### Correlation coefficient

The covariance depends on the scale of $z^{(1)}$ and $z^{(2)}$. In order to have a scale invariant measure, we can use the correlation coefficient 
$$\rho = \frac{\textrm{cov}(z^{(1)},z^{(2)})}{\sqrt{\textrm{var}(z^{(1)})\textrm{var}(z^{(2)})}}$$

The correlation coefficient lies within $[-1,1]$.

When it is equal to $-1$ or $1$, the variables are linked by a linear relationship

$$z^{(2)}=a.z^{(1)}+b$$

where the sign of $a$ corresponds to the sign of $\rho$.

When $\rho=0$, we say that the variables are uncorrelated. But they can still have a link (not linear).