Pi (pi) and kappa (kappa) statistics are widely used in the areas of psychiatry and psychological testing to compute the extent of agreement between raters on nominally scaled data. It is a fact that these coefficients occasionally yield unexpected results in situations known as the paradoxes of kappa. This paper explores the origin of these limitations, and introduces an alternative and more stable agreement coefficient referred to as the AC1 coefficient. Also proposed are new variance estimators for the multiple-rater generalized pi and AC1 statistics, whose validity does not depend upon the hypothesis of independence between raters. This is an improvement over existing alternative variances, which depend on the independence assumption. A Monte-Carlo simulation study demonstrates the validity of these variance estimators for confidence interval construction, and confirms the value of AC1 as an improved alternative to existing inter-rater reliability statistics.