# An Interface to Monitor Process Variability Using the Binomial ATTRIVAR SS Control Chart
## 1 Introduction  
Control charts were initially introduced by [1] and are the primary and most technically sophisticated tools of statistical process control [2]. They are widely employed in process monitoring and serve to detect nonconformities and special causes of variation, thus enabling the early identification of disturbances aimed to minimize negative adverse financial impacts [3]. Among control charts, some utilize variable inspection for monitoring continuous quality characteristics, while others employ attribute inspection, which was initially designed for monitoring non-continuous quality characteristics. One of which is grounded on the numerical measurements of quality characteristics, and another is based on the attributes defined by [4] as quality characteristics measured in a nominal scale or categorized according to a predetermined scheme of labels. For instance, classifying fruits as being good or rotten, and nails or screws as defective or not defective. Although the primary purpose of attribute inspection was outlined about 70 years ago, researchers have made contributions by suggesting their use for controlling continuous quality characteristics, and more recent studies have proposed better means for such [5]. There are numerous comparisons between variable and attribute charts in the literature, such as [5; 6; 7; 8; 9; 10]. These authors emphasize the superior efficiency and informativenessof variable inspection due to its reliance on measurements, as opposed to a simpler, faster, and cheaper application of attribute inspection. Several researchers explore the use of attribute inspections for continuous characteristics, such as [8; 9; 11], in an attempt to continuously seek to improve the efficiency of resource utilization and improve the performance of control charts. Ergo, what once meant having to use a sample 6.667 times larger in attribute charts to achieve the same level of performance as variable charts [2] now requires a sample less than twice as large using the \(np_{S^{2}}\) proposed by [9]. The strategy of using variable and attribute inspections combined into a single chart had initially been proposed by [12] and is similar to one which was later on called as ATTRIVAR (ATTRIbute + VARIable) by [5]. Thenceforth, these mixed charts combine the advantages of both forms of inspection initially called as such. A version of [12] consisted in designing the \(np_{x}\) (attribute) and \(\overline{X}\) (variable) charts separately and dividing the collected sample into two subsamples. Then, the former was submitted to \(np_{x}\) chart evaluation, and the latter was submitted to \(\overline{X}\) chart evaluation if the former had been rejected. Ref. [7] also proposed a mixed chart without subdividing the sample and used the same sample in both stages of inspection (as in [13; 14], and to the proposal herein). Ref. [5] proposed ATTRIVAR-1 chart, which uses the same sample in both stages, in addition to ATTRIVAR-2 chart, which makes use of different samples. In this work, the terminologies SS (same sample) and DS (different sample) refer to these strategies, respectively. Unlike these proposals, which addressed only mean monitoring, Ref. [10] proposed a mixed chart to monitor variability through variance. The authors coupled the \(np_{S^{2}}\) chart from [9] in the stage of attribute inspection with the \(S^{2}\) chart in the stage of variable inspection using different samples. Afterwards, Ref. [15] proposed the Trinomial version of the ATTRIVAR (T-ATTRIVAR) chart aimed at mean monitoring but using three attributes in the first inspection stage instead of two as in other strategies. Thus, this paper explores the Binomial version of ATTRIVAR Same Sample \(S^{2}\) (B-ATTRIVAR SS \(S^{2}\)) chart that makes use of the first inspection stage with two attributes (\"binomial\") using the same sample in both inspection stages (\"same sample\") to monitor the process variance (\(S^{2}\)).