Properties of averages When we were considering the average of a sum, eqn 1.15, we saw that the average of the sum was the sum of the average: $$\langle \sum_{i=1}^N x_i \rangle = \sum_{i=1}^N \langle x_i \rangle$$ (1.21) Or more simply $\langle x + y \rangle = \langle x \rangle + \langle y \rangle$. This is just because an average is itself a sum, and one overall multiplication, so the above equation is just rearranging the order you sum things in. Similarly if you multiply by any constant $c$ $$\langle c x \rangle = c \langle x \rangle$$ (1.22) Another thing that you see a lot is the average of a constant. For example $\langle 2 \rangle$. What's that? If every time you do the experiment you get $2$. Then on average the result is 2. So for a constant $c$ $$\langle c \rangle = c$$ (1.23) This means that the average of an average is just an average. I mean: $$\langle \langle x \rangle\rangle = \langle x \rangle$$ (1.24)