On the Subrange and Its Application to the R-Chart

Abstract

The conventional sample range is widely used for the construction of an R-chart. In an R-chart, the sample range estimates the standard deviation, especially in the case of a small sample size. It is well known that the performance of the sample range degrades in the case of a large sample size. In this paper, we investigate the sample subrange as an alternative to the range. This subrange includes the range as a special case. We recognize that we can improve the performance of estimating the standard deviation by using the subrange, especially in the case of a large sample size. Note that the original sample range is biased. Thus, the correction factor is used to make it unbiased. Likewise, the original subrange is also biased. In this paper, we provide the correction factor for the subrange. To compare the sample subranges with different trims to the conventional sample range or the sample standard deviation, we provide the theoretical relative efficiency and its values, which can be used to select the best trim of the subrange with the sense of maximizing the relative efficiency. For a practical guideline, we also provide a simple formula for the best trim amount, which is obtained by the least-squares method. It is worth noting that the breakdown point of the conventional sample range is always zero, while that of the sample subrange increases proportionally to a trim amount. As an application of the proposed method, we illustrate how to incorporate it into the construction of the R-chart.

Keywords

• subrange
• distribution
• unbiasing factors
• relative efficiency
• breakdown point