Quality & Quantity

, Volume 51, Issue 1, pp 435–458 | Cite as

Fundamental characteristics and statistical analysis of ordinal variables: a review

  • Michele Lalla


The measurement of several concepts used in social sciences generates an ordinal variable, which is characterized by rawness of the output values and presents some much debated problems in data analysis. In fact, the need for effective analysis is easily satisfied with parametric models that deal with quantitative variables. However, the peculiarities of the ordinal scales, and the crude values produced by them, limit the use of parametric models, which has generated conflicting favourable and unfavourable views of the parametric approach. The main distinctive features of ordinal scales, some of which are critical points and nodal issues, are illustrated here along with the construction processes. Among the traditional procedures, the most common ordinal scales are described, including the Likert, semantic differential, feeling thermometers, and the Stapel scale. A relative new method, based on fuzzy sets, can be used to handle and generate ordinal variables. Therefore, the structure of a fuzzy inference system is exemplified in synthetic terms to show the treatment of ordinal variables to obtain one or more response variables. The nature of ordinal variables influences the interpretation and selection of many strategies used for their analysis. Four approaches are illustrated (nonparametric, parametric, latent variables, and fuzzy inference system), highlighting their potential and drawbacks. The modelling of an ordinal dependent variable (loglinear models, ordinary parametric models or logit and probit ordinal models, latent class models and hybrid models) is affected by the various approaches.


Measurement Fuzzy sets Feeling thermometer Semantic differential Likert scale Stapel scale 



Part of this paper, specifically Sect. 2, has been previously published (Lalla 2015) in the volume edited by Sefano Campostrini, Giulio Ghellini, and Arjuna Tuzzi (2015), a collection of papers written by pupils and colleagues in honour of Professor Lorenzo Bernardi, who was a fine, versatile, and brilliant academic and social statistician. Overall, he was a master, a mentor and a friend for many of us. This paper is dedicated to his memory.


