Overall conversion fell while every acquisition channel's own conversion rate rose — how?

## The blended rate is a weighted average An overall conversion rate is never a number in its own right. It is a **weighted average** of segment rates, with weights equal to each segment's share of the denominator: ``` R = sum over segments i of w_i * r_i where sum of w_i = 1 ``` Because the weights are part of the formula, the blend has two independent ways to move: the segment rates `r_i` can change, and the shares `w_i` can change. Once you see that, "every part rose but the total fell" stops being paradoxical and becomes ordinary arithmetic. ## A worked example | period | channel | visits | share | conversion | conversions | |---|---|---|---|---|---| | before | organic | 800 | 0.80 | 10% | 80 | | before | paid | 200 | 0.20 | 4% | 8 | | before | **total** | 1000 | | **8.8%** | 88 | | after | organic | 400 | 0.40 | 11% | 44 | | after | paid | 600 | 0.60 | 5% | 30 | | after | **total** | 1000 | | **7.4%** | 74 | Organic improved from 10% to 11%. Paid improved from 4% to 5%. The blend fell 1.4 points. Nothing is wrong with the data: traffic moved from the channel that converts at 10% into the one that converts at 5%, and the shift is larger than the improvements. ## The decomposition Write the change as `R' - R = sum (w_i' * r_i' - w_i * r_i)` and expand it. The exact identity has three terms: ``` within = sum w_i * (r_i' - r_i) rates move, mix held at baseline mix = sum (w_i' - w_i) * r_i mix moves, rates held at baseline interaction = sum (w_i' - w_i) * (r_i' - r_i) both move together ``` These three add up to the total delta exactly. In the example: within `= 0.80*0.01 + 0.20*0.01 = +1.0 point`; mix `= (0.40-0.80)*0.10 + (0.60-0.20)*0.04 = -4.0 + 1.6 = -2.4 points`; interaction `= (-0.40)(0.01) + (0.40)(0.01) = 0`. Total `+1.0 - 2.4 + 0 = -1.4 points`, matching 8.8% to 7.4%. The interaction term is an artefact of holding one factor at baseline, and it embarrasses people in presentations. A **symmetric** version removes it by evaluating each effect at the midpoint of the other: ``` within = sum ((w_i + w_i')/2) * (r_i' - r_i) mix = sum (w_i' - w_i) * ((r_i + r_i')/2) ``` Those two also sum exactly to `R' - R`, with no residual. Either convention is defensible; state which one you used. ## Attributing the mix effect to individual segments A subtlety: shares are constrained to sum to one, so a segment's share change is never independent — one segment gaining forces others to lose. Naively reporting `(w_i' - w_i) * r_i` per segment makes low-rate segments look responsible for gains they did not cause. The cleaner per-segment mix contribution measures each segment's rate **relative to the baseline overall rate**: ``` mix contribution of segment i = (w_i' - w_i) * (r_i - R) ``` This still sums exactly to the total mix effect, because the shares' changes sum to zero and so the `R` term drops out. It reads correctly too: gaining share in a segment that converts **above** the overall average helps; gaining share in one that converts below it hurts. In the example: organic contributes `(-0.40)(0.10 - 0.088) = -0.48 points` and paid contributes `(0.40)(0.04 - 0.088) = -1.92 points`, together the -2.4 point mix effect. ## Segments that appear or disappear A channel launched this period has no baseline share and no baseline rate, so neither the within nor the mix term is defined for it. Do not invent one. Give entrants and exits their own line in the reconciliation and say so explicitly; smearing them across the surviving segments is how a decomposition quietly stops adding up. ## What to do with the answer The decomposition tells you whose problem it is. A large negative mix term with a positive within term means the funnel got better and the traffic portfolio got cheaper — which may be entirely intentional, if the low-converting channel is also far cheaper per visit. That is a question about cost per acquisition, not about conversion. A large negative within term means the experience actually degraded, and the segment-level bars point at where. The wrong move is to accept the headline at face value in either direction. A flat blended rate can hide a real within-segment collapse offset by a favourable mix shift, and that is the more dangerous case, because nothing on the dashboard moves at all. ## What interviewers listen for That you reach for the weighted-average structure immediately rather than doubting the data; that you can write the two effects and show they reconcile; that you notice a change in shares is zero-sum across segments; and that you finish with an ownership statement rather than an arithmetic one.