History of observed samples: How much affects the ARL of <img src='/image/spc_char/X_bar.gif'> control charts

Abstract

This paper presents history of observed samples upon average run length (ARL) of <img src='/image/spc_char/X_bar.gif'><b style=""> </b>chart, when process shift takes place, because the performance characteristic of a control chart is measured in terms of ARL. The concept of proposed <img src='/image/spc_char/X_bar.gif'><b style=""> </b>chart is based upon the Chi square distribution; in which a Statistic, ‘U’ has been defined and calculated for the product of sample size (n) and history (H). ARL values of proposed chart are compared to the standard Shewhart chart, considering the control limits as ± 2σ and confidence level (CL) as 95%. The history of previous samples is considered to evaluate U and comparing it with the maximum permissible value of standard Chi Square Table to evaluate the status of running process. Sample sizes are taken as 4, 5, 6, 8 and 10, while previous history is considered from 4 to 10. For smaller shift (0.1-0.5σ), the length of history was found to affect ARL of <img src='/image/spc_char/X_bar.gif'><b style=""> </b>chart to some extent, but for moderate and large shifts (> 0.5 σ), the effect of length of history is negligible.