Question 22696: Undefined Null-Cell Mean and Variance Have Zero Outer Contribution

Original question and givens

Use the same complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1, 2], [3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).

• G001 — Use the same complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1, 2], [3]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-law-of-total-variance

Step-by-step derivation

source operation=totalvar-zero-mass-boundary;original snapshot and mathematics retained

explicit ordered source output fields=[cell_rows, within, between, variance, equal, null_cell_division_claim, boundary]

Total variance complete probability mass := sum(1/30, 1/15, 9/10, 0)

Total variance complete probability mass = 1

ORIGINAL masses=[1/30, 1/15, 9/10, 0];SIGNED original X=[20, 21, 23, 119];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(1/30, 20)

Total variance original atom0 weighted X = 2/3

Total variance original atom1 weighted X := mul(1/15, 21)

Total variance original atom1 weighted X = 7/5

Total variance original atom2 weighted X := mul(9/10, 23)

Total variance original atom2 weighted X = 207/10

Total variance original atom3 weighted X := mul(0, 119)

Total variance original atom3 weighted X = 0

Total variance ORIGINAL global mean := sum(2/3, 7/5, 207/10, 0)

Total variance ORIGINAL global mean = 683/30

Total variance original atom0 centered X := sub(20, 683/30)

Total variance original atom0 centered X = -83/30

Total variance original atom0 squared deviation := mul(-83/30, -83/30)

Total variance original atom0 squared deviation = 6889/900

Total variance original atom0 weighted squared deviation := mul(1/30, 6889/900)

Total variance original atom0 weighted squared deviation = 6889/27000

Total variance original atom1 centered X := sub(21, 683/30)

Total variance original atom1 centered X = -53/30

Total variance original atom1 squared deviation := mul(-53/30, -53/30)

Total variance original atom1 squared deviation = 2809/900

Total variance original atom1 weighted squared deviation := mul(1/15, 2809/900)

Total variance original atom1 weighted squared deviation = 2809/13500

Total variance original atom2 centered X := sub(23, 683/30)

Total variance original atom2 centered X = 7/30

Total variance original atom2 squared deviation := mul(7/30, 7/30)

Total variance original atom2 squared deviation = 49/900

Total variance original atom2 weighted squared deviation := mul(9/10, 49/900)

Total variance original atom2 weighted squared deviation = 49/1000

Total variance original atom3 centered X := sub(119, 683/30)

Total variance original atom3 centered X = 2887/30

Total variance original atom3 squared deviation := mul(2887/30, 2887/30)

Total variance original atom3 squared deviation = 8334769/900

Total variance original atom3 weighted squared deviation := mul(0, 8334769/900)

Total variance original atom3 weighted squared deviation = 0

Total variance ORIGINAL global variance := sum(6889/27000, 2809/13500, 49/1000, 0)

Total variance ORIGINAL global variance = 461/900

Total variance cell0 mass := sum(1/30, 1/15, 9/10)

Total variance cell0 mass = 1

Total variance cell0 weighted numerator := sum(2/3, 7/5, 207/10)

Total variance cell0 weighted numerator = 683/30

Total variance positive cell0 mean := div(683/30, 1)

Total variance positive cell0 mean = 683/30

Total variance cell0 atom0 conditional centering := sub(20, 683/30)

Total variance cell0 atom0 conditional centering = -83/30

Total variance cell0 atom0 conditional square := mul(-83/30, -83/30)

Total variance cell0 atom0 conditional square = 6889/900

Total variance cell0 atom0 normalized conditional mass := div(1/30, 1)

Total variance cell0 atom0 normalized conditional mass = 1/30

Total variance cell0 atom0 conditional variance term := mul(1/30, 6889/900)

Total variance cell0 atom0 conditional variance term = 6889/27000

Total variance cell0 atom0 weighted residual := mul(1/30, -83/30)

Total variance cell0 atom0 weighted residual = -83/900

Total variance cell0 atom1 conditional centering := sub(21, 683/30)

Total variance cell0 atom1 conditional centering = -53/30

Total variance cell0 atom1 conditional square := mul(-53/30, -53/30)

Total variance cell0 atom1 conditional square = 2809/900

Total variance cell0 atom1 normalized conditional mass := div(1/15, 1)

Total variance cell0 atom1 normalized conditional mass = 1/15

Total variance cell0 atom1 conditional variance term := mul(1/15, 2809/900)

Total variance cell0 atom1 conditional variance term = 2809/13500

Total variance cell0 atom1 weighted residual := mul(1/15, -53/30)

Total variance cell0 atom1 weighted residual = -53/450

Total variance cell0 atom2 conditional centering := sub(23, 683/30)

Total variance cell0 atom2 conditional centering = 7/30

Total variance cell0 atom2 conditional square := mul(7/30, 7/30)

Total variance cell0 atom2 conditional square = 49/900

Total variance cell0 atom2 normalized conditional mass := div(9/10, 1)

Total variance cell0 atom2 normalized conditional mass = 9/10

Total variance cell0 atom2 conditional variance term := mul(9/10, 49/900)

