# 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. subrange; distribution; sub biasing factors; relative efficiency; breakdown point 1111632. 11.  
used an alternative to the standard deviation to construct the control charts in Phase-I when the data is contaminated. [PERSON] and [PERSON] modified the S-control chart based on the decile mean standard deviation, which possessed a better performance when the sample size was small in most realistic applications [13]. When the process data in Phase-I diverge from normality and outliers are present, the robust S-chart based on robust scale estimators is more preferable than a traditional S-chart [10; 13]. The reweighted robust standard deviation estimator is adopted to modify the S-chart by [PERSON] and [PERSON] [14]. To improve robustness and be more efficient, in the case of the presence of outliers in the data set, the trimmed mean of mean, standard deviation, and range are adopted to construct the control charts [15; 16; 17]. [PERSON] et al. [18] and [PERSON] and [PERSON] [19] pointed out that the control chart based on the robust scale estimator provides good performance not only without outliers, but also within outliers in the data set. Cumulative sum (CUSUM) control charts for special causes are also under the assumption of normal distribution. Given this, many attentions are paid to the robust CUSUM chart when the ideal assumption is violated [20; 21; 22]. Considering the multivariate non-normality problems, the distribution-free method [23] and Bayesian seemingly unrelated regression model [24] can give us some inspirations. The trimmed mean (also called truncated mean) is the mean of the part after removing the \(x\%\) largest and smallest observations [25]. For example, to reduce and mitigate the influence of bias in subjective judgements, the highest and lowest scores are discarded and the middle three are summed for a five-judge panel [26; 27] for many Olympics games (e.g., diving and gymnastics etc.). However, there is seldom research about the \(R\)-chart under data contamination in the literature. This motivates us to develop a robust method for control charts when the data is contaminated. In this paper, we propose a robust scale estimator subrange as an alternative to the sample range--as an estimator of the standard deviation to improve the control chart in the case of data contamination. The calculation of subrange is very similar to the trimmed mean. However, the subrange is not unbiased. As we know, order statistics are widely used in nonparametric inferences and robust procedures.