第 22619 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-full-identity；資料={"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"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-full-identity；資料={"masses":[[1,28],[2,28],[25,28],[0,1]],"values":[18,19,21,117],"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-full-identity;original snapshot and mathematics retained

explicit ordered source output fields=[variance, within, between, sum, equal]

Total variance complete probability mass := sum(1/28, 1/14, 25/28, 0)

Total variance complete probability mass = 1

ORIGINAL masses=[1/28, 1/14, 25/28, 0];SIGNED original X=[18, 19, 21, 117];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(1/28, 18)

Total variance original atom0 weighted X = 9/14

Total variance original atom1 weighted X := mul(1/14, 19)

Total variance original atom1 weighted X = 19/14

Total variance original atom2 weighted X := mul(25/28, 21)

Total variance original atom2 weighted X = 75/4

Total variance original atom3 weighted X := mul(0, 117)

Total variance original atom3 weighted X = 0

Total variance ORIGINAL global mean := sum(9/14, 19/14, 75/4, 0)

Total variance ORIGINAL global mean = 83/4

Total variance original atom0 centered X := sub(18, 83/4)

Total variance original atom0 centered X = -11/4

Total variance original atom0 squared deviation := mul(-11/4, -11/4)

Total variance original atom0 squared deviation = 121/16

Total variance original atom0 weighted squared deviation := mul(1/28, 121/16)

Total variance original atom0 weighted squared deviation = 121/448

Total variance original atom1 centered X := sub(19, 83/4)

Total variance original atom1 centered X = -7/4

Total variance original atom1 squared deviation := mul(-7/4, -7/4)

Total variance original atom1 squared deviation = 49/16

Total variance original atom1 weighted squared deviation := mul(1/14, 49/16)

Total variance original atom1 weighted squared deviation = 7/32

Total variance original atom2 centered X := sub(21, 83/4)

Total variance original atom2 centered X = 1/4

Total variance original atom2 squared deviation := mul(1/4, 1/4)

Total variance original atom2 squared deviation = 1/16

Total variance original atom2 weighted squared deviation := mul(25/28, 1/16)

Total variance original atom2 weighted squared deviation = 25/448

Total variance original atom3 centered X := sub(117, 83/4)

Total variance original atom3 centered X = 385/4

Total variance original atom3 squared deviation := mul(385/4, 385/4)

Total variance original atom3 squared deviation = 148225/16

Total variance original atom3 weighted squared deviation := mul(0, 148225/16)

Total variance original atom3 weighted squared deviation = 0

Total variance ORIGINAL global variance := sum(121/448, 7/32, 25/448, 0)

Total variance ORIGINAL global variance = 61/112

Total variance cell0 mass := sum(1/28, 1/14)

Total variance cell0 mass = 3/28

Total variance cell0 weighted numerator := sum(9/14, 19/14)

Total variance cell0 weighted numerator = 2

Total variance positive cell0 mean := div(2, 3/28)

Total variance positive cell0 mean = 56/3

Total variance cell0 atom0 conditional centering := sub(18, 56/3)

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(1/28, 3/28)

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(1/28, -2/3)

Total variance cell0 atom0 weighted residual = -1/42

Total variance cell0 atom1 conditional centering := sub(19, 56/3)

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(1/14, 3/28)

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(1/14, 1/3)

Total variance cell0 atom1 weighted residual = 1/42

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(-1/42, 1/42)

Total variance positive cell0 residual mean zero = 0

ORIGINAL cell index=0;actual atoms=[0, 1];mass=3/28;numerator=2

conditional mean=56/3;conditional variance=2/9;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell1 mass := sum(25/28, 0)

Total variance cell1 mass = 25/28

Total variance cell1 weighted numerator := sum(75/4, 0)

Total variance cell1 weighted numerator = 75/4

Total variance positive cell1 mean := div(75/4, 25/28)

Total variance positive cell1 mean = 21

Total variance cell1 atom2 conditional centering := sub(21, 21)

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(25/28, 25/28)

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(25/28, 0)

Total variance cell1 atom2 weighted residual = 0

Total variance cell1 atom3 conditional centering := sub(117, 21)

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, 25/28)

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=25/28;numerator=75/4

conditional mean=21;conditional variance=0;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell0 weighted WITHIN term := mul(3/28, 2/9)

Total variance cell0 weighted WITHIN term = 1/42

Total variance cell0 mean minus global mean := sub(56/3, 83/4)

Total variance cell0 mean minus global mean = -25/12

Total variance cell0 BETWEEN squared deviation := mul(-25/12, -25/12)

Total variance cell0 BETWEEN squared deviation = 625/144

Total variance cell0 weighted BETWEEN term := mul(3/28, 625/144)

Total variance cell0 weighted BETWEEN term = 625/1344

Total variance cell1 weighted WITHIN term := mul(25/28, 0)

Total variance cell1 weighted WITHIN term = 0

Total variance cell1 mean minus global mean := sub(21, 83/4)

Total variance cell1 mean minus global mean = 1/4

Total variance cell1 BETWEEN squared deviation := mul(1/4, 1/4)

Total variance cell1 BETWEEN squared deviation = 1/16

Total variance cell1 weighted BETWEEN term := mul(25/28, 1/16)

Total variance cell1 weighted BETWEEN term = 25/448

Total variance sum WITHIN terms := sum(1/42, 0)

Total variance sum WITHIN terms = 1/42

Total variance sum BETWEEN terms := sum(625/1344, 25/448)

Total variance sum BETWEEN terms = 25/48

Total variance WITHIN plus BETWEEN := sum(1/42, 25/48)

Total variance WITHIN plus BETWEEN = 61/112

ORIGINAL SAME-law mean=83/4;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/28, 56/3, 2/9], [1, [2, 3], 25/28, 21, 0]]

Var(X)=61/112;within=1/42;between=25/48;sum=61/112;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

(61/112, 1/42, 25/48, 61/112, 是)

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。