Given sample A and sample B, sets of two-dimensional measurements: Are their (two dimensional) means (or ``centroids'') genuinely different? To take a slightly more concrete case, Is the effect of a linguistic factor on F1-F2 measurements significantly at a 5% level of confidence? The statistical problem is that of numerically estimating the significance of the difference between means of two (or possibly more) bivariate samples (which cannot be assumed to have similar (co)variances).

The method used here was the following. Consider the line between the means of the two sets A and B. Project all the points from both sets onto that line (that is, find the point on the line from which a perpendicular will intersect the datum). Thus the data are reduced to one dimension, namely, distance along that line from some arbitrary reference point. The standard test for statistically distinguishing two one-dimensional data samples, namely the t-test, is applied. Since the t-test assumes equal variances between the two sets, this assumption is initially tested using the usual F-ratio test. If the variances are significantly different, then the unequal-variances t-test is used instead of the standard t-test. These tests are discussed in Press, et al, (1988, Numerical Recipes in C, Chapter 13.) Software in the S data analysis language was written to compute these statistics.

All the information about the differences between two distributions is not retained when the two-dimensional data are reduced to one dimension. Nonetheless if this test indicates that the difference between the means is significant, then there is a significant difference between the two sets of measurements in the direction of the line between the two means. There may also be other differences as well, which are not pointed out by this test.