Abstract
The control charts based on X ¯ and S are widely used to monitor the mean and variability of variables and can help quality engineers identify and investigate causes of the process variation. The usual requirement behind these control charts is that the sample sizes from the process are all equal, whereas this requirement may not be satisfied in practice due to missing observations, cost constraints, etc. To deal with this situation, several conventional methods were proposed. However, some methods based on weighted average approaches and an average sample size often result in degraded performance of the control charts because the adopted estimators are biased towards underestimating the true population parameters. These observations motivate us to investigate the existing methods with rigorous proofs and we provide a guideline to practitioners for the best selection to construct the X ¯ and S control charts when the sample sizes are not equal.
Highlights
Control charts, known as Shewhart control charts [1,2,3], have been used to determine if a manufacturing process is in a state of control
We have considered several unbiased location and scale estimators for the process parameters of the Xand S control charts when the sample sizes are not necessarily equal
A natural question is: among these unbiased estimators, which one should be recommended in practical applications? We clarified this question by providing the inequality relations among the variances of these estimators through the rigorous proofs
Summary
Known as Shewhart control charts [1,2,3], have been used to determine if a manufacturing process is in a state of control. Variables include continuous measurement process data such as length, pressure, width, temperature, and volume, in a time-ordered sequence Due to their importance and usefulness in real life applications, these traditional types of univariate and control charts have still received much attention in the literature. The second approach uses fixed-width control limits which is based on the average of the sample sizes. Given that the average of the sample sizes is not necessarily an integer in general, a practical alternative to the second approach is the use of a modal sample size
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