  1. Agresti, A.: Categorical Data Analysis. John Wiley & Sons, New York (1990)Google Scholar
  2. Aiello, F., Attanasio, M.: How to transform a batch of simple indicators to make up a unique one?. In: Atti della XLII Riunione Scientifica: Sessioni Plenarie e Specializzate, pp. 327–338. SIS, 9–11 giugno. University of Bari, Bari, IT (2004)Google Scholar
  3. Aitchison, J., Silvey, S.D.: The generalization of probit analysis to the case of multiple responses. Biometrika 44(1/2), 131–140 (1957)CrossRefGoogle Scholar
  4. Alvarez, R.M., Bailey, D., Katz, J.N.: An empirical bayes approach to estimating ordinal treatment effects. Polit. Anal. 19(1 Winter), 20–31 (2011)CrossRefGoogle Scholar
  5. Amemiya, T.: Qualitative response models: a survey. J. Econ. Lit. 19(4), 1483–1538 (1981)Google Scholar
  6. ANES, American National Election Studies: Pre- post- election study. Survey Research Center (S473) (1964)
  7. Babbie, E.: The Practice of Social Research. Cengage Learning, 12th edn. Wadsworth, Belmont (2010)Google Scholar
  8. Bernardi, L.: Misurazione e valutazione: le difficoltà di una coppia alle prime esperienze in comune. In: Bertin, G. (ed.) Valutazione e sapere sociologico. Metodi e tecniche di gestione dei processi decisionali, pp. 69–82. Franco Angeli, Milano (1995)Google Scholar
  9. Bernardi, L., Capursi, V., Librizzi, L.: Measurement awareness: the use of indicators between expectations and opportunities. In: Atti della XLII Riunione Scientifica: Sessioni Plenarie e Specializzate, pp. 315–326. SIS, 9–11 giugno. University of Bari, Bari, IT (2004)Google Scholar
  10. Bernberg, R.E.: Socio-psychological factors in individual morale: I. The prediction of specific indicators. J. Soc. Psychol. 36(1), 73–82 (1952)CrossRefGoogle Scholar
  11. Bollen, K.A.: Structural Equations with Latent Variables. John Wiley & Sons, New York (1989)CrossRefGoogle Scholar
  12. Cacciola, S., Marradi, A.: Contributo al dibattito sulle scale Likert basato sull’analisi di interviste registrate. In: Marradi, A. (ed.) Costruire il dato. Sulle tecniche di raccolta delle informazioni nelle scienze sociali, pp. 63–102. Franco Angeli, Milano (1988)Google Scholar
  13. Campostrini, S., Ghellini, G., Tuzzi, A.: Con senso di misura. Riflessi statistici da alcuni allievi di Lorenzo Bernardi. CLEUP, Padova (2015)Google Scholar
  14. Cantril, H., Free, L.A.: Hopes and fears for self and country: the self-anchoring striving scale in cross-cultural research. Am. Behav. Sci. 6(2, Supplement: Oct.), 1–30 (1962)Google Scholar
  15. Conover, W.J.: Practical Nonparametric Statistics, 3rd edn. John Wiley & Sons, New York (1999)Google Scholar
  16. Coombs, C.H.: Psychological scaling without a unit of measurement. Psychol. Rev. 57(3), 145–158 (1950)CrossRefGoogle Scholar
  17. Coombs, C.H.: Theory and method of social measurement. In: Festinger, L., Katz, D. (eds.) Research Methods in the Behavioral Sciences, pp. 471–535. Dryden, New York (1953)Google Scholar
  18. Crespi, I.: Use of a scaling technique in surveys. J. Market. 25(July), 69–72 (1961)CrossRefGoogle Scholar
  19. Crespi, L.P.: Public opinion toward conscientious objectors: II. Measurement of national approval-disapproval. J. Psychol. 19(2), 209–250 (1945a)CrossRefGoogle Scholar
  20. Crespi, L.P.: Public opinion toward conscientious objectors: III. Intensity of social rejection in stereotype and attitude. J. Psychol. 19(2), 251–276 (1945b)CrossRefGoogle Scholar
  21. Das, S.: Quantifying fuzziness due to the scale of measurement in response systems. Fuzzy Sets Syst. 132(3), 317–333 (2002)CrossRefGoogle Scholar
  22. Das, S.: On measuring imprecision in human response due to respondent and attribute and its utility in questionnaire design. Int. J. Uncertain. Fuzz. 14(2), 155–173 (2006)CrossRefGoogle Scholar
  23. Domingo-Ferrer, J., Torra, V.: Extending microaggregation procedures using defuzzification methods for categorical variables. Proceedings of first International IEEE symposium on intelligent systems, Verna, Sept. pp. 44–49 (2002)Google Scholar
  24. Dubois, D., Prade, H. (eds.): Fundamentals of Fuzzy Sets. Kluwer Academic Publ, Boston (2000)Google Scholar
  25. Dubois, D., Prade, H., Gil, M.A., Grzegorzewsky, P., Hryniewicz, O. (eds.): Soft Methods for Handling Variability and Imprecision. Springer-Verlag, Heidelberg (2008)Google Scholar
  26. Farebrother, R.W.: A class of shrinkage estimators. J. R. Stat. Soc. B 40(1), 47–49 (1977)Google Scholar
  27. Gilula, Z., Krieger, A.M., Ritov, Y.: Ordinal association in contingency tables: some interpretive aspects. J. Am. Stat. Assoc. 83(402), 540–545 (1988)CrossRefGoogle Scholar
  28. Goodman, L.A.: Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61(2), 215–231 (1974)CrossRefGoogle Scholar
