A method of testing for significant relationships between multidimensional configurations is presented. The ‘index of invariance” is defined as the correlation between the inter‐point distances from one configuration and the corresponding inter‐point distances from a second configuration. Empirical distributions of the index are developed for varying numbers of points and dimensions. Fitting the Gram‐Charlier family of curves to these data, additional distributions of the index are estimated based on two parameters, namely, the number of points and the number of dimensions. Tables of significance are presented for the index in terms of the two parameters. Applications based on ordinal multidimensional scaling and other multivariate techniques are