A Study on the X ¯ and S Control Charts with Unequal Sample Sizes
Abstract
The control charts based on <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </semantics> </math> </inline-formula> and <i>S</i> 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 <inline-formula> <math display="inline"> <semantics> <mover accent="true"> <mi>X</mi> <mo stretchy="false">¯</mo> </mover> </semantics> </math> </inline-formula> and <i>S</i> control charts when the sample sizes are not equal.