smc

We can consider each statement independently and see which ones are true. The average of two numbers $x$ and $y$ is $avg(x,y)=\frac{x+y}{2}$. Statement I $$avg(avg(x,y),z)=avg(x,avg(y,z))$$ $$\frac{\frac{x+y}{2}+z}{2}=\frac{x+\frac{y+z}{2}}{2}$$ $$\frac{x+y+2z}{4}=\frac{2x+y+z}{4}$$ The above statement is not true for all $(x,y,z)$, so Statement I is false. Statement II $$avg(x,y)=avg(y,x)$$ $$\frac{x+y}{2}=\frac{y+x}{2}$$ Since addition is commutative, the above is true for all $(x,y)$, so Statement II is true. Statement III $$avg(x,y+z)=avg(x,y)+avg(x,z)$$ $$\frac{x+y+z}{2}=\frac{x+y}{2}+\frac{x+z}{2}$$ $$\frac{x+y+z}{2}=\frac{2x+y+z}{2}$$ The above statement is not true for all $(x,y,z)$, so Statement III is false. Statement IV $$x+avg(y,z)=avg(x+y,x+z)$$ $$x+\frac{y+z}{2}=\frac{2x+y+z}{2}$$ $$\frac{2x+y+z}{2}=\frac{2x+y+z}{2}$$ The left side and the right side are equivalent expressions, so Statement IV is true. Statement V Let $i$ be the identity element. If such $i$ exists, then the following is true: $$avg(x,i)=x$$ $$\frac{x+i}{2}=x$$ $$x+i=2x$$ $$i=x$$ However, since $x$ is a variable, but $i$ must be a constant, fixed number, there is no identity element for the binary operation of averaging. Therefore, Statement V is false. Statements II and IV are true, so the answer is $\boxed{\text{II and IV only}}$.