Grad Checking, verify by average?
Let me give you an example where Andrew's recommendation works better than yours:

Let's say that the real gradient is $(0, 0, 0)$ and the gradient you have computed is $(10^{-4}, 10^{-4}, 10^{-4})$. Then your average would return $10^{-8}$, and Andrew's recommendation would return $1$. Your metric could fool you into thinking that your gradient is computed propperly and the error is just due to a numeric issue, while Andrew's cannot fool you into that, due to the fact that it considers the fact that the gradient can be very small.

To wrap up, if your gradient doesn't have norm close to zero, it wouldn't really matter. However, when the gradient is close to zero you can be fooled into thinking that your gradient is right when it is not.