docs: document mergeable population standard deviation

[v0y4g3r]

What changed

• document stddev_pop_state, stddev_pop_merge, and stddev_pop_calc
• explain their input, output, NULL, non-finite input, and malformed-state behavior
• add an end-to-end example that persists minute-level states and verifies the merged result against direct stddev_pop aggregation
• update both English and Chinese Nightly documentation

This documents the functions introduced by GreptimeTeam/greptimedb#8972.

Closes #2802

Scope

• Documentation versions: Nightly
• Languages: English, Chinese

The feature is introduced for GreptimeDB 1.3.0, so released documentation versions are not changed.

Verification

• pnpm build
• DOC_LANG=zh pnpm build
• DOC_LANG=en pnpm check:links
• DOC_LANG=zh pnpm check:links
• git diff --check
• manually compared the documented behavior and SQL result with GreptimeDB PR #8972

Checklist

• I verified the content against the applicable GreptimeDB version.
• I updated the relevant documentation versions and languages, or explained why not.
• I checked changed links and anchors.
• Navigation does not require an update because this extends an existing page.

[Copilot]

Pull request overview

Adds Nightly documentation for GreptimeDB’s mergeable population standard deviation state functions (stddev_pop_state, stddev_pop_merge, stddev_pop_calc) on the existing “Approximate Functions” reference page, including behavioral notes (NULL handling, malformed state behavior, non-finite inputs) and an end-to-end SQL example. The same content is added to both the English and Chinese Nightly docs.

Changes:

• Documented stddev_pop_state, stddev_pop_merge, and stddev_pop_calc, including input/output types and NULL / malformed-state semantics.
• Added an end-to-end SQL example that persists minute-level states, merges them, and compares the result to direct stddev_pop.
• Applied the same documentation update to both English and Chinese Nightly pages.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated no comments.

File	Description
docs/reference/sql/functions/approximate.md	Adds English docs section for mergeable Welford-based stddev_pop_* state/merge/calc functions with a full SQL example.
i18n/zh/docusaurus-plugin-content-docs/current/reference/sql/functions/approximate.md	Adds the corresponding Chinese section and example, aligned with the English Nightly docs.

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View logs

[fengjiachun]

Both stddev_pop_state and stddev_pop_merge explicitly reject DISTINCT in the merged implementation (Welford DISTINCT aggregations are not available). Since these are introduced here as aggregate functions, please document this limitation and mirror it in the Chinese page; otherwise users may reasonably expect stddev_pop_state(DISTINCT value) to work.