[alphacarbon]

To calculate the pI value, we take the average of the pKa values from the 2 groups that result in a net charge of 0.

For that to happen, the pKa values should be at the midpoint of titration where the charge is either +0.5 or -0.5.

Since pKa can also be calculated from -log(ka), but the pKa value from that calculation doesn't necessary mean the group is half titrated (+/-0.5 charge). So are the published pKa values for amino acids all at half titrated position (pH=pKa -> +/-0.5 charge).

[Babcock_Hall]

Your first sentence does not look correct.  The correct formula for the calculation of pI is different for different side chains, depending on whether or not the side chain ionizes and if it does ionize, whether the side chain is a neutral versus a cationic acid.

[hollytara]

You should go back and look at how the Ka is defined and measured, and how that impacts a titration.