The average of Ct values in real-time PCR

The Ct value is the number of cycles when the "PCR amplification curve with the horizontal axis as the number of cycles" and the "predetermined threshold" intersect, and is not necessarily an integer value.

Real-time PCR was performed on n series of samples under the same conditions, and the Ct values, ${Ct_1},{Ct_2},cdots {Ct_n}$, were obtained for each. Then, in many instructions, the average of these Ct values is determined as follows;

$$bar{Ct}:=frac{{Ct_1}+{Ct_2}+cdots +{Ct_n}}{n}tag{1}$$

My question:
What is the meaning of such an average? How do I justify taking such an average?

The amplification curves are ideally considered to consist of the following when the initial template quantities are ${N}_{1}(0), {N}_{2}(0), cdots N_{n}(0)$, respectively;

$${N}_{i}(k)={N}_{i}(0){a}^{k}tag{2}$$

It is my understanding that the Ct value is for calculating ${N}_{i}(0)$ backwards.

Let the threshold value be X, then, for all $i=1,2,cdots n$,

$$X={N}_{i}(0){a}^{{Ct}_{i}} tag{3}$$

therefore, for all $i=1,2,cdots n$,

$$X{a}^{-{Ct}_{i}}={N}_{i}(0) tag{4}$$

therefore,

$$bar{{N}(0)}
:=frac{{N}_{1}(0)+ {N}_{2}(0)+ cdots + N_{n}(0)}{n}
=Xfrac{{a}^{-{Ct}_{1}}+ cdots + {a}^{-{Ct}_{n}}}{n} tag{5}$$

therefore, a better Ct average , $Ct_{avg}$ shall satisfy

$$X=bar{{N}(0)}{a}^{Ct_{avg}} tag{6}$$

therfore ,
$$X{a}^{-{Ct}_{avg}}=Xfrac{{a}^{-{Ct}_{1}}+ cdots + {a}^{-{Ct}_{n}}}{n} tag{7}$$

$${a}^{-{Ct}_{avg}}=frac{{a}^{-{Ct}_{1}}+ cdots + {a}^{-{Ct}_{n}}}{n} tag{8}$$

Based on the above discussion, wouldn’t it be more natural to use the following?
$$Ct_{avg}=-{log}_{a}(frac{{a}^{-{Ct}_{1}}+ cdots + {a}^{-{Ct}_{n}}}{n}) tag{9}$$