This paper is concerned with the treatment of residuals associated with principal component analysis. These residuals are the difference between the original observations and the predictions of them using less than a full set of principal components. Specifically, procedures are proposed for testing the residuals associated with a single observation vector and for an overall test for a group of observations. In this development, it is assumed that the underlying covariance matrix is known; this is reasonable for many quality control applications where the proposed procedures may be quite useful in detecting outliers in the data. A numerical example is included.