Strictly speaking, the formulae given on the main page apply only to PCA done with the covariance matrix of the normalized data. In the Bio2900 lab analysis, we use the correlation matrix. Coefficients calculated from either the covariance or correlation matrices are usually quite similar. With the latter, the data are standardized before calculation of the PCA. For each measurement of each variable, MINITAB subtracts the mean and divides by the standard deviation of all measurements for that variable: all variables will then have identical distributions, with a mean of zero and a variance of one. As a consequence, in the actual analysis the relationship of matrices [2] and [3] to [4] is not quite as direct as described, because the standardization of matrix [2] in effect multiplies each variable by a (hidden) constant. In principal, though, the relationships among the matrices are identical. The correlation matrix is used because of its other useful properties, as discussed.

    An important practical consequence of using the correlation matrix is that the component scores calculated from standardized data will be 'positive' or 'negative', depending whether they are above or below the mean value, respectively, unlike the example presented here, in which the extreme cases are less or more negative. A score of zero can then be interpreted as 'neutral': in the present example, 'neither cat-like nor dog-like'. The resemblance of other individuals to either extreme is then easier to assess.

   In the notation of matrix algebra, the entire procedure can be written   [2][3] = [4]