# Products of shares: funnels and metric trees > Overall conversion as a product of step rates, working a funnel backwards from a goal, ARPU = paying share × ARPPU, revenue per visitor, stickiness, diluted effects, ownership dilution and RICE. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Averages and shares of shares - Lesson 11/36, ~25 min - Source: https://productovi.com/courses/math-basics/products-of-shares ## A funnel multiplies **Sofia:** Klasna's trial funnel: 6% of site visitors book a trial, 75% of those attend, 25% of attendees buy a package. Is overall conversion 6% + 75% + 25% divided by something? **Alex Tutor:** No: each share is taken **of the previous step**, so they multiply. 20,000 visitors × 0.06 = 1,200 bookings; × 0.75 = 900 attend; × 0.25 = **225** buyers. Overall: 0.06 × 0.75 × 0.25 = 0.01125, or **1.125%**. ![Kittens in a stack of sieves: five in the top one, two in each of the next two, and a single kitten on an orange cushion at the bottom.](https://productovi.com/courses/math-basics/inline/m3-l3-i1.webp) *Each step keeps a share of what reached it.* ![20,000 visitors × 6% = 1,200 bookings × 75% = 900 attend × 25% = 225 buyers; the overall conversion is 0.06 × 0.75 × 0.25 = 1.125%.](https://productovi.com/courses/math-basics/diagrams/m3-l3-funnel.png) *Step rates multiply into the overall rate.* **Sofia:** So a step rate and the overall rate are different numbers with different bases. **Alex Tutor:** Exactly: 25% is "of attendees", 1.125% is "of visitors". And because they multiply, a relative lift at any step passes straight through: attendees buying 20% more often lifts overall conversion by 20% too. > **Exercise:** 15,000 visitors come to the site. 5% book a trial, 80% of those attend, and 30% of attendees buy. How many buyers is that? **Alex Tutor:** Now a lift at one step: what does it do to the total? > **Exercise:** In the trial funnel (6% book, 75% attend, 25% buy), attendees start buying 20% more often: 25% → 30%. Overall conversion from visitors also rises by 20% relative. ## Backwards from a goal **Alex Tutor:** Run the funnel backwards to plan. Leo wants 300 buyers a month. Divide by each rate from the bottom: 300 ÷ 0.25 = 1,200 attendees; ÷ 0.75 = 1,600 bookings; ÷ 0.06 ≈ **26,667 visitors**. **Alex Tutor:** At about 5 UAH per visit from ads, that's a budget of roughly 133,000 UAH. The same as one division by the overall rate: 300 ÷ 0.01125 ≈ 26,667. **Sofia:** So reporting multiplies and planning divides. **Alex Tutor:** Yes. And the rounding rule from module 1 applies: people come in whole numbers, so round a plan up, never down. > **Exercise:** The goal is 240 buyers a month. 30% of attendees buy, 80% of bookings attend, and 5% of visitors book. How many visitors are needed? ## Metric trees **Alex Tutor:** A metric tree is the same idea for revenue. Revenue per active learner, **ARPU**, = paying share × **ARPPU**, revenue per paying learner. If 8% of active learners pay for Klasna Club at 350 UAH, ARPU = 0.08 × 350 = **28 UAH**. **Alex Tutor:** Then revenue = active learners × ARPU: 12,000 × 28 = **336,000 UAH** a month. Every node is the product of the nodes under it, so when revenue moves you can ask which factor moved. ![A metric tree: revenue 336,000 UAH = 12,000 active learners × ARPU 28 UAH, and ARPU = 8% paying share × ARPPU 350 UAH.](https://productovi.com/courses/math-basics/diagrams/m3-l3-tree.png) *Each branch is a factor of the node above it.* > **Exercise:** 6% of active learners pay, and a paying learner brings 400 UAH a month. What is ARPU, revenue per active learner, in UAH? **Alex Tutor:** The same pattern gives revenue per visitor, **RPV** = conversion × price, and **stickiness** = average DAU ÷ MAU, the share of the month's users who show up on a typical day. **Alex Tutor:** Compute one stickiness. > **Exercise:** In April an average day had 1,800 active learners (average DAU), and 6,000 learners were active at least once in the month (MAU). What is stickiness, DAU ÷ MAU, in percent? **Sofia:** And roadmap scores like RICE? **Alex Tutor:** Also a product, with one division: RICE = Reach × Impact × Confidence ÷ Effort. 