# What do we divide by? Denominator traps > ARPU vs ARPPU, LTV per cohort user vs per survivor, win rate with or without "no decision", gross vs net margin, concentration with the item in the total, and incremental results against a control group. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Averages and shares of shares - Lesson 12/36, ~34 min - Source: https://productovi.com/courses/math-basics/choosing-the-denominator ## Same top, different bottom **Alex Tutor:** Module 3 ends with the quietest source of wrong numbers: the denominator. The numerator is usually obvious; what you divide by decides the question you're answering. Today's example comes from Klasna Club. **Sofia:** Klasna Club brought 84,000 UAH in June. 3,000 learners were active and 240 of them paid. So revenue per learner is 84,000 ÷ 240 = 350? **Alex Tutor:** That's **ARPPU**, per paying learner. Per active learner, **ARPU**, it's 84,000 ÷ 3,000 = **28 UAH**. Both are right; they answer different questions. Just never compare one company's ARPU with another's ARPPU. ![84,000 UAH of Club revenue divided by 3,000 active learners gives ARPU of 28 UAH; divided by 240 paying learners it gives ARPPU of 350 UAH.](https://productovi.com/courses/math-basics/diagrams/m3-l4-denominators.png) *One numerator, two denominators, two different questions.* **Alex Tutor:** Lifetime value hides the same trap. A cohort of 1,000 learners has brought 1,800,000 UAH of margin by month 6, and 300 are still active. Per cohort learner: **1,800 UAH**. Per survivor: 6,000. **Sofia:** And the 6,000 looks much better against acquisition cost. **Alex Tutor:** Which is why it's wrong for that comparison. You paid to acquire all 1,000, so the value must be spread over all 1,000. Dividing by survivors counts only the winners. > **Exercise:** In July 2,500 learners were active, 200 paid for Klasna Club, and the Club brought 76,000 UAH. Which pair is correct? **Alex Tutor:** Then a slide that puts two of these numbers side by side. > **Exercise:** A slide compares two schools: school A's ARPU is 30 UAH, school B's ARPPU is 300 UAH. What can you conclude? > **Exercise:** A cohort of 800 learners has brought 1,040,000 UAH of cumulative margin by month 12; 200 of them are still active. What is the margin per cohort learner, the number to compare with acquisition cost, in UAH? ## Is the item in the total? **Alex Tutor:** Klasna for Teams earned 900,000 UAH last quarter; the largest client paid 240,000. Its share is 240,000 ÷ 900,000 = **26.7%**. Someone divided by the other clients only, 660,000, and got 36.4%, a share of nothing in particular. > **Exercise:** Klasna for Teams, the school's lessons for companies, earned 750,000 UAH this quarter, 180,000 of it from the largest client. What share of revenue does that client bring, in percent? **Alex Tutor:** Win rates have the same issue. Of 20 proposals, 6 were won, 4 lost to a rival and 10 ended with no decision. Against decided deals the win rate is 6 ÷ 10 = 60%; against all proposals, 6 ÷ 20 = 30%. Say which. **Alex Tutor:** Try the decided-deals version on other numbers. > **Exercise:** Of 15 proposals, 6 were won, 3 were lost to a rival and 6 ended with no decision. What is the win rate against decided deals, in percent? **Sofia:** And margins: gross margin over revenue, net margin over revenue too? **Alex Tutor:** Same denominator, different numerator: gross profit or net profit, both over revenue. The trap there is a marketplace dividing by everything customers paid instead of its own revenue. **Alex Tutor:** Check the marketplace version. > **Exercise:** A marketplace keeps 20% of each lesson booked through it. To report its gross margin, it should divide gross profit by everything customers paid for the lessons. ## Compared with what would have happened **Alex Tutor:** The hardest denominator is incremental. A referral campaign ran in one region: referrals went from 400 to 560. A holdout region without the campaign grew from 300 to 330, +10%, on its own. **Alex Tutor:** Build the baseline first, on other numbers. > **Exercise:** Before a campaign the test region had 250 referrals a month. A holdout region without the campaign grew by 12% over the same period. How many referrals would the test region likely have had without the campaign? **Alex Tutor:** So without the campaign the test region would likely have had 400 × 1.10 = **440**. Incremental = actual − expected = 560 − 440 = **120** referrals. At a cost of 36,000 UAH, that's 300 UAH per incremental referral. ![The holdout grew 300 → 330 (+10%). The test region would have reached 400 × 1.1 = 440; it reached 560, so the campaign added 120 referrals.](https://productovi.com/courses/math-basics/diagrams/m3-l4-incremental.png) *Only the part above the expected baseline belongs to the campaign.* **Sofia:** And the report said 36,000 ÷ 560 = 64 UAH per referral. That counts the 440 that would have come anyway. **Alex Tutor:** Exactly. **Incremental CAC** works the same way: extra spend divided by the customers the spend actually added. > **Exercise:** Before a campaign the test region had 500 sign-ups a month; during it, 620. The holdout region grew by 8% over the same time. The campaign cost 24,000 UAH. What is the cost per incremental sign-up, in UAH? **Alex Tutor:** Before the case, a quick one in words: the win rate from the proposals above, for a board slide. **Sofia:** I'll show both numbers and say which is which. > **Exercise:** Of 20 proposals, 6 were won, 4 were lost to a rival and 10 ended with no decision. Give the win rate both ways, say which one you would put on a board slide and why, and how you would label it. ## Case: fix the marketing report **Alex Tutor:** Now the module case. Klasna's August marketing report makes four claims, and each one divides or averages the wrong way. Use everything from module 3. **Sofia:** Weighted means, funnels and denominators. Got it. > **Exercise:** Correct the four claims in the August report and say which correction matters most for the next budget. ## Where you will meet this > **In short** > > - Name the denominator every time: **ARPU** divides by active users, **ARPPU** by paying ones. > - Value per learner divides by **everyone acquired**, not by survivors. > - A share of a total keeps the item **in the total**. A win rate says whether "no decision" is in the denominator. > - **Incremental** = actual − expected, where expected = baseline × the holdout's growth. Cost per incremental result = cost ÷ incremental, and incremental CAC = extra spend ÷ extra customers. > **These lessons use it** > > - [How to calculate ARPU and ARPPU](/courses/product-analytics/revenue-metrics): two denominators for the same revenue. > - [The LTV curve](/courses/product-analytics/ltv-curve): cumulative revenue divided by everyone in the cohort. > - [Channels and marketing attribution](/courses/product-analytics/channels): a holdout to measure what a channel really adds. > - [GMV vs revenue](/courses/finance-for-product/gmv-vs-revenue): what a marketplace should divide by. > - [One customer too big](/courses/bootstrapped-product/one-customer-too-big): a customer's share with that customer in the total. > - [Referrals and community](/courses/bootstrapped-product/referrals-and-community): cost per customer the program really added. **Alex Tutor:** That's module 3. Next: squares, roots and distances, starting with why statistics squares deviations at all. - Previous lesson: [Products of shares: funnels and metric trees](https://productovi.com/courses/math-basics/products-of-shares.md) - Next lesson: [Why we square: variance, sums of squares and n − 1](https://productovi.com/courses/math-basics/why-we-square.md) --- Productovi Learning. Free, no sign-up needed to read.