Tamar Gadrich, Emil Bashkansky

Consider an object that is measured using an ordinal scale with *m* ordered categories. The common approach when dealing with features measured using an ordinal scale is to convert the ordinal estimation into a numerical one by assigning numerical values to each category of the ordinal variable. This procedure is undesirable because it can lead to misunderstanding and misinterpretation of the measurement results [1-3]. Since only comparisons of “greater than”, “greater than”, “greater than” and “greater than” can be made among ordinal variable values, all statistical measures of such ordinal variables must be based on these limitations. We focus on an ordinal dispersion measure that allows practical engineering applications such as quality/failure classification, uncertainty evaluations, repeatability and reproducibility (R&R) analysis and so on. Desirable properties of such a variation measure include: (1) it equals zero when all the data relate to a single category; (2) it has a maximal value for the most polarized distribution; (3) it is reversible, i.e., invariant to category order inversion; (4) it is similar to a Bernoulli distribution variation when applied to only two categories; (5) it is similar to existing measures of inequality (Gini, for example); (6) the measure can be decomposed to “greater than” and “greater than” components.

Based on a literature survey we assembled a set of ordinal dispersion measures. Our study showed that all the above mentioned properties are best satisfied by Blair and Lacy's [1] measure. The advantages of this measure allowed us to:

1. Develop simplified expressions for the variations of quality measures [4], using a multivariate delta method for large samples. These expressions allow statistical quality control (SQC) for items whose quality is measured on an ordinal ternary scale.

2. Decompose the total variation to between and within components, and study the meanings of such decomposition.

3. Analyze the possible engineering applications such as hypothesis testing for small ordinal samples, R&R study [5, 6], key characteristics identification, etc.

**References:**

[1] Blair J, Lacy MG. Statistics of ordinal variation. Sociological Methods & Research 2000; 28: 251-280.

[2] Franceschini F, Galetto M, Varetto M. Ordered samples control charts for ordinal variables. Quality and Reliability Engineering International 2005; 21:177-195.

[3] Bashkansky E, Gadrich T. Evaluating quality measured on a ternary ordinal scale. Quality and Reliability Engineering International 2008; 24:957-971.

[4] Bashkansky E, Gadrich T. Statistical Quality Control for Ternary Ordinal Quality Data. Applied stochastic models in business and industry DOI: 10.1002/asmb.868

[5] Bashkansky E, Gadrich T. Some Metrological Aspects of Ordinal Measurements. Accreditation and Quality Assurance 2010; 15(6): 331-336.

[6] Bashkansky E, Gadrich T, Knani D. Some metrological aspects of the comparison between two ordinal measuring systems. Accreditation and Quality Assurance 2011; 16:63-72.

**Keywords:** Ordinal data; ANOVA; Dispersion measures

**Biography:** Dr. Tamar Gadrich serves as the Head of the Industrial Engineering and Management Department at Ort Braude College, Karmiel, Israel. Tamar has a D.Sc. in statistics from the Technion–Israel Institute of Technology. Her current research interests are in the area of evaluating quality measured on an ordinal scale and applied probability. Dr. Gadrich has published her research in journals such as Quality and Reliability Engineering International, Methodology and Computing in Applied Probability, and Applied Stochastic Models in Business and Industry.