Revenue fell 3%: how do you build a contribution waterfall attributing that delta to country segments?

## Why an additive metric decomposes cleanly If a metric is a **sum** over segments — total revenue, total orders, total minutes watched — then the change in the total is the sum of the changes: ``` Total' - Total = sum over segments i of (x_i' - x_i) ``` This is an identity, not an approximation. There is no interaction term, no weighting choice, no residual. That is what makes a contribution waterfall a legitimate chart for additive metrics: each bar is a real quantity of money, and the bars have to land on the headline number. If they do not, you have a partition problem, not a rounding problem. ## Step one: the partition The segmentation must be **mutually exclusive and exhaustive** over the same universe as the headline. Two failure modes recur: - **Overlap.** A customer assigned to both "United Kingdom" and "EMEA" gets counted twice, and the bars overshoot the total. - **Gaps.** Rows with a null or unmapped country silently vanish. If the headline includes them and the chart does not, the reconciliation fails by exactly that amount. The fix is a visible "unmapped" bar, not a filter. Always print the reconciliation line: sum of bars, headline delta, difference. A difference of zero is the chart's licence to be believed. ## Step two: gross versus net A net delta of -1.5M can be produced by many different worlds: one country down 1.5M with everything else flat, or four countries down 2.4M against three up 0.9M. Those call for different responses, and only the gross figures distinguish them. Report both: ``` gross decline = sum of negative segment deltas gross increase = sum of positive segment deltas net = gross decline + gross increase ``` A large gross with a small net means the business is churning underneath a calm headline — usually more urgent than a clean single-segment decline. ## Step three: share of the decline, handled honestly The natural summary is `segment delta / total delta`. Two properties surprise people: 1. **Shares can exceed 100%.** If one country lost 2.4M and the net was 1.5M, that country represents 160% of the net decline, offset by gains elsewhere. This is correct arithmetic, and it is why the sentence needs the word "net" in it. 2. **The ratio is unstable near zero.** As the net delta approaches zero, every share blows up. When the headline barely moved, drop the percentage entirely and quote absolute currency amounts. A safe convention: percentages of the **gross** movement in each direction, plus the absolute values. ## Step four: entrants and exits A country launched this period has no prior value; a country shut down has no current value. Both are genuine parts of the delta, but they are not "a segment that changed" — they are the appearance or disappearance of a segment. Give them their own bars, labelled as such. Folding a launch into the general improvement makes an expansion decision look like organic growth. ## Step five: how deep to drill Each extra dimension multiplies the number of bars and shrinks each one. Two disciplines keep it useful: - **Drill only along dimensions someone owns.** Country, platform, plan tier and acquisition channel usually map to a team that can act. Splitting by an arbitrary hash of user id produces bars that are arithmetically valid and decision-free. - **Stop when the bars stop implying an action.** Below a certain size, a segment's delta is a number on a chart rather than a thing to do, and the temptation to narrate whichever tiny bar fits the story grows with the bar count. ## Step six: know when the tool does not apply All of the above rests on additivity. A **rate** — conversion, retention, margin percent — is not a sum over segments; it is a share-weighted blend. Its segment shares are constrained to sum to one, so a segment's share change is never independent of the others', and a naive waterfall of segment rate differences will not reconcile. Rates need a mix-versus-within decomposition first, and only then can the resulting effects be charted as bars. A quick tell that someone has confused the two: a conversion-rate waterfall whose bars are expressed in percentage points of each segment's own rate. Those quantities cannot sum to a change in the overall rate. ## What good looks like in the room State the identity. Name the MECE requirement and the reconciliation check. Distinguish gross from net before quoting any share. Handle entrants explicitly. Say where you would stop drilling and why. Then, unprompted, note that the whole approach changes for a ratio metric — that last sentence is usually what separates a senior answer from a competent one.