第 22666 題：全變異數公式

原題與已知條件

同一有限law與完整disjoint partition生成的G。操作=totalvar-within-between-components；資料={"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 同一有限law與完整disjoint partition生成的G。操作=totalvar-within-between-components；資料={"masses":[[1,25],[2,25],[22,25],[0,1]],"values":[15,16,18,114],"partition":[[0,1],[2,3]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-law-of-total-variance

逐步推導

source operation=totalvar-within-between-components;original snapshot and mathematics retained

explicit ordered source output fields=[cell_rows, within, between, certificate]

Total variance complete probability mass := sum(1/25, 2/25, 22/25, 0)

Total variance complete probability mass = 1

ORIGINAL masses=[1/25, 2/25, 22/25, 0];SIGNED original X=[15, 16, 18, 114];ORIGINAL cell order=[[0, 1], [2, 3]]

ALL cells NONEMPTY and every atom occurs EXACTLY ONCE,including null atoms;normalized mass=1;no independence premise

Total variance original atom0 weighted X := mul(1/25, 15)

Total variance original atom0 weighted X = 3/5

Total variance original atom1 weighted X := mul(2/25, 16)

Total variance original atom1 weighted X = 32/25

Total variance original atom2 weighted X := mul(22/25, 18)

Total variance original atom2 weighted X = 396/25

Total variance original atom3 weighted X := mul(0, 114)

Total variance original atom3 weighted X = 0

Total variance ORIGINAL global mean := sum(3/5, 32/25, 396/25, 0)

Total variance ORIGINAL global mean = 443/25

Total variance original atom0 centered X := sub(15, 443/25)

Total variance original atom0 centered X = -68/25

Total variance original atom0 squared deviation := mul(-68/25, -68/25)

Total variance original atom0 squared deviation = 4624/625

Total variance original atom0 weighted squared deviation := mul(1/25, 4624/625)

Total variance original atom0 weighted squared deviation = 4624/15625

Total variance original atom1 centered X := sub(16, 443/25)

Total variance original atom1 centered X = -43/25

Total variance original atom1 squared deviation := mul(-43/25, -43/25)

Total variance original atom1 squared deviation = 1849/625

Total variance original atom1 weighted squared deviation := mul(2/25, 1849/625)

Total variance original atom1 weighted squared deviation = 3698/15625

Total variance original atom2 centered X := sub(18, 443/25)

Total variance original atom2 centered X = 7/25

Total variance original atom2 squared deviation := mul(7/25, 7/25)

Total variance original atom2 squared deviation = 49/625

Total variance original atom2 weighted squared deviation := mul(22/25, 49/625)

Total variance original atom2 weighted squared deviation = 1078/15625

Total variance original atom3 centered X := sub(114, 443/25)

Total variance original atom3 centered X = 2407/25

Total variance original atom3 squared deviation := mul(2407/25, 2407/25)

Total variance original atom3 squared deviation = 5793649/625

Total variance original atom3 weighted squared deviation := mul(0, 5793649/625)

Total variance original atom3 weighted squared deviation = 0

Total variance ORIGINAL global variance := sum(4624/15625, 3698/15625, 1078/15625, 0)

Total variance ORIGINAL global variance = 376/625

Total variance cell0 mass := sum(1/25, 2/25)

Total variance cell0 mass = 3/25

Total variance cell0 weighted numerator := sum(3/5, 32/25)

Total variance cell0 weighted numerator = 47/25

Total variance positive cell0 mean := div(47/25, 3/25)

Total variance positive cell0 mean = 47/3

Total variance cell0 atom0 conditional centering := sub(15, 47/3)

Total variance cell0 atom0 conditional centering = -2/3

Total variance cell0 atom0 conditional square := mul(-2/3, -2/3)

Total variance cell0 atom0 conditional square = 4/9

Total variance cell0 atom0 normalized conditional mass := div(1/25, 3/25)

Total variance cell0 atom0 normalized conditional mass = 1/3

Total variance cell0 atom0 conditional variance term := mul(1/3, 4/9)

Total variance cell0 atom0 conditional variance term = 4/27

Total variance cell0 atom0 weighted residual := mul(1/25, -2/3)

Total variance cell0 atom0 weighted residual = -2/75

Total variance cell0 atom1 conditional centering := sub(16, 47/3)

Total variance cell0 atom1 conditional centering = 1/3

Total variance cell0 atom1 conditional square := mul(1/3, 1/3)

Total variance cell0 atom1 conditional square = 1/9

Total variance cell0 atom1 normalized conditional mass := div(2/25, 3/25)

Total variance cell0 atom1 normalized conditional mass = 2/3

Total variance cell0 atom1 conditional variance term := mul(2/3, 1/9)

Total variance cell0 atom1 conditional variance term = 2/27

Total variance cell0 atom1 weighted residual := mul(2/25, 1/3)

Total variance cell0 atom1 weighted residual = 2/75

Total variance positive cell0 conditional variance := sum(4/27, 2/27)

Total variance positive cell0 conditional variance = 2/9

Total variance positive cell0 residual mean zero := sum(-2/75, 2/75)

Total variance positive cell0 residual mean zero = 0

ORIGINAL cell index=0;actual atoms=[0, 1];mass=3/25;numerator=47/25

conditional mean=47/3;conditional variance=2/9;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell1 mass := sum(22/25, 0)

Total variance cell1 mass = 22/25

Total variance cell1 weighted numerator := sum(396/25, 0)

Total variance cell1 weighted numerator = 396/25

Total variance positive cell1 mean := div(396/25, 22/25)

Total variance positive cell1 mean = 18

Total variance cell1 atom2 conditional centering := sub(18, 18)

Total variance cell1 atom2 conditional centering = 0

Total variance cell1 atom2 conditional square := mul(0, 0)

