pKa value of an indicator
Visually, we can only estimate the  $\mathrm{p}K_\mathrm{a}$ value to be near the middle of the indicator range, shifted to more intense colour.

The $\mathrm{p}K_\mathrm{a}$ of an indicator could be determined more precisely by  combining $\mathrm{pH}$ meter with photometry, where we would get the well known " round step function" of absorbance versus $\mathrm{pH}$.



E.g phenolphthalein has indicating range $\mathrm{pH}=8.2 - 10.0$ for reaction
$$\ce{HInd(clear) &lt;=&gt; H+ + Ind^-(violet)}$$

(I will not fight over the colour name. It may be magenta, carmine, purpur, dark pink -  you know the 16 colours of Windows jokes. )

Let suppose the phenolphthalein $\mathrm{p}K_\mathrm{a}$ is in the middle of the range, i.e. $\mathrm{p}K_\mathrm{a}=9.1$.
Let suppose the ratio of concentrations of clear and violet forms at $\mathrm{pH}=8.2$ is $\ce{[Ind-]/[HInd]}=X$.    

The equation for indicates are the same as for any other weak acid:


$$\mathrm{pH}=\mathrm{p}K_\mathrm{a} + \log \frac {\ce{[Ind-]}}{\ce{[HInd]}}$$

If we consider the mentioned border $\mathrm{pH}$ values,
then
$$\begin{align}
8.2 &amp;=9.1+\log X \\
10.0 &amp;=9.1+\log \frac 1X \\
X&amp;=0.126 \\
\end{align}$$

The problem is the relative eye sensibility to small additions  of one form of an indicator to the other form.
 We notice much easier a small amount of violet substance in clear substance, than a small amount of clear substance in violet substance.

$$\begin{align}
\mathrm{pH_{low}}=\mathrm{p}K_\mathrm{a} + \log \frac {\ce{[Ind^-_{noticeble}]}}{\ce{[HInd_{abundance}]}} \\
\mathrm{pH_{high}}=\mathrm{p}K_\mathrm{a} + \log \frac {\ce{[Ind^-_{abundance}]}}{\ce{[HInd_{noticeble}]}} \\
\end{align}$$

As the consequence, the $\mathrm{pH}=10.0$ is significantly closer to $\mathrm{p}K_\mathrm{a}$ than $\mathrm{pH}=8.2$, therefore

$$\mathrm{p}K_\mathrm{a} \gt \frac {8.2+10.0}{2}$$

Let suppose for now the real $\mathrm{p}K_\mathrm{a}=9.7$.    

Then $\ce{[Ind-]/[HInd]}$ is 


2 for pH 10.0, i.e full violet weakened by 1/3
0.03 for pH 8.2, i.e 3% of the full violet.


The similar visual shifts happen to other indicators as well.