I am currently trying to figure out whether it is possible to "decompose" a standard deviation. I'd like to illustrate my question with an example: Say you have 10 countries, and you compute the standard deviation of labor productivity. Each country has 3 sectors which it uses to produce, namely agriculture, manufacturing, and services. You can compute total economy productivity then as a weighted sum of the sector labour productivities, where the weight used is a sector's employment share (shares of agriculture, manufacturing, and services logically add up to 1). My question is then the following. Is there a way to compute the standard deviation of these three sectors, such that they "aggregate" to the standard deviation of the economy-wide labour productivity?
What I am trying to do is compare labour productivity dispersion for the different sectors, and their role in accounting for total dispersion of labour productivity, hence why I am interested in finding out whether a standard deviation can be decomposed (into sectoral components in this case).
Any help is greatly appreciated!
Best,
Oz