2,000 learners a quarter × impact 2 × confidence 0.8 ÷ 4 person-weeks = **800**. **Alex Tutor:** Score one item. > **Exercise:** A roadmap item reaches 1,500 learners a quarter, with impact 3, confidence 0.5 and effort 5 person-weeks. What is its RICE score, Reach × Impact × Confidence ÷ Effort? ## Diluted effects **Alex Tutor:** Products also shrink effects. A new booking page lifts conversion by 3 pp, but only 15% of visitors ever see it. The overall effect is 3 pp × 0.15 = **0.45 pp**: reach multiplies the effect. **Sofia:** So a test on a page few people see can't move the company total much. **Alex Tutor:** Not by much. That's why a test is analyzed only on those who reach the page: inside that group the effect is 3 pp, not 0.45. **Alex Tutor:** Ownership dilutes the same way. A founder with 50% who sells 20% of the company in each of two rounds keeps 50% × 0.8 × 0.8 = **32%**. Each round shrinks everyone's share by the same factor. > **Exercise:** A feature lifts conversion by 4 pp for the users who see it, and 20% of users see it. What is the effect on overall conversion? > **Exercise:** A founder owns 60%. The company raises two rounds, selling 25% of the company in the first and 10% in the second. What share does the founder own afterwards, in percent? ## Your turn: which step to fix **Alex Tutor:** The team can do one project this quarter. A raises attendance from 75% to 80%; B raises buying from 25% to 28%. Which gives more buyers? **Sofia:** A gains 5 pp, B only 3 pp... but I bet that's the trap. > **Exercise:** Funnel: 20,000 visitors, 6% book, 75% attend, 25% buy (225 buyers). Project A raises attendance to 80%; project B raises buying to 28%. Compute the buyers for each and explain in one or two sentences why the step with the smaller gain in points can win. ## Where you will meet this > **In short** > > - Funnel steps **multiply**: overall = step₁ × step₂ × … A step rate has the previous step as its base. > - A relative lift at one step lifts the overall rate by the same relative amount. Compare projects by relative change, not points. > - To plan, go **backwards**: goal ÷ each rate, or goal ÷ overall rate. > - **ARPU** = paying share × **ARPPU**; **RPV** = conversion × price; **stickiness** = average DAU ÷ MAU; **RICE** = R × I × C ÷ E. > - **Diluted effect** = effect × reach (3 pp × 0.15 = 0.45 pp). Ownership after rounds = share × (1 − sold₁) × (1 − sold₂). > **These lessons use it** > > - [The north star metric and its KPI tree](/courses/product-analytics/north-star): a metric as a product of its drivers. > - [Funnels](/courses/product-analytics/funnels): step conversion vs overall conversion. > - [DAU, WAU, MAU and stickiness](/courses/product-analytics/active-users): average DAU ÷ MAU. > - [Who is in the test](/courses/product-analytics/who-is-in-the-test): an effect diluted by the share who see the change. > - [How to calculate ARPU and ARPPU](/courses/product-analytics/revenue-metrics): ARPU = paying share × ARPPU. > - [Debt vs equity](/courses/finance-for-product/debt-vs-equity): dilution of ownership. > - [Scoring frameworks](/courses/outcome-based-roadmaps/scoring-frameworks): RICE = R × I × C ÷ E. **Alex Tutor:** Next time: what do we divide by? The same numerator over different denominators gives different stories. - Previous lesson: [Weighted mean and mix](https://productovi.com/courses/math-basics/weighted-mean-and-mix.md) - Next lesson: [What do we divide by? Denominator traps](https://productovi.com/courses/math-basics/choosing-the-denominator.md) --- Productovi Learning. Free, no sign-up needed to read.