) number of sample points with the corresponding expected number, obtained under the (hypothesized) pdf, for subsequent data intervals. An example of such tests is the Chi Square GoF test. The other type compares (vertical) distances between empirical, Fn and theoretical, F0 CDF values, for the ordered sample Material E A S E