I have vectors $x_i$ which obtained by the sigmoid function, $$ \{x_1, x_2, \cdots , x_n \}, x_i \in (0,1) $$ and I can get inverse sigmoid of $x_i$, such that $z_i = - \ln (\frac{1}{x_i} -1 ) $ .
What I'm cursious about is, let $\bar{x}$ is the mean of $x_i$ and $\bar{z}$ is the mean of $z_i$,
Are $\bar{z}$ and the inverse sigmoid of $\bar{x}$ the same?
To clarify, I'd like to ask $\bar{z}= \frac{1}{n}\sum_i^n (- \ln( \frac{1}{x_i } -1)) = - \ln (\frac{1}{\bar{x}}-1)$ ?
If not, how about the inequality between them?