Total variance cell1 atom2 conditional square = 0

Total variance cell1 atom2 normalized conditional mass := div(22/25, 22/25)

Total variance cell1 atom2 normalized conditional mass = 1

Total variance cell1 atom2 conditional variance term := mul(1, 0)

Total variance cell1 atom2 conditional variance term = 0

Total variance cell1 atom2 weighted residual := mul(22/25, 0)

Total variance cell1 atom2 weighted residual = 0

Total variance cell1 atom3 conditional centering := sub(114, 18)

Total variance cell1 atom3 conditional centering = 96

Total variance cell1 atom3 conditional square := mul(96, 96)

Total variance cell1 atom3 conditional square = 9216

Total variance cell1 atom3 normalized conditional mass := div(0, 22/25)

Total variance cell1 atom3 normalized conditional mass = 0

Total variance cell1 atom3 conditional variance term := mul(0, 9216)

Total variance cell1 atom3 conditional variance term = 0

Total variance cell1 atom3 weighted residual := mul(0, 96)

Total variance cell1 atom3 weighted residual = 0

Total variance positive cell1 conditional variance := sum(0, 0)

Total variance positive cell1 conditional variance = 0

Total variance positive cell1 residual mean zero := sum(0, 0)

Total variance positive cell1 residual mean zero = 0

ORIGINAL cell index=1;actual atoms=[2, 3];mass=22/25;numerator=396/25

conditional mean=18;conditional variance=0;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell0 weighted WITHIN term := mul(3/25, 2/9)

Total variance cell0 weighted WITHIN term = 2/75

Total variance cell0 mean minus global mean := sub(47/3, 443/25)

Total variance cell0 mean minus global mean = -154/75

Total variance cell0 BETWEEN squared deviation := mul(-154/75, -154/75)

Total variance cell0 BETWEEN squared deviation = 23716/5625

Total variance cell0 weighted BETWEEN term := mul(3/25, 23716/5625)

Total variance cell0 weighted BETWEEN term = 23716/46875

Total variance cell1 weighted WITHIN term := mul(22/25, 0)

Total variance cell1 weighted WITHIN term = 0

Total variance cell1 mean minus global mean := sub(18, 443/25)

Total variance cell1 mean minus global mean = 7/25

Total variance cell1 BETWEEN squared deviation := mul(7/25, 7/25)

Total variance cell1 BETWEEN squared deviation = 49/625

Total variance cell1 weighted BETWEEN term := mul(22/25, 49/625)

Total variance cell1 weighted BETWEEN term = 1078/15625

Total variance sum WITHIN terms := sum(2/75, 0)

Total variance sum WITHIN terms = 2/75

Total variance sum BETWEEN terms := sum(23716/46875, 1078/15625)

Total variance sum BETWEEN terms = 1078/1875

Total variance WITHIN plus BETWEEN := sum(2/75, 1078/1875)

Total variance WITHIN plus BETWEEN = 376/625

ORIGINAL SAME-law mean=443/25;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/25, 47/3, 2/9], [1, [2, 3], 22/25, 18, 0]]

Var(X)=376/625;within=2/75;between=1078/1875;sum=376/625;X-mu=(X-c_j)+(c_j-mu),and each positive-cell cross term2(c_j-mu)*sum_i p_i(X_i-c_j)=0

逐步反向核對

• S065 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S064 · premise_and_result_audit — 以精確有理數重算 Total variance WITHIN plus BETWEEN，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Total variance sum BETWEEN terms，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Total variance sum WITHIN terms，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 weighted BETWEEN term，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 BETWEEN squared deviation，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 mean minus global mean，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 weighted WITHIN term，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 weighted BETWEEN term，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 BETWEEN squared deviation，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 mean minus global mean，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 weighted WITHIN term，不以小數近似替代。
• S053 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S052 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell1 residual mean zero，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell1 conditional variance，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom3 weighted residual，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom3 conditional variance term，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom3 normalized conditional mass，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom3 conditional square，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom3 conditional centering，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom2 weighted residual，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom2 conditional variance term，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom2 normalized conditional mass，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom2 conditional square，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 atom2 conditional centering，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell1 mean，不以小數近似替代。
• S039 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 weighted numerator，不以小數近似替代。
• S038 · premise_and_result_audit — 以精確有理數重算 Total variance cell1 mass，不以小數近似替代。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell0 residual mean zero，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell0 conditional variance，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom1 weighted residual，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom1 conditional variance term，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom1 normalized conditional mass，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom1 conditional square，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom1 conditional centering，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom0 weighted residual，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom0 conditional variance term，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom0 normalized conditional mass，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom0 conditional square，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 atom0 conditional centering，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Total variance positive cell0 mean，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 weighted numerator，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Total variance cell0 mass，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Total variance ORIGINAL global variance，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Total variance original atom3 weighted squared deviation，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Total variance original atom3 squared deviation，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Total variance original atom3 centered X，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Total variance original atom2 weighted squared deviation，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Total variance original atom2 squared deviation，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Total variance original atom2 centered X，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Total variance original atom1 weighted squared deviation，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Total variance original atom1 squared deviation，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Total variance original atom1 centered X，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Total variance original atom0 weighted squared deviation，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Total variance original atom0 squared deviation，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Total variance original atom0 centered X，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Total variance ORIGINAL global mean，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Total variance original atom3 weighted X，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Total variance original atom2 weighted X，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Total variance original atom1 weighted X，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Total variance original atom0 weighted X，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 以精確有理數重算 Total variance complete probability mass，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([[0, [0, 1], 3/25, 47/3, 2/9], [1, [2, 3], 22/25, 18, 0]], 2/75, 1078/1875, same_law_same_partition_within_and_between_terms)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。