DTM.5:4.3 - Decompose the aggregate change

For a scalar property, let π_g be the source population’s weight in group g, z_g its initial mean, w_g its contribution per source unit to the covered part of the receiving population, and Δz_g the change in the corresponding transmitted mean. Weights π_g sum to one; the mean contribution w̄ is positive. The source-attributed contributions must cover that part without omission or double counting.

The covered mean and its difference from the initial mean satisfy:

z_cov = Σ_g π_g w_g (z_g + Δz_g) / w̄
z_cov − z_old
  = Cov_π(w,z) / w̄ + Σ_g π_g w_g Δz_g / w̄.

The first term expresses the effect of different contribution weights associated with the earlier property. The second expresses changes along those contributions. If those contributions cover the whole receiving population, z_new = z_cov.

Otherwise let c be the covered fraction of the receiving population and z_ext the mean property in the rest, using the same counted unit and weighting convention. Include that contribution:

z_new = c z_cov + (1−c) z_ext
z_new − z_old = c (z_cov − z_old) + (1−c)(z_ext − z_old).

An unknown external composition remains unknown; it is not zero. For a binary variant share and known c, the possible new share lies between c z_cov and c z_cov + 1−c until the external composition is constrained. When no receiving unit derives from the identified sources, omit their decomposition and calculate from the other contribution. MMP and the relevant mathematical methods supply the calculation.

This identity does not tell why w or Δz has the observed value. Call them between-source and within-contribution changes until the causal interpretation is established. If the units are collectives, redefine z and w for collective continuation; do not carry over a particle count while silently changing the claim.

For a question needing no scalar aggregate, retain the corresponding source and change table. Do not invent a total score solely to use the formula.