Behavior Research Methods

, Volume 37, Issue 4, pp 677–683

A mean for all seasons

Article

Abstract

The averaging of scores differs from the averaging of numbers in that behavioral issues are built into scores. The behavioral issues are the weight attached to a score and the metric on which the scores have been gathered. A single equation is proposed, derived from Aczél’s (1966) model of the quasilinear mean, that encompasses the standard measures of central tendency. The equation allows for differential weighting of scores and also addresses the metric issue by incorporating response transformation.

References

  1. Aczél, J. (1966).Lectures on functional equations and their applications. New York: Academic Press.Google Scholar
  2. Anderson, N. H. (1961). Scales and statistics: Parametric and nonparametric.Psychological Bulletin,58, 305–316.CrossRefPubMedGoogle Scholar
  3. Anderson, N. H. (1962). On the quantification of Miller’s conflict theory.Psychological Review,69, 400–414.CrossRefPubMedGoogle Scholar
  4. Anderson, N. H. (1976). Equity judgments in information integration theory.Journal of Personality & Social Psychology,33, 291–299.CrossRefGoogle Scholar
  5. Anderson, N. H. (1979). Algebraic rules in psychological measurement.American Scientist,67, 555–563.Google Scholar
  6. Anderson, N. H. (2001).Empirical direction in design and analysis. Mahwah, NJ: Erlbaum.Google Scholar
  7. Bartlett, M. S. (1947). The use of transformations.Biometrics,3, 39–52.CrossRefPubMedGoogle Scholar
  8. Birnbaum, M. H., &Veit, C. T. (1974). Scale convergence as a criterion for rescaling: Information integration with difference, ratio, and averaging tasks.Perception & Psychophysics,15, 7–15.CrossRefGoogle Scholar
  9. Box, G. E. P., &Cox, D. R. (1964). An analysis of transformations (with discussion).Journal of the Royal Statistical Society: Series B,26, 211–252.Google Scholar
  10. Breault, K. D. (1983). Psychophysical measurement and the validity of the modern economic approach: A presentation of methods and preliminary experiments.Social Science Research,12, 187–203.CrossRefGoogle Scholar
  11. Clemen, R. T. (1989). Combining forecasts: A review and annotated bibliography.International Journal of Forecasting,5, 559–583.CrossRefGoogle Scholar
  12. Cochran, W. G. (1940). The analysis of variance when experimental errors follow the Poisson or binomial laws.Annals of Mathematical Statistics,14, 335–347.CrossRefGoogle Scholar
  13. Doksum, K. A., &Wong, C. W. (1983). Statistical tests based on transformed data.Journal of the American Statistical Association,78, 411–417.CrossRefGoogle Scholar
  14. Edwards, W. (1966). Introduction.IEEE Transactions on Human Factors in Electronics,7, 1–6.CrossRefGoogle Scholar
  15. Edwards, W., Lindman, H., &Savage, L. J. (1963). Bayesian statistical inference for psychological research.Psychological Review,70, 193–242.CrossRefGoogle Scholar
  16. Galanter, E. (1990). Utility functions for nonmonetary events.American Journal of Psychology,103, 449–470.CrossRefGoogle Scholar
  17. Hinkley, D. V., &Runger, G. (1984). The analysis of transformed data.Journal of the American Statistical Association,79, 302–309.CrossRefGoogle Scholar
  18. Krantz, D. H., Luce, R. D., Suppes, P., &Tversky, A. (1971).Foundations of measurement (Vol. 1). New York: Academic Press.Google Scholar
  19. Krueger, L. E. (1989). Reconciling Fechner and Stevens: Toward a unified psychophysical law.Behavioral & Brain Sciences,12, 251–329.CrossRefGoogle Scholar
  20. Kruskal, J. B., &Carmone, F. L. (1969). MONANOVA: A FORTRAN IV program for monotone analysis of variance.Behavioral Science,14, 165–166.Google Scholar
  21. Levine, D. W., &Dunlap, W. P. (1982). Power of theF test with skewed data: Should one transform or not?Psychological Bulletin,92, 272–280.CrossRefGoogle Scholar
  22. Levine, D. W., &Dunlap, W. P. (1983). Data transformations, power, and skew: A rejoinder to Games.Psychological Bulletin,93, 596–599.CrossRefGoogle Scholar
