Compute Day-N Retention from Raw Events Under Three Definitions
Six users all signed up on day 0. The table lists the days (since signup) on which each user performed the core action; 20 event rows in total.
| user | active days |
|---|---|
| p | 0, 1, 3, 7 |
| q | 0, 1 |
| r | 0, 2, 7 |
| s | 0 |
| t | 0, 1, 2, 3, 4, 5, 6, 7 |
| u | 0, 6 |
- Compute Day-1, Day-3 and Day-7 retention under the exact (classic) definition.
- Compute Day-3 and Day-7 retention under the unbounded (rolling) definition, and bracketed retention for days 4 to 7.
- Explain why the exact curve is not monotone here, what that says about the product, and which definition you would use for a product people are expected to use roughly weekly.
- A teammate reports "Day-7 retention is 50 percent" in a deck that earlier showed Day-7 retention of 20 percent. List what you would check before celebrating.
1. Exact. Day-1: users active on day 1 are p, q, t = 3 of 6 = 50.0%. Day-3: p and t = 2 of 6 = 33.3%. Day-7: p, r, t = 3 of 6 = 50.0% (all verified by hand and by code).
2. Unbounded and bracketed. Day-3 unbounded counts users active on day 3 or later: p (3, 7), r (7), t, u (6) = 4 of 6 = 66.7%; s and q are out. Day-7 unbounded is the same as exact Day-7 here because day 7 is the last observed day: 50.0%. Bracketed days 4 to 7: r (7), p (7), t, u (6) = 4 of 6 = 66.7%; q and s never returned in the window.
3. Non-monotone curve. Exact retention goes 50, 33, 50 percent because users come back after a gap (r skipped days 1 and 3-6 and returned on day 7; p returned on day 7 after a quiet stretch). That is informative: usage is intermittent, so a daily-habit metric misrepresents the product. The unbounded definition can never rise, so it hides resurrection but also overstates "kept" users (everyone who ever came back counts, even if they are gone now). For a weekly-use product use bracketed retention such as "active at least once in days 8 to 14" (or week-N retention), which tolerates the natural gap between uses; report the definition next to every number.
4. Before celebrating. (a) Did the definition change (exact versus unbounded versus bracketed, day 7 versus a 7-day window, cohort anchored on signup versus first use)? On the table above, Day-7 is 50 percent for both exact and unbounded, but Day-3 jumps from 33 to 67 percent when you switch; a "jump" from 20 to 50 percent should make you suspect a changed definition first. (b) Did "active" change (opened app versus core action)? (c) Are cohort sizes large enough: with 6 users one person is 16.7 points. (d) Cohort composition: a new acquisition channel or market, or a different mix of platforms, can shift the number (check the triangle by channel). (e) Instrumentation: a tracking change, bot traffic, or identity stitching that now merges devices. (f) Time-zone or day-boundary changes. Only after those checks would you attribute the change to the product, ideally confirmed by an experiment.
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