# On the Subrange and Its Application to the R-Chart
###### Abstract
Order statistics theories can also be used to compute simple statistics such as median, the empirical cumulative distribution function (cdf), and the sample range [28; 29; 30]. The range of a sample dispersion tells us how far apart the largest and smallest values are. The traditional sample range is the distance between the largest and the smallest observation values from a sample. However, estimating the standard deviation by means of the sample range is not unbiased. Thus, it is necessary to use the unbiasing factor \(d_{2}\) for the original sample range to estimate the standard deviation properly, which is a function of the sample size [31]. The method is widely seen in quality control literature [32; 33]. Based on the joint pdf, the distribution of the sample range has been extensively studied [34; 35; 36]. However, in the earlier literature, many papers focus on the sample range. If there is an outlier in a sample, obviously the sample range will be easily influenced by even a single outlier. Given this, we can say that the sample range clearly has a zero-breakdown point, which is a popular measure for the robustness property of a statistical estimator. The breakdown point of a statistical estimator is the fraction of data that can be given arbitrary values without making the estimator arbitrarily bad [10; 36]. The sample range is almost as efficient as the sample standard deviation for a small sample under the normal distribution, which is generally assumed in many engineering applications. However, [PERSON] [37] pointed out that when the sample size is larger than 20, even a small departure from normality in the tail of the parental distribution has a considerable effect on the distribution of the sample range. In order to make a good performance of the sample range for a large sample, [PERSON] [38] proposed a method of dividing the sample into a few subgroups and used the average of the range of these subgroups. Since the sample subrange depends on the trim amount, there may be several subranges to consider under different trims. Therefore, natural questions arise: which of the subranges is preferred? Then, by what criteria, shall we make the choice? [PERSON] [39] and [PERSON] [40] proposed that the smaller the variance in its sampling distribution, the more \"efficient\" is that estimator. The relative efficiency (RE) value is considered in order to decide how to choose the optimal trim amount of the subrange by comparing the variance of the sample. Firstly, we give the distribution of the sample subrange and provide the unbiasing factors, which depend on the sample size and the number of trims to make the estimator unbiased. Next, we provide a criterion that is based on the RE value to choose the optimal trims in the sample. Lastly, we consider the breakdown point of the subrange. As we know, the inter-quartile range (IQR) is widely used to estimate the standard deviation when the sample has an outlier or extreme values. The proposed subrange includes the IQR as a special case. As well, the subrange is robust to outliers. The contribution of this article is as follows. We give the distribution of the sample subrange and provide the unbiasing factor for the subrange of a sample through a Monte Carlo simulation. We assume the data are a normal distribution in order to calculate the unbiasing factor and the distribution for the subrange because the commonly used control charts are under the independent and normality assumption [1; 12; 41; 42]. The RE values are calculated by using two different kinds of baseline estimators, from which we discover that the results coincide with choosing the best performance of subranges. Some interesting points are found through comparing the breakdown points of a range and a subrange, by which we conclude that the subrange has a positive breakdown point of \(k/n\), and the asymptotic breakdown point of the subrange can be increased to around \(1/2\) in theory.