Discussion Papers no. 944

An exact additive decomposition of the weighted arithmetic mean

Several multiplicative decompositions of the unit value index are described in literature.

The weighted arithmetic mean is used in a wide variety of applications. An infinite number of possible decompositions of the weighted mean are available, and it is therefore an open question which of the possible decompositions should be applied. In this paper, we derive a decomposition of the weighted mean that is rooted in functional analysis. Our proposed decomposition is easy to employ and interpret, and we show that it satisfies the difference counterpart to the index number time reversal test. We illustrate the framework by decomposing aggregate earnings growth from 2020Q1 to 2021Q1 in Norway and compare it with some of the main decompositions proposed in the literature. We find that the wedge between the identified compositional effects from the proposed decomposition and the Bennet decomposition is substantial, and for some industries, the compositional effects are of opposite signs.

About the publication


An exact additive decomposition of the weighted arithmetic mean


Thomas von Brasch, Håkon Grini, Magnus Berglund Johnsen and Trond Christian Vigtel

Series and number

Discussion Papers no. 944


Statistisk sentralbyrå


Discussion Papers



Number of pages


About Discussion Papers

Discussion papers comprise research papers intended for international journals and books. A preprint of a Discussion Paper may be longer and more elaborate than a standard journal article as it may include intermediate calculations, background material etc.