  29. Goodman, L.A.: Simple models for the analysis of association in cross-classifications having ordered categories. J. Am. Stat. Assoc. 74(367), 537–552 (1979)CrossRefGoogle Scholar
  30. Greene, W.H.: Econometric Analysis, 5th edn. Prentice Hall, Upper Saddle River (2003)Google Scholar
  31. Guttman, L.A.: The basis for scalogram analysis. In: Stouffer, S. (ed.) Measurement and Prediction, pp. 60–90. Princeton University Press, New York (1950)Google Scholar
  32. Guttman, L.A.: A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika 33(4), 469–506 (1968)CrossRefGoogle Scholar
  33. Hagenaars, J.A., McCutcheon, A.L.: Applied Latent Class Analysis. Kluwer, Dordrecht (2002)CrossRefGoogle Scholar
  34. Hand, D.J.: Measurement theory and practice. The world through quantification. Arnold, London (2004)Google Scholar
  35. Hoaglin, D.C., Mosteller, F., Tukey, J.W.: Understanding Robust and Exploratory Data Analysis. John Wiley & Sons, New York (1983)Google Scholar
  36. Hofacker, C.F.: Categorical judgment scaling with ordinal assumptions. Multivar. Behav. Res. 19(1), 91–106 (1984)CrossRefGoogle Scholar
  37. Hofmans, J., Theuns, P., Van Acker, F.: Combining quality and quantity. A psychometric evaluation of the self-anchoring scale. Qual. Quant. 43(5), 703–716 (2009)CrossRefGoogle Scholar
  38. Jöreskog, K.G.: A general method for estimating a linear structural equation system. In: Goldberger, A.S., Duncan, O.D. (eds.) Structural Equation Models in the Social Sciences, pp. 85–112. Seminar Press, New York (1973)Google Scholar
  39. Jöreskog, K.G.: New developments in LISREL: analysis of ordinal variables using polychoric correlations and weighted least squares. Qual. Quant. 24(4), 387–404 (1990)CrossRefGoogle Scholar
  40. Jöreskog, K.G.: On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika 59(3), 381–389 (1994)CrossRefGoogle Scholar
  41. Jöreskog, K.G., Sörbom, D.: Advances in Factor Analysis and Structural Equation Models. Abt books, Cambridge (1979)Google Scholar
  42. Juster, F.T.: Prediction and consumer buying intentions. Am. Econ. Rev. 50(2), 604–617 (1960)Google Scholar
  43. Juster, F.T.: Consumer buying intentions and purchase probability: an experiment in survey design. J. Am. Stat. Assoc. 61(315), 658–696 (1966)CrossRefGoogle Scholar
  44. Kampen, J., Swyngedouw, M.: The ordinal controversy revisited. Qual. Quant. 34(1), 87–102 (2000)CrossRefGoogle Scholar
  45. Khurshid, A., Sahai, H.: Scales of measurement: an introduction and selected bibliography. Qual. Quant. 27(3), 303–323 (1993)CrossRefGoogle Scholar
  46. Kilpatrick, F.P., Cantril, H.: Self-anchoring scaling: a measure of individuals’ unique reality worlds. J. Individ. Psychol. 16(2), 158–173 (1960)Google Scholar
  47. Lalla, M.: Le scale ordinali e i relativi problemi operativi. In: Campostrini, S., Ghellini, G., Tuzzi, A. (eds.) Con senso di misura Riflessi statistici da alcuni allievi di Lorenzo Bernardi, pp. 35–52. CLEUP, Padova (2015)Google Scholar
  48. Lalla, M., Facchinetti, G., Mastroleo, G.: Ordinal scales and fuzzy set systems to measure agreement: an application to the evaluation of teaching activity. Qual. Quant. 38(5), 577–601 (2004)CrossRefGoogle Scholar
  49. Lalla, M., Ferrari, D., Pirotti, T.: Fuzzy inference systems to analyze ordinal variables—the case of evaluating teaching activity. In: Proceedings of the international conference on fuzzy computation theory and applications. SciTePress—Science and Technology Publications Digital Library, Setubal, pp. 25–36 (2014)Google Scholar
  50. Landenna, G., Marasini, D.: Metodi statistici non parametrici. il Mulino, Bologna (1990)Google Scholar
  51. Likert, R.: A technique for the measurement of attitudes. Archives of Psychology 140, 1–55 (1932)Google Scholar
  52. Lord, F.M.: On the statistical treatment of football members. Am. Psychol. 8(12), 750–751 (1953)CrossRefGoogle Scholar
  53. Marradi, A.: Termometri con vincolo di ordinabilità: il «gioco della torre» consente di aggirare la tendenza alla desiderabilità sociale? Sociologia e ricerca sociale 57, 49–59 (1998)Google Scholar
  54. Marradi, A.: Metodologia delle scienze sociali. il Mulino, Bologna (2007)Google Scholar
  55. Mantel, N.: Chi square tests with one degree of freedom: extensions of the Mantel-Haenszel procedure. J. Am. Stat. Assoc. 58(303), 690–700 (1963)Google Scholar
  56. McCullagh, P.: Regression models for ordinal data (with discussion). J. R. Stat. Soc. B 42(2), 109–142 (1980)Google Scholar
  57. McKelvey, R.D., Zavoina, W.: A statistical model for the analysis of ordinal level dependent variables. J. Math. Sociol. 4(1), 103–120 (1975)CrossRefGoogle Scholar
