Allan std deviation: why do we need to read the values at specific sample times?
In IEEE Std 952-1997 (Annex C) we find the answer to this.  They have a fancy integral relation between Allan Variance (\$\sigma^{2}\$) and Power Spectral Density.  If \$K\$ is the random walk coefficient of which you speak, they show that

$$\sigma^{2}(\tau) = \frac{K^{2} \tau}{3}$$

So 

$$\sigma(\tau) = \frac{K \tau^{0.5}}{3^{0.5}}$$

The 0.5 power on \$\tau\$ is why you want the part of slope with 0.5 gradient, the fact that when \$\tau\$ is 3 we are left with

$$\sigma(3) = K$$

is why we care about the intercept with \$\tau==3\$.