INDIVIDUAL CONTROL CHARTS FOR NON-NORMAL DATA

Abstract

Individual control charts are a suitable option to consider when the processes being studied change slowly. If obtaining standard samples for Shewhart control charts is expensive or in the case of studying statistical characteristics of the technological process, they are used. Control charts are used to control technological processes in the case of a normal distribution of the possible values of the controlled parameter. Sometimes a quasi-normal distribution of the studied parameter is allowed. In practice, it is often necessary to deal with technological processes in which samples are not distributed according to a normal distribution. In such cases, using individual control charts can lead to type I and II errors. This is because the control limits of the charts are calculated for the mean and standard deviation of the normal distribution. Control charts of individual values are most sensitive to the deviation of the distribution from the normal one since individual values are used to construct the limits, to which the central limit theorem does not apply, as in the case of control charts of mean values. Therefore, it is extremely important to develop methods of applying the individual control charts for random samples, the general population of which is not normally distributed. The study examines the application of individual control charts for a random process that is non-normally distributed. This is achieved by using the normalization method to transform samples of the studied parameter into samples that are distributed normally or quasi-normally. The normalization is performed using the Box-Cox method with further construction of control charts based on the transformed samples. The mathematical apparatus of the individual control chart application for samples with non-normal distribution is described. Numerical modeling of the application of individual control charts for the chi-square distributed samples, which have a significant distribution asymmetry for small values of the shape factor, and for the samples obtained by normalizing the chi-square one, was performed. According to the research findings, the utilization of individual control charts for samples exhibiting a significantly asymmetric distribution results in a higher likelihood of type I errors being detected. However, when the method of sample normalization is employed, this issue is almost entirely eliminated. Modeling process disruption, by changing a certain individual value, also demonstrated the emergence of false signals of process disruption.