A brief, easily applicable test for determining the number of factors to extract in factor analytic experiments has long been in demand. Unfortunately, as pointed out elsewhere in a general theoretical examination of the number of factors problem (Cattell, 1966 b), a test does not exist-even a long or complex one-which is both mathematically precise and logically satisfying. The present paper does not touch the theoretical problem, except explicitly to state acceptance of the position there reached, but concerns itself with an empirical procedure for reaching decisions. That theoretical position, insofar as it can be briefly stated in familiar terms, is:(1) That the “true” number of factors (considered either as the number of substantive determining influences in the physical world, or as these plus the number of factors from correlated error) must in general be greater than the number of variables used in the experiment. Consequently, the decision in choosing a point at which to cut off extraction must aim merely at encompassing what may be called the “NCV”(or non-trivial common variance, ie, covariance). The decision as to what shall be considered trivial must depend on circumstances, 2 but in the sense of any fixed percentage of the total substantive variance it can be made only after rotation, when substantive and error common factors have been separated by rotation. Consequently the tentative decision at extraction is made only so as to permit final decision after rotation to be made most reliably. If “trivial” is defined in the sense of, say, 1% of substantive (non-error) factor variance, then it means 1% of the non-error portion of that variance which would be extracted if we had the correct communality for the> n factors needing to be extracted.(This variance can be very closely approximated in most studies by taking the uniquely determined communalities for n/2 factors; though special cases will arise where n/2 does not cover the number of substantial substantive and error factors.)(2) That common (experimental) error factor variance-notably the large error factors which may be larger than the smaller substantive factors-should also be considered part of the NCV

1. This investigation was supported by Public Health Service Research Grant No. 1027401 from the National Institute of Mental Health. The writer wishes to acknowledge his indebtedness to Malcolm Coulter, John Horn, and John Nesselroade for helpful discussions on this problem. 2. By “circumstances” is meant:(1) whether the experiment is an exploratory one (hypothesis producing) or hypothesis testing;(2) the size of sample and measurement reliability of tests;(3) whether the factor correlations (through fixing hyperplanes) need to be exact, for a higher order anal-