第 22622 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":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"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":[[1,31],[2,31],[28,31],[0,1]],"values":[21,22,24,120],"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(1/31, 2/31, 28/31, 0)

Total variance complete probability mass = 1

ORIGINAL masses=[1/31, 2/31, 28/31, 0];SIGNED original X=[21, 22, 24, 120];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(1/31, 21)

Total variance original atom0 weighted X = 21/31

Total variance original atom1 weighted X := mul(2/31, 22)

Total variance original atom1 weighted X = 44/31

Total variance original atom2 weighted X := mul(28/31, 24)

Total variance original atom2 weighted X = 672/31

Total variance original atom3 weighted X := mul(0, 120)

Total variance original atom3 weighted X = 0

Total variance ORIGINAL global mean := sum(21/31, 44/31, 672/31, 0)

Total variance ORIGINAL global mean = 737/31

Total variance original atom0 centered X := sub(21, 737/31)

Total variance original atom0 centered X = -86/31

Total variance original atom0 squared deviation := mul(-86/31, -86/31)

Total variance original atom0 squared deviation = 7396/961

Total variance original atom0 weighted squared deviation := mul(1/31, 7396/961)

Total variance original atom0 weighted squared deviation = 7396/29791

Total variance original atom1 centered X := sub(22, 737/31)

Total variance original atom1 centered X = -55/31

Total variance original atom1 squared deviation := mul(-55/31, -55/31)

Total variance original atom1 squared deviation = 3025/961

Total variance original atom1 weighted squared deviation := mul(2/31, 3025/961)

Total variance original atom1 weighted squared deviation = 6050/29791

Total variance original atom2 centered X := sub(24, 737/31)

Total variance original atom2 centered X = 7/31

Total variance original atom2 squared deviation := mul(7/31, 7/31)

Total variance original atom2 squared deviation = 49/961

Total variance original atom2 weighted squared deviation := mul(28/31, 49/961)

Total variance original atom2 weighted squared deviation = 1372/29791

Total variance original atom3 centered X := sub(120, 737/31)

Total variance original atom3 centered X = 2983/31

Total variance original atom3 squared deviation := mul(2983/31, 2983/31)

Total variance original atom3 squared deviation = 8898289/961

Total variance original atom3 weighted squared deviation := mul(0, 8898289/961)

Total variance original atom3 weighted squared deviation = 0

Total variance ORIGINAL global variance := sum(7396/29791, 6050/29791, 1372/29791, 0)

Total variance ORIGINAL global variance = 478/961

Total variance cell0 mass := sum(1/31, 2/31)

Total variance cell0 mass = 3/31

Total variance cell0 weighted numerator := sum(21/31, 44/31)

Total variance cell0 weighted numerator = 65/31

Total variance positive cell0 mean := div(65/31, 3/31)

Total variance positive cell0 mean = 65/3

Total variance cell0 atom0 conditional centering := sub(21, 65/3)

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(1/31, 3/31)

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(1/31, -2/3)

Total variance cell0 atom0 weighted residual = -2/93

Total variance cell0 atom1 conditional centering := sub(22, 65/3)

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(2/31, 3/31)

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(2/31, 1/3)

Total variance cell0 atom1 weighted residual = 2/93

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(-2/93, 2/93)

Total variance positive cell0 residual mean zero = 0

ORIGINAL cell index=0;actual atoms=[0, 1];mass=3/31;numerator=65/31

conditional mean=65/3;conditional variance=2/9;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell1 mass := sum(28/31, 0)

Total variance cell1 mass = 28/31

Total variance cell1 weighted numerator := sum(672/31, 0)

Total variance cell1 weighted numerator = 672/31

Total variance positive cell1 mean := div(672/31, 28/31)

Total variance positive cell1 mean = 24

Total variance cell1 atom2 conditional centering := sub(24, 24)

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(28/31, 28/31)

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(28/31, 0)

Total variance cell1 atom2 weighted residual = 0

Total variance cell1 atom3 conditional centering := sub(120, 24)

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, 28/31)

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=28/31;numerator=672/31

conditional mean=24;conditional variance=0;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell0 weighted WITHIN term := mul(3/31, 2/9)

Total variance cell0 weighted WITHIN term = 2/93

Total variance cell0 mean minus global mean := sub(65/3, 737/31)

Total variance cell0 mean minus global mean = -196/93

Total variance cell0 BETWEEN squared deviation := mul(-196/93, -196/93)

Total variance cell0 BETWEEN squared deviation = 38416/8649

Total variance cell0 weighted BETWEEN term := mul(3/31, 38416/8649)

Total variance cell0 weighted BETWEEN term = 38416/89373

Total variance cell1 weighted WITHIN term := mul(28/31, 0)

Total variance cell1 weighted WITHIN term = 0

Total variance cell1 mean minus global mean := sub(24, 737/31)

Total variance cell1 mean minus global mean = 7/31

Total variance cell1 BETWEEN squared deviation := mul(7/31, 7/31)

Total variance cell1 BETWEEN squared deviation = 49/961

Total variance cell1 weighted BETWEEN term := mul(28/31, 49/961)

Total variance cell1 weighted BETWEEN term = 1372/29791

Total variance sum WITHIN terms := sum(2/93, 0)

Total variance sum WITHIN terms = 2/93

Total variance sum BETWEEN terms := sum(38416/89373, 1372/29791)

Total variance sum BETWEEN terms = 1372/2883

Total variance WITHIN plus BETWEEN := sum(2/93, 1372/2883)

Total variance WITHIN plus BETWEEN = 478/961

ORIGINAL SAME-law mean=737/31;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1], 3/31, 65/3, 2/9], [1, [2, 3], 28/31, 24, 0]]

Var(X)=478/961;within=2/93;between=1372/2883;sum=478/961;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

(478/961, 2/93, 1372/2883, 478/961, 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。