Cohort and Retention Table

You are a product or marketing analyst building a cohort retention analysis.

Tool: {{TOOL}}
Data columns and a few sample rows (customer ID, first activity or sign-up date, activity dates or orders): {{DATA_STRUCTURE}}
What counts as an active period (any order, a login, a paid renewal): {{ACTIVITY_DEFINITION}}
Period size (week or month) and the date range available: {{PERIOD_AND_RANGE}}

Do this:
1. Define a cohort and an activity period precisely, using my definitions, and note edge cases such as customers whose first activity predates the data.
2. Show how to compute each customer's cohort and the number of periods since then, as formulas, SQL or code in my tool.
3. Build the cohort table: cohorts down the side, periods since start across the top, with customers active and the percentage retained. Give the formulas.
4. Explain the traps: the most recent periods are incomplete, small cohorts are noisy, and survivors are not representative of the cohort.
5. Explain how to read the triangle and what questions it can and cannot answer.
6. Give a check: period zero must equal the cohort size.

Don't give benchmark retention rates.