Calculating the (Almost) Exact Control Limits for a C-Chart

Abstract

A power transformation using an exponent of 2/3 for Poisson-distributed data, with a small constant added, achieves symmetry for improved statistical process control (SPC) applications whether it is for an individual, a cumulative sum, or an exponentially weighted moving average chart. Two simple equations are proposed for calculating the lower control limit (LCL) and the upper control limit (UCL) for Poisson type data. Agreement between the exact LCL and UCL, as determined by the lower and upper tail area, is excellent. The square-root transformation that stabilizes the variance produces a negatively skewed distribution and tends to give false SPC signals.

Keywords:

• Control chart
• Poisson distribution
• Symmetry
• Transformation

Notes

Table 1 was calculated using a Texas Instruments TI-83®, with internal accuracy to 14 digits and a 2-digit exponent.

Table 2 was calculated using a Texas Instruments TI-83®, with internal accuracy to 14 digits and a 2-digit exponent. Note: CLCL—Calculated Lower Control Limit using Eq. (3). CUCL—Calculated Upper Control Limit using Eq. (4).

Table 3 was calculated using a Texas Instruments TI-83®, with internal accuracy to 14 digits and a 2-digit exponent. Note: TLCC—Traditional Lower Control Limit using Eq. (1). TUCL—Traditional Upper Control Limit using Eq (2).

Table 4 was calculated using a Texas Instruments TI-83®, with internal accuracy to 14 digits and a 2-digit exponent.

Table 5 was calculated using a Texas Instruments TI-83®, with internal accuracy to 14 digits and a 2-digit exponent. Note: CLWL—Calculated Lower Warning Limit using Eq. (5) and CUWL—Calculated Upper Warning Limit using Eq. (6).