  58. Niederée, R.: There is more to measurement than just measurement: measurement theory, symmetry, and substantive theorizing. J. Math. Psychol. 38(4), 527–594 (1994)CrossRefGoogle Scholar
  59. Nowakowska, M.: Methodological problem of measurement of fuzzy concepts in the social sciences. Behav. Sci. 22(2), 107–115 (1977)CrossRefGoogle Scholar
  60. Osgood, C.E.: The nature of measurement and meaning. Psychol. Bull. 49(3), 197–237 (1952)CrossRefGoogle Scholar
  61. Osgood, C.E., Suci, G.J., Tannenbaum, R.H.: The Measurement of Meaning. University of Illinois Press, Urbana (1957)Google Scholar
  62. Pearson, K.: On a new method of determining correlation between a measured character A, and a character B, of which only the percentage of cases wherein B exceeds (or falls short of) a given intensity is recorded for each grades of A. Biometrika 7(1/2), 96–105 (1909)CrossRefGoogle Scholar
  63. Pesarin, F.: Multivariate Permutation Tests: With Application in Biostatistics. John Wiley & Sons, Chichester (2001)Google Scholar
  64. Pesarin, F., Salmaso, L.: Permutation Tests for Complex Data: Theory, Applications and Software. John Wiley & Sons, Chichester (2010)CrossRefGoogle Scholar
  65. Prytulac, L.S.: A critique of S. S. Stevens’ theory of measurement scale classification. Percept. Motor Skills 41(1), 3–28 (1975)CrossRefGoogle Scholar
  66. Ricolfi, L.: Operazioni di ricerca e scale. Rassegna italiana di sociologia XXVI(2), 189–227 (1985)Google Scholar
  67. Savage, I.R.: Nonparametric statistics. J. Am. Stat. Assoc. 52(279), 331–344 (1957)CrossRefGoogle Scholar
  68. Smithson, M.J.: Fuzzy Set Analysis for Behavioral and social sciences. Springer, Heidelberg (1987)CrossRefGoogle Scholar
  69. Smithson, M.J.: Fuzzy set theory and the social sciences: the scope for applications. Fuzzy Sets Syst. 26(1), 1–21 (1988)CrossRefGoogle Scholar
  70. Siegel, S., Castellan Jr, N.J.: Nonparametric Statistics for the Behavioral Science. McGraw-Hill, New York (1988)Google Scholar
  71. Stevens, S.S.: On the theory of scale measurement. Science 103(2684), 677–680 (1946)CrossRefGoogle Scholar
  72. Stevens, S.S.: Mathematics, measurement, and psychophysics. In: Stevens, S.S. (ed.) Handbook of Experimental Psychology, pp. 1–49. Wiley & Sons, New York (1951)Google Scholar
  73. Symeonaki, M., Michalopoulou, C., Kazani, A.: A fuzzy set theory solution to combining likert items into a single overall scale (or subscales). Qual. Quant. 49(2), 739–762 (2015)CrossRefGoogle Scholar
  74. Tanaka, H.: Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst. 24(3), 363–375 (1987)CrossRefGoogle Scholar
  75. Thurstone, L.L.: A law of comparative judgment. Psychol. Rev. 34(4), 273–286 (1927a)CrossRefGoogle Scholar
  76. Thurstone, L.L.: The method of paired comparison for social values. J. Abnorm. Soc. Psychol. 21(4), 384–397 (1927b)CrossRefGoogle Scholar
  77. Thurstone, L.L.: Attitudes can be measured. Am. J. Sociol. 33(4), 529–554 (1928)CrossRefGoogle Scholar
  78. Tukey, J.W.: Exploratory Data Analysis. Addison-Wesley, Reading (1977)Google Scholar
  79. Van Leekwijck, W., Kerre, E.E.: Defuzzification: criteria and classification. Fuzzy Sets Syst. 108(1), 159–178 (1999)CrossRefGoogle Scholar
  80. Velleman, P.F., Wilkinson, L.: Ordinal, interval, and ratio typologies are misleading. Am. Stat. 47(1), 65–72 (1993)Google Scholar
  81. Viertl, R.: Statistical Methods for Fuzzy Data. John Wiley and Sons, New Delhi (2011)CrossRefGoogle Scholar
  82. Von Altrock, C.: Fuzzy logic and neurofuzzy applications in business and finance. Prentice Hall PTR, Upper Saddle River (1997)Google Scholar
  83. Weisberg, H.F., Rusk, J.G.: Dimensions of candidate evaluation. Am. Polit. Sci. Rev. 64(4), 1167–1185 (1970)CrossRefGoogle Scholar
  84. Westermann, R.: Interval-scale measurement of attitudes: some theoretical conditions and empirical testing methods. Br. J. Math. Stat. Psychol. 36(2), 228–239 (1983)CrossRefGoogle Scholar
  85. White, M.: Psychological technique and social problems. Southwest. Polit. Soc. Sci. Q. 7, 58–73 (1926)Google Scholar
  86. Wilson, T.P.: Critique of ordinal variables. Soc. Forces 49(3), 432–444 (1971)CrossRefGoogle Scholar
  87. Yang, M., Lin, T.: Fuzzy least-squares linear regression analysis for fuzzy input–output data. Fuzzy Sets Syst. 126(3), 389–399 (2002)CrossRefGoogle Scholar
  88. Yu, J.H., Albaum, G., Swenson, M.: Is a central tendency error inherent in the use of semantic differential scales in different cultures? Int. J. Market Res. 45(2), 213–228 (2003)Google Scholar
  89. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Economics “Marco Biagi”, CAPP (Centre for the Analysis of Public Policies)University of Modena and Reggio EmiliaModenaItaly

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