& I_0 \left[
  1 - \sigma_{\rm HI} \langle {\cal N} \rangle 
  + \frac{\sigma_{\rm HI}^2}{2} \langle {\cal N} \rangle^2  + \cdots \right], 
  \label{eq:mean}\end{aligned}$$ where $\langle {\cal N} \rangle$ is the mean ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ column density given by $$\begin{aligned}
\langle {\cal N} \rangle = \frac{\sum^{N_y}_j {\cal N}_j}{N_y} 
=  \frac{\Delta L \sum_j^{N_y} \sum_i^{N_x} n_{i, j}}{N_y} = \Delta L N_x \langle n \rangle. 
\label{eq:column}\end{aligned}$$ On the other hand, the Taylor series expansion of Eqn. (\[eq:exact\]) is $$\begin{aligned}
\langle I_{\rm out} \rangle &=& \frac{I_0}{N_y} 
\sum_j^{N_y} \left[1 - \sigma_{\rm HI} {\cal N}_j + \frac{1}{2} (\sigma_{\rm HI} {\cal N}_j)^2  
+ \cdots \right] \nonumber  \\
    &=& \frac{I_0}{N_y} \left[N_y - \sigma_{\rm HI} \sum^{N_y}_j {\cal N}_j 
    + \frac{\sigma_{\rm HI}^2}{2} \sum^{N_y}_j {\cal N}_j^2  + \cdots \right]  \nonumber \\
    &=& I_0 \left[1 - \sigma_{\rm HI} \langle {\cal N} \rangle 
    + \frac{\sigma_{\rm HI}^2}{2} \langle {\cal N}^2 \rangle + \cdots \right].  
\label{eq:tayler}\end{aligned}$$ The difference between $\langle I_{\rm out} \rangle$ and $\langle I_{\rm out} \rangle_{\rm mean}$ is the second order in $\tau$. From Eqn. (\[eq:exact\]) and (\[eq:mean\]), the leading error in $\langle I_{\rm out} \rangle_{\rm mean}$ is $$\begin{aligned}
\langle I_{\rm out} \rangle - \langle I_{\rm out} \rangle_{\rm mean} &=& I_0 \frac{\sigma_{\rm HI}^2}{2}
  \left(\langle {\cal N}^2 \rangle  - \langle {\cal N} \rangle^2 \right). 
  \label{eq:error}\end{aligned}$$ Since the 