Nd6 Dice Sum Mean & Variance

Compute mean/variance for N independent identically weighted d6 from caller face weights; unlike generic sample statistics, weights describe a probability model and no correlated-dice assumption is hidden.

Description

Loaded-die weights define a probability model only after normalization.

Loaded-die weights define a probability model only after normalization. Independent dice add expectations and variances.^[1]

Enter integer dice count 1–1000000 and exactly six nonnegative face weights with positive total. Return face probabilities, single-die mean and modeled total moments.

When to use Nd6 Dice Sum Mean & Variance

Use Nd6 Dice Sum Mean & Variance when you need to evaluate the supplied records under the calculation and evidence rules below. No observed fairness test or correlated-roll model is performed.

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.

How to use Nd6 Dice Sum Mean & Variance and understand its results

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.
Enter independentD6Count, callerFaceProbabilityWeights using the field table and calculation rules, then run the tool.
Read singleDieMean, sumMean, sumVariance. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
independentD6Count	Independent d6 count	number	Required
callerFaceProbabilityWeights	Caller face probability weights	list of number	Required

Enter integer dice count 1–1000000 and exactly six nonnegative face weights with positive total. Return face probabilities, single-die mean and modeled total moments.

p_i=weight_i/sum(weights); totalMean=n×mean; totalVariance=n×variance

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
singleDieMean	Single die mean	number
sumMean	Sum mean	number
sumVariance	Sum variance	number

json

{
  "input": {
    "independentD6Count": 2,
    "callerFaceProbabilityWeights": [
      1,
      1,
      1,
      1,
      1,
      1
    ]
  },
  "output": {
    "singleDieMean": 3.5,
    "sumMean": 7,
    "sumVariance": 5.833333333333334
  }
}

Limitations and assumptions

No observed fairness test or correlated-roll model is performed.

Numeric inputs, where present, must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Numeric lists are limited to 10,000 entries; string lists to 128 strings of at most 128 characters each. The rules above may impose smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use 2d6 Sum Probability Table: Two fair six-sided dice have 36 equally likely ordered outcomes. Different sums have different counts. For a complementary task, use Risk 3v2 Attack Odds Lookup: Risk compares sorted attacking and defending dice, with ties favoring the defender. Exhaustive enumeration gives an exact fair-dice casualty model.

References

Nd6 Dice Sum Mean & Variance — tool.io

Similar or alternative tools

2d6 Sum Probability Table
Two fair six-sided dice have 36 equally likely ordered outcomes. Different sums have different counts.
Risk 3v2 Attack Odds Lookup
Risk compares sorted attacking and defending dice, with ties favoring the defender. Exhaustive enumeration gives an exact fair-dice casualty model.