Control charts for paired differences: d ¯ and S d charts

Control chart procedures for monitoring paired variables are sparse in the literature. After considering the average run length properties of the d ¯  chart, which monitors the mean of the differences between paired variables, we propose a new chart based on S d , the subgroup standard deviations of the differences. Our findings show that the d ¯  chart is powerful for monitoring the changes in the means and the S d  chart is suitable for monitoring changes in the covariance structure. Furthermore, we show that the d ¯ and S d  charts perform better than existing bivariate control charts for detecting shifts in mean and variance/covariance, respectively, when standards are known. The difference charts also performed well compared to common alternatives when the standards are unknown arising from a limited amount of Phase I data. An application of these difference charts in a finance context is illustrated using the returns of Apple Inc's stock and the S&P 500 index.