Advanced SQL: Metric Monitoring, Anomaly Detection, and Data Correctness at Scale Questions

SQL for monitoring a business or operational metric over time and catching when it is wrong or behaving abnormally, as distinct from the mechanics of the SQL constructs used to compute it (window function syntax, frame semantics, CTEs, and subqueries are covered by the companion topic Advanced SQL: Window Functions, CTEs, and Subqueries), from defining and computing cohort retention, funnel conversion, DAU/WAU/MAU, and lifetime value themselves (covered by the companion topics User Retention & Engagement, Conversion Funnel Optimization, and SQL for Data Analysis), and from sessionization via inactivity-gap detection (also covered by the companion window-functions topic). Covers rolling-window anomaly and change-point detection (z-score and threshold-based, including seasonality-aware baselines), testing whether a period-over-period change in a rate metric is statistically significant versus noise, decomposing a metric's change into which segments drove it, average revenue per user (ARPU) at both a simple and a rolling/at-scale altitude, keeping cumulative and rolling metrics correct against missing dates, late-arriving or out-of-order events, and incremental (not full-recompute) materialization at very large scale, approximate-distinct (HyperLogLog) computation, standalone timezone/daylight-saving-aware day bucketing, Kaplan-Meier-style survival curves computed in pure SQL, and multi-state subscription churn and retention (renewals, expansions, downgrades, and pauses at the plan level, not just customer presence/absence).