An agreement table with [n as an element of N is greater than or equal to] 3 ordered categories can be collapsed into n - 1 distinct 2 x 2 tables by combining adjacent categories. Vanbelle and Albert ("Stat. Methodol." 6:157-163, 2009c) showed that the components of Cohen's weighted kappa with linear weights can be obtained from these n - 1 collapsed 2 x 2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2 x 2 tables, where the weights are the denominators of the 2 x 2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.