  23. Luce, R. D. (1996). The ongoing dialog between empirical science and measurement theory.Journal of Mathematical Psychology,40, 78–98.CrossRefGoogle Scholar
  24. Luce, R. D., &Tukey, J. W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement.Journal of Mathematical Psychology,1, 1–27.CrossRefGoogle Scholar
  25. Mandel, J. (1976). Models, transformations of scale, and weighting.Journal of Quality Technology,8, 86–97.Google Scholar
  26. Mellers, B., &Hartka, E. (1989). Test of a subtractive model of “fair” allocations.Journal of Personality & Social Psychology,56, 691–697.CrossRefGoogle Scholar
  27. Myung, I. J., Kim, C., &Pitt, M. A. (2000). Toward an explanation of the power law artifact: Insights from response surface analysis.Memory & Cognition,28, 832–840.CrossRefGoogle Scholar
  28. Ovchinnikov, S. (1996). Means on ordered sets.Mathematical Social Sciences,32, 39–56.CrossRefGoogle Scholar
  29. Ovchinnikov, S., &Dukhovny, A. (2002). On order invariant aggregation functionals.Journal of Mathematical Psychology,46, 12–18.CrossRefGoogle Scholar
  30. Ratcliff, R. (1993). Methods for dealing with reaction time outliers.Psychological Bulletin,114, 510–532.CrossRefPubMedGoogle Scholar
  31. Rosenbaum, M. E., &Levin, I. P. (1969). Impression formation as a function of source credibility and the polarity of information.Journal of Personality & Social Psychology,12, 34–37.CrossRefGoogle Scholar
  32. Silver, N. C., &Dunlap, W. P. (1987). Averaging correlation coefficients: Should Fisher’sz transformation be used?Journal of Applied Psychology,72, 146–148.CrossRefGoogle Scholar
  33. Singh, R. (1995). “Fair” allocations of pay and workload: Tests of a subtractive model with nonlinear judgment function.Organizational Behavior & Human Decision Processes,62, 70–78.CrossRefGoogle Scholar
  34. Stanley, J. C., &Porter, A. C. (1972). ANOVA analysis of unweighted and weighted Fisherz’s.Social Science Research,1, 237–241.CrossRefGoogle Scholar
  35. Stevens, S. S. (1955). On the averaging of data.Science,121, 113–116.CrossRefPubMedGoogle Scholar
  36. Stevens, S. S. (1958). Problems and methods of psychophysics.Psychological Bulletin,55, 177–196.CrossRefPubMedGoogle Scholar
  37. Stigler, S. M. (1986).The history of statistics: The measurement of uncertainty before 1900. Cambridge, MA: Belknap.Google Scholar
  38. Strube, M. J. (1988). Averaging correlation coefficients: Influence of heterogeneity and set size.Journal of Applied Psychology,73, 550–568.CrossRefGoogle Scholar
  39. Tversky, A., &Russo, J. E. (1969). Substitutability and similarity in binary choices.Journal of Mathematical Psychology,6, 1–12.CrossRefGoogle Scholar
  40. Von Winterfeldt, D., &Edwards, W. (1986).Decision analysis and behavioral research. New York: Cambridge University Press.Google Scholar
  41. Weiss, D. J. (1972). Averaging: An empirical validity criterion for magnitude estimation.Perception & Psychophysics,12, 385–388.CrossRefGoogle Scholar
  42. Weiss, D. J. (1975). Quantifying private events: A functional measurement analysis of equisection.Perception & Psychophysics,17, 351–357.CrossRefGoogle Scholar
  43. Weiss, D. J., &Gardner, G. S. (1979). Subjective hypotenuse estimation: A test of the Pythagorean theorem.Perceptual & Motor Skills,48, 607–615.Google Scholar
  44. Weiss, D. J., &Shanteau, J. (2003). Empirical assessment of expertise.Human Factors,45, 104–116.CrossRefPubMedGoogle Scholar
  45. Wilcox, R. R. (2003).Applying contemporary statistical techniques. San Diego: Academic Press.Google Scholar
  46. Zalinski, J., &Anderson, N. H. (1991). Parameter estimation for averaging theory. In N. H. Anderson (Ed.),Contributions to information integration theory: Vol. 1. Cognition (pp. 353–394). Hillsdale, NJ: Erlbaum.Google Scholar

Copyright information

© Psychonomic Society, Inc. 2005

Authors and Affiliations

  1. 1.Department of PsychologyCalifornia State University, Los AngelesLos Angeles
  2. 2.University of Southern CaliforniaLos Angeles

Personalised recommendations