Knuth TAOCP 1.2.11.2 People often abuse O-notation by assuming that it gives an exact order of growth; they use it as if it specifies a lower bound as well as an upper bound. [It looks to me like Knuth does this on page 389 2.3.4.4 right after equation 14.] For example, an algorithm to sort n numbers might be called inefficient "because its running time is O(n^2)." But a running time of O(n^2) dos not necessarily imply that the running time is not also O(n). There's another notation, Big Omega, for lower bounds

I find this a bit confusing because it's unclear to me what Knuth actually endorses for use in average case description.

Wikipedia: Although developed as a part of pure mathematics, this notation is now frequently also used in the analysis of algorithms to describe an algorithm's usage of computational resources: the worst case or average case running time or memory usage of an algorithm is often expressed as a function of the length of its input using big O notation. Wikipedia and the web in general have many examples of "worst case O(...)" which would be redundant under your convention.

So clearly one needs to be vigilant about assumptions when discussing average case, but common usage does not agree that big-oh notation implies worst case.