Total variance cell0 atom2 conditional variance term = 49/1000

Total variance cell0 atom2 weighted residual := mul(9/10, 7/30)

Total variance cell0 atom2 weighted residual = 21/100

Total variance positive cell0 conditional variance := sum(6889/27000, 2809/13500, 49/1000)

Total variance positive cell0 conditional variance = 461/900

Total variance positive cell0 residual mean zero := sum(-83/900, -53/450, 21/100)

Total variance positive cell0 residual mean zero = 0

ORIGINAL cell index=0;actual atoms=[0, 1, 2];mass=1;numerator=683/30

conditional mean=683/30;conditional variance=461/900;positive-cell weighted residual sum0,so cross term vanishes

Total variance cell1 mass := sum(0)

Total variance cell1 mass = 0

Total variance cell1 weighted numerator := sum(0)

Total variance cell1 weighted numerator = 0

ORIGINAL cell index=1;actual atoms=[3];mass=0;numerator=0

conditional mean=null;conditional variance=null;NONEMPTY null cell:all atom masses0,no conditional division;BOTH quotients null

Total variance cell0 weighted WITHIN term := mul(1, 461/900)

Total variance cell0 weighted WITHIN term = 461/900

Total variance cell0 mean minus global mean := sub(683/30, 683/30)

Total variance cell0 mean minus global mean = 0

Total variance cell0 BETWEEN squared deviation := mul(0, 0)

Total variance cell0 BETWEEN squared deviation = 0

Total variance cell0 weighted BETWEEN term := mul(1, 0)

Total variance cell0 weighted BETWEEN term = 0

Total variance sum WITHIN terms := sum(461/900)

Total variance sum WITHIN terms = 461/900

Total variance sum BETWEEN terms := sum(0)

Total variance sum BETWEEN terms = 0

Total variance WITHIN plus BETWEEN := sum(461/900, 0)

Total variance WITHIN plus BETWEEN = 461/900

ORIGINAL SAME-law mean=683/30;ALL [index,NONEMPTYcell,mass,mean_or_null,var_or_null]=[[0, [0, 1, 2], 1, 683/30, 461/900], [1, [3], 0, null, null]]

Var(X)=461/900;within=461/900;between=0;sum=461/900;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

ACTUAL NONEMPTY null cell indices=[1];within=461/900,between=0,Var=461/900

null_cell_division_claim fixed No;both conditional quotients remain null;total-variance identity remains TRUE

Reverse checks

• S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S053 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S052 · premise_and_result_audit — Recompute Total variance WITHIN plus BETWEEN with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Total variance sum BETWEEN terms with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Total variance sum WITHIN terms with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Total variance cell0 weighted BETWEEN term with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Total variance cell0 BETWEEN squared deviation with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Total variance cell0 mean minus global mean with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Total variance cell0 weighted WITHIN term with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Recompute Total variance cell1 weighted numerator with exact rational arithmetic, not rounded decimals.
• S043 · premise_and_result_audit — Recompute Total variance cell1 mass with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute Total variance positive cell0 residual mean zero with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Recompute Total variance positive cell0 conditional variance with exact rational arithmetic, not rounded decimals.
• S039 · premise_and_result_audit — Recompute Total variance cell0 atom2 weighted residual with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional variance term with exact rational arithmetic, not rounded decimals.
• S037 · premise_and_result_audit — Recompute Total variance cell0 atom2 normalized conditional mass with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional square with exact rational arithmetic, not rounded decimals.
• S035 · premise_and_result_audit — Recompute Total variance cell0 atom2 conditional centering with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Recompute Total variance cell0 atom1 weighted residual with exact rational arithmetic, not rounded decimals.
• S033 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional variance term with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Recompute Total variance cell0 atom1 normalized conditional mass with exact rational arithmetic, not rounded decimals.
• S031 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional square with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Recompute Total variance cell0 atom1 conditional centering with exact rational arithmetic, not rounded decimals.
• S029 · premise_and_result_audit — Recompute Total variance cell0 atom0 weighted residual with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional variance term with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Total variance cell0 atom0 normalized conditional mass with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional square with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Total variance cell0 atom0 conditional centering with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Total variance positive cell0 mean with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Total variance cell0 weighted numerator with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Total variance cell0 mass with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Total variance ORIGINAL global variance with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Total variance original atom3 weighted squared deviation with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Total variance original atom3 squared deviation with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Total variance original atom3 centered X with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Total variance original atom2 weighted squared deviation with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Total variance original atom2 squared deviation with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Total variance original atom2 centered X with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Total variance original atom1 weighted squared deviation with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Total variance original atom1 squared deviation with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Total variance original atom1 centered X with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Total variance original atom0 weighted squared deviation with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Total variance original atom0 squared deviation with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Total variance original atom0 centered X with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Total variance ORIGINAL global mean with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Total variance original atom3 weighted X with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Total variance original atom2 weighted X with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Total variance original atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Total variance original atom0 weighted X with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Recompute Total variance complete probability mass with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[0, [0, 1, 2], 1, 683/30, 461/900], [1, [3], 0, null, null]], 461/900, 0, 461/900, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.