Skip to main content

Multidimensional scaling: I. Theory and method


Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.

This is a preview of subscription content, access via your institution.


  1. Gulliksen, Harold. Paired comparisons and the logic of measurement.Psychol. Rev. 1946,53, 199–213.

    Google Scholar 

  2. Klingberg, F. L. Studies in measurement of the relations among sovereign states.Psychometrika, 1941,6, 335–352.

    Google Scholar 

  3. Richardson, M. W. Multidimensional psychophysics.Psychol. Bull., 1938,35, 659. (Abstract).

    Google Scholar 

  4. Thurstone, L. L. Psychophysical analysis.Amer. J. Psychol., 1927,38, 368–389.

    Google Scholar 

  5. Young, G., and Householder, A. S. Discussion of a set of points in terms of their mutual distances.Psychometrika, 1938,3, 19–22.

    Google Scholar 

Download references

Author information

Authors and Affiliations


Additional information

This study was carried out while the author was an Educational Testing Service Psychometric Fellow at Princeton University. The author expresses his appreciation to his thesis adviser, Dr. H. Gulliksen, for his guidance throughout the study and to Dr. B. F. Green, Jr., for valuable assistance on several of the derivations.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Torgerson, W.S. Multidimensional scaling: I. Theory and method. Psychometrika 17, 401–419 (1952).

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI:


  • Public Policy
  • Triad
  • Statistical Theory
  • Comparison Method
  • Multidimensional Scaling