The main theme of this dissertation deals with the impact and consequences of non-normal distribution on the process capability index Cpm. In this thesis, much work has been done in this area including the properties of C^pm, the estimate of Cpm, under normality, its sensitivity to non-normality and also the relationship of Cpm to squared error loss. Related to Cpm is the unifying measure of process capability index Cpw. Several properties of C^pw are investigated. Much of the controversy surrounding the Cp index involves 6[sigma] in the denominator. It carries particular physical meaning when the process characteristic is normally distributed. A new index Cpo is proposed which is based on the difference between two order statistics. The sampling distribution of C^po is obtained for those cases where the process characteristic is uniform, exponential and normal distributions. The behavior of C^p, when n = 2, under non-normal situations such as uniform and exponential distributions is also investigated as a special case of C^po. Another major issue addressed in this dissertation is the Inverted Probability Loss Functions (IPLFs). It is a modified loss function found by inverting a probability density function which was first invented by my supervisor Dr. F.A. Spiring in 1993. The first loss function I studied is the inverted beta loss function (IBLF). I have found certain interesting properties that this class of loss function possesses such as the shape, the loss function and its associated risk function of the IBLF are scale invariant under linear transformation. Finally, I have investigated a few more IPLFs satisfying the usual loss function properties and developed some theorems in this portion of the study.