{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-descriptive-statistics",
  "semantic_family_id": "sf-descriptive-statistics",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "有限觀測列的描述統計"
  },
  "summary": {
    "zh-Hant": "保留重數的非空觀測列，精確計算均值、中位數、全距與所有最大頻率值。"
  },
  "statement": {
    "zh-Hant": "給非空有限實數觀測列 D=(d1,…,dn)，算術平均為 Σdi/n；排序後奇數 n 的中位數是中央值，偶數 n 是中間兩值的平均；全距為最大值減最小值。本輪眾數集合約定為所有出現頻率達最大值的數，按遞增列出；各數均只出現一次時，全部不同觀測值都是此約定的頻率最大者。"
  },
  "formula": {
    "tex": "M(D)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>M</mi><mo>(</mo><mi>D</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "M",
        "glyph": "M"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "D",
          "glyph": "D"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "a0304df2c68d7596b64a508b4ced7f6fa87623b36ff3834e221d335ee8c3711b"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "觀測列保留重複值，不化成去重集合；排序只改順序。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫使用原創合成整數資料，均值及中位數為最簡正分母分數，n>0。"
    },
    {
      "id": "a3",
      "zh-Hant": "本輪明示眾數集合慣例，不否定其他教材對無眾數的用語；不從描述摘要推論母體、機率律或因果。"
    }
  ],
  "symbols": [
    {
      "token": "M",
      "role": "descriptive_summary",
      "role_zh": "指定描述摘要函數",
      "scope": "representative_formula"
    },
    {
      "token": "D",
      "role": "nonempty_observation_list",
      "role_zh": "保留重數的非空觀測列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-arithmetic-operations",
      "title_zh": "四則運算",
      "reason_zh": "求和、差與平均。"
    },
    {
      "id": "mko-rational-number",
      "title_zh": "有理數",
      "reason_zh": "精確有理摘要表示。"
    },
    {
      "id": "mko-order-relation",
      "title_zh": "實數大小與順序",
      "reason_zh": "排序、極值及中央位置。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "均值不一定是某筆觀測；偶數個資料的中位數可能在中間兩值之間。單筆觀測的均值、中位數等於本身，全距零。",
      "眾數集合可能有多個元素；輸出按值排序且每值只列一次，但計頻率必須保留觀測重數。全相同資料的眾數集合是單元素。",
      "描述統計與上一輪的有限機率期望、變異數物件不同。沒有把觀測列假定成獨立樣本或已知母體，也不計 n−1 樣本變異數；其他分位數和圖表仍是後續切片。"
    ],
    "common_misconception_zh": "先去重再算均值／中位數，漏掉並列頻率最大者，或把樣本摘要當機率模型證明。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限觀測列摘要可重算；統計推論、一般分位數與機率模型均未新增形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "橋接有限列表求和、排序中位及頻數最大集合；保留樣本摘要與機率矩數的型別區別。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-descriptive-statistics.md 整理；2026-10-05 脈衝 R07（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}