# How do I calculate the isoelectric point of amino acids, each of which has more than two values of pKa?

For most amino acids, the $$\mathrm{pI}$$ is simply the arithmetic mean of the amino and carboxyl $$\mathrm pK_\mathrm a$$s. However, for tyrosine and cysteine, which have more than one $$\mathrm pK_\mathrm a$$ value, this rule of thumb doesn't apply.

I see that for tyrosine, it's the $$\mathrm pK_\mathrm a$$s of the carboxyl and amino groups that are averaged, but for cysteine it's those of the carboxyl group and the side chain.

I haven't been able to find an explanation of why this is the case, or what the reasoning behind the calculations is?

Since the $\mathrm{pI}$ is the $\mathrm{pH}$ at which the amino acid has no overall net charge, you need to average the $\mathrm pK_\mathrm a$ values relevant to the protonation/deprotonation of the form with no net charge. Here are the acid-base equilibria for tyrosine:
The form with no net charge is in red (+1 and -1 cancel out to give no net charge). It is the $\mathrm pK_\mathrm a$ values on either side of this form (in blue) that matter, hence the $\mathrm{pI}$ of tyrosine is $5.66$ (the average of $2.20$ and $9.11$).
It just so happens that $2.20$ is the carboxyl $\mathrm pK_\mathrm a$ and $9.11$ is the amino $\mathrm pK_\mathrm a$. If the side chain $\mathrm pK_\mathrm a$ were lower than $9.11$, then you should average the carboxyl and side chain $\mathrm pK_\mathrm a$'s instead.
The same logic applies to cysteine (look up the $\mathrm pK_\mathrm a$ values and draw out the differently protonated forms). You'll find that since the side chain has a lower $\mathrm pK_\mathrm a$ than the amino group, you average the carboxyl and the side chain $\mathrm pK_\mathrm a$'s.