Consider a survey of firms of size $n$. This survey includes, among other variables, the average wage of workers in firm $i$ ($x_i$) and the number of workers in the firm ($L_i$). Both are random variables.
I want to study the properties of the average wage in this economy. This average wage is defined as
$$ \chi = \sum_i \delta_i x_i $$
where $\delta_i$ is the proportion of workers in firm $i$ with respect to all workers in the sample. This is, $\delta_i=\frac{L_i}{\sum_i L_i}$. These weights add up to one.
As part of my analysis of this weighted average, I am interested in two decompositions which came into my mind:
Mathematical decomposition:
Define the arithmetic (i.e. unweighted) average as $\bar x$. We can write:
$$ \chi = \bar x \left(\frac{\sum_i \delta_i x_i}{\bar x}\right) $$
In words, we can think of $\chi$ as a decomposition between the arithmetic mean and a measure of "dispersion around that mean". In effect, if $x_i=c$ (no dispersion), $\chi = \bar x = c$. So whether $\chi$ is above or below $\bar x$ tell us something about dispersion of $x_i$.
Statistical decomposition:
Assuming the sample is iid, the sample equivalent of the population moment $E(\delta_i x_i)$ is
$$\hat E(\delta_i x_i) = \dfrac{\sum_i \delta_i x_i}{n} = \dfrac{\chi}{n}$$
But we know the properties of the covariance:
$$ E(\delta_i x_i) = E(\delta_i)E(x_i) + cov(\delta_i,x_i) $$
Which sample equivalents are:
$$ \hat E(\delta_i x_i) = \hat E(\delta_i) \hat E(x_i) + \hat{cov}(\delta_i,x_i) $$
Using the above, and noting that $n\hat E(\delta_i)=1$ (because of the definition of weights), we get:
$$ \chi \approx \hat E(x_i) + n \ \hat{cov}(\delta_i,x_i) $$
This is also a form of mean and dispersion decomposition, since in the case of no dispersion ($x_i=x$), the covariance term is zero.
Now, the two above seem to me very straighforward, and almost obvious. As such, I imagine these decompositions have been studied in the literature already. Maybe they have a name, and a whole set of properties around them.
Thus, what I want is to find our about these decompositions. However, I cannot find anywhere a literature related to them. For what I can gather, this is unrelated to mean-variance decomposition literature, Blinder-Oaxaca decomposition, standardisation, and so on.
My questions are: are these common decompositions? Do they have a name? Can you refer me to literature where I can read more about them?