Article
On the Subrange and Its Application to the R-Chart
En Xie
1
, Yizhong Ma
1
, Linhan Ouyang
2
and Chanseok Park
3,
*
Citation: Xie, E.; Ma, Y.; Ouyang, L.;
Park, C. On the Subrange and Its
Application to the R-Chart. Appl. Sci.
2021, 11, 11632. https://doi.org/
10.3390/app112411632
Academic Editor: Pentti Nieminen
Received: 13 October 2021
Accepted: 2 December 2021
Published: 8 December 2021
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1
School of Economics & Management, Nanjing University of Science and Technology, Nanjing 210094, China;
jinyu2016@njust.edu.cn (E.X.); yzma@njust.edu.cn (Y.M.)
2
College of Economics and Management, Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China; ouyang@nuaa.edu.cn
3
Department of Industrial Engineering, Pusan National University, Busan 46241, Korea
* Correspondence: cp@pusan.ac.kr
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
1. Introduction
The control chart is a widely used and powerful graphical tool in quality control that
is used to measure, monitor, and control a process over time. Usually, the control charts
are in pairs. For example, an
X
-chart monitors the average of the manufacturing process
while an
R
-chart monitors the variation of the process [
1
]. Generally, there are two phases
for constructing control charts [
2
]. In Phase-I the goal is to obtain reliable control limits
from the process data. Then, in Phase-II monitor the process by comparing the statistical
properties of the future observation to the control limits, which are achieved in Phase-
I [
3
,
4
]. The performance of the control charts constructed in Phase-I will determine the
performance of the results in Phase-II. Thus, the data quality in Phase-I plays an important
role in statistical process control (SPC). However, for
X − R
charts, the sample mean and
range are susceptible to the outliers, which is also called data contamination. Thus, the
conventional control charts may be invalidated in the case of data contamination. To solve
this problem, we use a robust estimator to construct control charts in Phase-I.
Robust statistics can provide good performance when there is the presence of outliers
and departures from the model assumption. In robust design, when the collected data are
contaminated, the robust estimators are employed to reduce or even avoid the influence of
outliers on the results [
5
–
9
]. In statistical process control, Park et al. [
10
] proposed the use
of robust scale estimators (e.g., median absolute deviation (MAD)) [
11
] and Shamos [
12
])
Appl. Sci. 2021, 11, 11632. https://doi.org/10.3390/app112411632 https://www.mdpi.com/journal/applsci
