Analysis of the Effect of Subgroup Size on the X-Bar Control Chart Using Forensic Science Laboratory Sample Influx Data

Abstract

This paper analyzes the effect of subgroup size on the x-bar chart characteristics using sample influx (SIF) into forensic science laboratory (FSL). The characteristics studied include changes in out-or-control points (OCP), upper control limit UCL x , and zonal demarcations. Multi-rules were used to identify the number of out-of-control-points, N ocp as violations using five control chart rules applied separately. A sensitivity analysis on the N ocp was applied for subgroup size, k, and number of sigma above the mean value to determine the upper control limit, UCL x . A computer code was implemented using a FORTRAN code to create x-bar control-charts and capture OCP and other control-chart characteristics with increasing k from 2 to 25. For each value of k, a complete series of average values, Q ( p ), of specific length, N sg , was created from which statistical analysis was conducted and compared to the original SIF data, S ( t ). The variation of number of out-of-control points or violations, N ocp , for different control-charts rules with increasing k was determined to follow a decaying exponential function, N ocp = A e – α , for which, the goodness of fit was established, and the R 2 value approached unity for Rule #4 and #5 only. The goodness of fit was established to be the new criteria for rational subgroup-size range, for Rules #5 and #4 only, which involve a count of 6 consecutive points decreasing and 8 consecutive points above the selected control limit ( σ /3 above the grand mean), respectively. Using this criterion, the rational subgroup range was established to be 4 ≤ k ≤ 20 for the two x-bar control chart rules.