QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL Qual. Reliab. Engng. Int. 2005; 21:177–195 Published online 27 January 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/qre.614 Research Ordered Samples Control Charts for Ordinal Variables Fiorenzo Franceschini ∗ ,† , Maurizio Galetto and Marco Varetto Dipartimento di Sistemi di Produzione ed Economia dell’Azienda, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy The paper presents a new method for statistical process control when ordinal variables are involved. This is the case of a quality characteristic evaluated by an ordinal scale. The method allows a statistical analysis without exploiting an arbitrary numerical conversion of scale levels and without using the traditional sample synthesis operators (sample mean and variance). It consists of a different approach based on the use of a new sample scale obtained by ordering the original variable sample space according to some specific ‘dominance criteria’ fixed on the basis of the monitored process characteristics. Samples are directly reported on the chart and no distributional shape is assumed for the population (universe) of evaluations. Finally, a practical application of the method in the health sector is provided. Copyright c  2005 John Wiley & Sons, Ltd. KEY WORDS: ordered samples control charts; ordinal variables; linguistic variables; ordinal scales; quality monitoring; service quality; dominance criteria 1. INTRODUCTION M any quality characteristics are evaluated on linguistic or ordinal scales. This is the case when performing visual controls on manufactured products or when evaluating some characteristics of the quality of a service. The levels of these scales are terms such as ‘good’, ‘bad’, ‘medium’, etc., which can be ordered according to the specific meaning of the quality characteristic at hand. Ordered linguistic scales mainly differ from numerical or cardinal scales because the concept of distance is not defined. The ordering is the main property associated to such scales 1,2 . The problem of on-line monitoring of an ordinal quality characteristic requires the development of techniques able to deal with ordinal data. The assignment of weights, demerits and so on, to reflect the degree of severity of product non-conformity, has been adopted in many circumstances 3,4 . Different numbers of demerits are assigned to each class and the total number of demerits is monitored by some control chart for defectives. This is a subjective approach that requires the ability to uniquely classify each state into one of several mutually exclusive classes, with well-defined boundaries among them. Although the numerical conversion of verbal information simplifies the subsequent analysis, it also gives rise to two basic problems. The first is concerned with the validity of encoding a discrete verbal scale into a numerical form. The numerical codification implies fixing the distances among scale levels, thus converting the ordinal scale into a cardinal one. The second is related to the absence of consistent criteria for the selection of the type of numerical conversion. It is obvious that changing the numerical ∗ Correspondence to: Fiorenzo Franceschini, Dipartimento di Sistemi di Produzione ed Economia dell’Azienda, Politecnico di Torino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy. † E-mail: fiorenzo.franceschini@polito.it Copyright c  2004 John Wiley & Sons, Ltd. Received 14 June 2002 Revised 3 September 2003