## 4 Results  
The results of this research are related to three main subjects. These subjects are the quality of generating multivariate data with a desired correlation, skewness reduction of the generated data, and the effectiveness of the proposed method in estimating the actual PCI. The generated data should meet both the given parameters and correlation coefficients. Figure 2 shows the distribution fitting of the generated data. One variable's data from each distribution was fitted to show the ability of the software code (Matlab R2023a) to generate data with the desired distribution and parameters. The upper right and left graphs of Figure 2 are for the same variable generated from the gamma distribution with shape and scale parameters of one and two, respectively. The upper left graph is the density function, whereas the graph in the upper right corner is the cumulative density function of the same variable data. Also, one variable from both the Beta and Weibull distributions were plotted using the cumulative density function. All the fitted data show very close fit and parameters. Furthermore, the correlation coefficients between generated variables in each sample should be the same as the reference data in Table 2. The correlation between variables has a significant effect on the estimation of the PCI; that is, the greater the positive correlation, the less variation there is between variables. Consequently, the correlation between generated variables in each sample is presented in Table 3. The results shows that the correlations between the generated variables and the actual correlations are very close to each other. Figure 2: Distribution fitting of generated data. However, the data were generated in four different sample sizes (50, 100, 500, and 1000), and the correlations of all different sample sizes are close to the actual correlation. It is noticed that the samples of two variables have more accurate correlations than those of three variables. Moreover, estimating the PCI required the data to be normally distributed. In this regard, the results of applying root transformation for skewness reduction are presented here. Figure 3 shows the skewness of what was generated before and after applying root transformation. The results ensure the effectiveness of root transformation in reducing the skewness of the data. Most of the skewness of the data after transforming is close to zero. The most skewness of the generated data before transformation was associated with the first variable of the gamma bivariate sample. However, after transforming the data using the root transformation method, the skewness was reduced to almost zero. It is worth noting that these are samples of size 1000 data. The ultimate goal of this research is to come up with a capability index that evaluates process capability through multivariate quality characteristics (QCs). Multivariate process capability analysis is conducted to evaluate the capability of a given process, which produces products with multivariate quality characteristics (QCs) to meet customer and/or engineering requirements. In this research, the process performance is measured using the process capability index, in which a process with a process capability index of one or more is considered capable of producing products with given specification limits, whereas a process capability index of less than one indicates that the process may produce products outside of the specification limits. That is because the process capability index divides the  
Figure 3: Skewness of the generated data before and after root transformation.