AHSME 1973 Problem 5

Of the following five statements, I to V, about the binary operation of averaging (arithmetic mean), $\text{I. Averaging is associative }$ $\text{II. Averaging is commutative }$ $\text{III. Averaging distributes over addition }$ $\text{IV. Addition distributes over averaging }$ $\text{V. Averaging has an identity element }$ those which are always true are $\textbf{(A)}\ \text{All} \textbf{(B)}\ \text{I and II only} \textbf{(C)}\ \text{II and III only}$ $\textbf{(D)}\ \text{II and IV only} \textbf{(E)}\ \text{II and V only}$