Comparisons of the symmetric and asymmetric control limits for $\overline{X}$ and R charts

Abstract

Though both symmetric and asymmetric control limits can be applied to $\overline{X}$ and R charts, no well-controlled comparison of the resulting performance has been conducted. Symmetric limits such as 3-sigma limits are the customary choice. However, previous researchers have proposed asymmetric control limits as a more appropriate choice than symmetric limits for skewed distributions. This paper examines the relative performance of symmetric and asymmetric limits. It compares the out-of-control average run length (ARL) for symmetric and asymmetric limits for fixed values of the in-control ARL. Two testing examples are employed: the exponential and Johnson unbounded distributions. The results of the performance comparison are mixed. For both $\overline{X}$ and R charts, the impact of the control limit choice (symmetric or asymmetric) depends on two factors: the skewness of the charting statistic (sample mean $\overline{X}$ for the $\overline{X}$ chart and sample range R for the R chart) and the shift direction. When the charting statistic has a right-skewed distribution, symmetric limits perform better if the monitored process property (the mean for the $\overline{X}$ chart and the standard deviation for the R chart) shifts upward, but worse when it shifts downward. When the charting statistic has a left-skewed distribution, the outcome is reversed. Although neither type of control limits dominates even for a skewed population, the asymmetric limits are more robust to the shift in the process mean or variation. The effect of the sample size is also discussed.

Introduction

In this paper, we compare the impact of control limit settings (symmetric versus asymmetric) on the performance of Shewhart control charts. Both $\overline{X}$ and R charts are used in statistical process control (SPC) to monitor the process mean and variance. A conventional choice for both charts is the symmetric control limits μ_Y ± kσ_Y, where μ_Y and σ_Y are the mean and variance of the charting statistic Y (the sample mean for the $\overline{X}$ chart and sample range for the R chart) and k is a positive constant, typically k = 3 (Montgomery, 2005).

Various methods of computing asymmetric control limits for skewed distributions have been proposed.¹ The probability-symmetric approach sets the lower control limit (LCL) and upper control limit (UCL) so that the charting statistic Y is equally likely to fall above the UCL or below the LCL when the process is in-control. That is, when the process is in control, P{Y ⩽ LCL} =  P{Y ⩾ UCL} = α/2, where α is the desired Type-I risk.² The probability-symmetric control limits are symmetric at μ_Y only if the distribution of Y is symmetric.

If the distribution function of the quality measurement X is unknown, the values of the probability-symmetric control limits must be approximated. A number of methods are available. The weighted variance (WV) method was developed by Choobineh and Ballard (1987) based on semivariance approximation (Choobineh & Branting, 1986). Heuristic variants of the WV method that requires no assumptions concerning the functional form of the distribution were later proposed by, e.g., Bai and Choi, 1995, Castagliola, 2000, Castagliola and Tsung, 2005. The skewness correction (SC) method (Chan & Cui, 2003) works for skewed distributions. Another approach is to fit a theoretical frequency curve, such as a member of the Pearson system, and obtain probability-symmetric control limits meeting the specified Type-I risk α value.

Some SPC approaches have been developed specifically for skewed distributions. The split distribution method of Cowden (1957) works for arbitrary skewed distributions. Other methods are distribution specific, including geometric midrange and range charts for monitoring the mean and variance of lognormal distributions (Ferrell, 1958) and median, range, scale, and location charts for Weibull distributions (Nelson, 1979).

Although the construction of asymmetric control limits for skewed distributions has been addressed in the literature, little is known about how the adoption of asymmetric control limits affects $\overline{X}$ and R chart performance. In particular, the performance impact of limit choice (symmetric or asymmetric) must be characterized in detail using a consistent benchmark. In this work, we compare the performance impact of symmetric and asymmetric limits (probability-symmetric control limits) for a quality measurement X with a known distribution. Our performance benchmark is the average run length (ARL) when the process goes out of control (called out-of-control ARL). Throughout the comparison, the in-control ARL remains fixed, where the in-control ARL is the ARL when the process is in control. That is, the values of the symmetric and asymmetric control limits are chosen to meet a specified value of the in-control ARL (or equivalently the Type-I risk).

The rest of this paper is organized as follows. Section 2 introduces the symmetric and asymmetric control limits for $\overline{X}$ and R charts. Section 3 presents example computation of the symmetric and asymmetric limits given quality measurements from two different distributions: the exponential and Johnson unbounded distributions (Johnson, 1949). Section 4 compares the performance outcome of symmetric and asymmetric limits with $\overline{X}$ and R charts. The ARL value for the exponential population is computed numerically while the ARL for the Johnson population is estimated via simulation experiments. Our results show that the performance impact of the control limits depends on two factors: the skewness of the charting statistic (right versus left) and the shift direction. We find that even when the distribution of the quality characteristic is skewed, symmetric limits sometimes outperform asymmetric limits. Section 5 concludes.