Meaning of "average correlation" in Var$(\bar X)$ From some process I got a series of values. I want to compute the variance of the mean from this series. The series is built with contiguous sub-series. In each sub-series the values are correlated. All sub-series follow the same pattern but they have different total lengths.
I read on Wikipedia that for correlated variables holds
$$\operatorname{Var}\left(\overline{X}\right) = \frac{\sigma^2}{n} + \frac{n - 1}{n}\rho\sigma^2 $$
where $\rho$ is the average correlation.
Q1: What does average correlation means or how it is computed?
Intuitively I would expect it to be something like the lag or the sum of all lags of the autocorrelation function of the series, but I don't know.
Q2: Is this approach reasonable?

EDIT: I checked these questions:
According to comments, I should use:
$$\operatorname{Var}\left(\sum_{i=1}^n X_i\right) =  \sum_{i=1}^n \operatorname{Var}\left(X_i\right) + 2\sum_{1\le i<j\le n}\operatorname{Cov}\left(X_i, X_j\right)$$
but I am not sure on how to compute the $\operatorname{Cov}\left(X_i, X_j\right)$.
If it helps, the data arises from an stationary process (although it always starts with a complete sub-series).
Variance of a sum of identically distributed random variables that are not independent
Variance of sum of dependent random variables
and this article.

The general formula is
$$\text{Var}\left(\sum_{k=1}^n a_k X_k\right) =  \sum_{k=1}^n a_k^2 \text{Var}\left(X_k\right) + \sum_{k=1,l=1 \\k\neq l}^n a_k a_l \text{Cov}\left(X_k,X_l\right) $$
And when all the variances are equal to $\sigma^2$ and all $a_k = \frac{1}{n}$ then
$$\sum_{k=1}^n a_k^2 \text{Var}\left(X_k\right) + \sum_{k=1,l=1 \\k\neq l}^n a_k a_l \text{Cov}\left(X_k,X_l\right) = \sum_{k=1}^n \frac{1}{n^2} \sigma^2 + \sum_{k=1,l=1 \\k\neq l}^n \frac{1}{n^2} \rho_{k,l}\sigma^2 = \frac{1}{n} \cdot \sigma^2  + \frac{1}{n^2} \left( \sum_{k=1,l=1 \\k\neq l}^n \rho_{k,l} \right) \cdot \sigma^2$$
It is this sum of all the correlations $\rho_{k,l}$ that is being replaced by the average.
$$\sum_{k=1,l=1 \\k\neq l}^n \rho_{k,l}  = \bar\rho  \cdot n \cdot (n-1)$$

but I am not sure on how to compute the $\operatorname{Cov}\left(X_i, X_j\right)$.

See the definition of the Pearson correlation coefficient $\rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{Var(X)\cdot Var(Y)}}$
