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While Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for two-parameter Weibull distribution that is useful in many life test data analyses.

In the theory of Six Sigma, most of the processes are assumed normal and Six Sigma metrics are determined for such a process of interest. In reliability studies non-normal distributions are more appropriate for life tests. In this paper, a theoretical procedure is developed for determining Six Sigma metrics when the underlying process follows two-parameter Weibull distribution. Numerical evaluations are also considered to study the proposed method.

In this paper, by matching the probabilities under different normal process-based sigma quality levels (SQLs), we first determined the Six Sigma specification limits (Lower and Upper Six Sigma Limits- LSSL and USSL) for the two-parameter Weibull distribution by setting different values for the shape parameter and the scaling parameter. Then, the lower SQL (LSQL) and upper SQL (USQL) values are obtained for the Weibull distribution with centered and shifted cases. We presented numerical results for Six Sigma metrics of Weibull distribution with different parameter settings. We also simulated a set of 1,000 values from this Weibull distribution for both centered and shifted cases to evaluate the Six Sigma performance metrics. It is found that the SQLs under two-parameter Weibull distribution are slightly lesser than those when the process is assumed normal.

The theoretical approach proposed for determining Six Sigma metrics for Weibull distribution is new to the Six Sigma Quality practitioners who commonly deal with normal process or normal approximation to non-normal processes. The procedure developed here is, in fact, used to first determine LSSL and USSL followed by which LSQL and USQL are obtained. This in turn has helped to compute the Six Sigma metrics such as defects per million opportunities (DPMOs) and the parts that are extremely good per million opportunities (EGPMOs) under two-parameter Weibull distribution for lower-the-better (LTB) and higher-the-better (HTB) quality characteristics. We believe that this approach is quite new to the practitioners, and it is not only useful to the practitioners but will also serve to motivate the researchers to do more work in this field of research.

The purpose of this paper is to propose an approach for studying the Six Sigma metrics when the underlying distribution is lognormal.

The Six Sigma metrics are commonly available for normal processes that are run in the long run. However, there are situations in reliability studies where non-normal distributions are more appropriate for life tests. In this paper, Six Sigma metrics are obtained for lognormal distribution.

In this paper, unlike the normal process, for lognormal distribution, there are unequal tail probabilities. Hence, the sigma levels are not the same for left-tail and right-tail defects per million opportunities (DPMO). Also, in life tests, while left-tail probability is related to DPMO, the right tail is considered as extremely good PMO. This aspect is introduced and based on which the sigma levels are determined for different parameter settings and left- and right-tail probability combinations. Examples are given to illustrate the proposed approach.

Though Six Sigma metrics have been developed based on a normality assumption, there have been no studies for determining the Six Sigma metrics for non-normal processes, particularly for life test distributions in reliability studies. The Six Sigma metrics developed here for lognormal distribution is new to the practitioners, and this will motivate the researchers to do more work in this field of research.

The purpose of this article is to study the link between mean shift and inflation coefficient when the underlying null hypothesis is rejected in the analysis of variance (ANOVA) application after the preliminary test on the model specification.

A new approach is proposed to study the link between mean shift and inflation coefficient when the underlying null hypothesis is rejected in the ANOVA application. First, we determine this relationship from the general perspective of Six Sigma methodology under the normality assumption. Then, the approach is extended to a balanced two-stage nested design with a random effects model in which a preliminary test is used to fix the main test statistic.

The features of mean-shifted and inflated (but centred) processes with the same specification limits from the perspective of Six Sigma are studied. The shift and inflation coefficients are derived for the two-stage balanced ANOVA model. We obtained good predictions for the process shift, given the inflation coefficient, which has been demonstrated using numerical results and applied to case studies. It is understood that the proposed method may be used as a tool to obtain an efficient variance estimator under mean shift.

In this work, as a new research approach, we studied the link between mean shift and inflation coefficients when the underlying null hypothesis is rejected in the ANOVA. Derivations for these coefficients are presented. The results when the null hypothesis is accepted are also studied. This needs the help of preliminary tests to decide on the model assumptions, and hence the researchers are expected to be familiar with the application of preliminary tests.

After studying the proposed approach with extensive numerical results, we have provided two practical examples that demonstrate the significance of the approach for real-time practitioners. The practitioners are expected to take additional care before deciding on the model assumptions by applying preliminary tests.

The proposed approach is original in the sense that there have been no similar approaches existing in the literature that combine Six Sigma and preliminary tests in ANOVA applications.