Pareto Power-Law Distribution (α, x_m), Tail Survival & Implied Gini Index Calculator

Evaluate Pareto tail survival (x_m/x)^α, power-law mean, and implied Gini inequality.

Pareto Tail Probability P(X > 250)
25.298 %
Expected Mean α·x_m / (α − 1) 300 units
Distribution Median (x_m · 2^(1/α)) 158.74 units
Implied Gini Inequality Coefficient 0.5 Gini (0–1)
Inputs
⚡ Scenarios:
x_m
Minimum scale threshold x_m (>0).
α
Pareto tail shape exponent α (α ≈ 1.16 gives the classic 80/20 rule).
x
Target threshold value x (≥ x_m) to evaluate P(X > x).
Results & Graph
Pareto Tail Probability P(X > 250)
25.298 %
Expected Mean α·x_m / (α − 1)
300 units
Distribution Median (x_m · 2^(1/α))
158.74 units
Implied Gini Inequality Coefficient
0.5 Gini (0–1)

Pareto Tail Probability P(X > 250)

Graph 1 • Pareto Power-Law & 80/20 • Domain Architecture & Live Telemetry Maps Scale Minimum Xm, Tail Exponent Alpha, Target Threshold X through Pareto Power-Law & 80/20 Formula.
Pareto Tail Probability P(X > : 25.298 % Pareto Power-Law & 80/20 • Invariants & Metrics Pareto Tail Probability P(X 25.298 % Expected Mean α·x_m / (α − 1 300 units Distribution Median (x_m · 2 158.74 units Implied Gini Inequality Coef 0.5 Gini (0–1)
Graph 2 • Pareto Tail Probability P(X > 250) Across Scale Minimum Xm Sweep (40% → 160%) Evaluates P(X > x) = (x_m / x)^α; E[X] = α·x_m / (α − 1) for α > 1; Gini = 1 / across 7 operating points (current input highlighted).
0 % 28.16 % 56.32 % 6.40 40x_m 11.76 60x_m 18.10 80x_m 25.30 100x_m 33.26 120x_m 41.91 140x_m 51.20 160x_m
Graph 3 • Joint Sensitivity: Scale Minimum Xm × Tail Exponent Alpha Compares Pareto Tail Probability P(X > 250) when Tail Exponent Alpha is shifted +25% (solid) vs. −25% (dashed).
0 % 26.36 % 52.71 % --- Tail Exponent Alpha @ 75% 9.19 -30% 13.23 -15% 17.94 +0% 23.32 +15% 29.34 +30%
Summary: With x_m = 100 and α = 1.5, tail survival P(X > 250) is 25.3%, expected mean is 300, and implied Gini coefficient is 0.5.
Summary: With x_m = 100 and α = 1.5, tail survival P(X > 250) is 25.3%, expected mean is 300, and implied Gini coefficient is 0.5.
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Formula & Mathematical Methodology: Pareto Power-Law & 80/20 Formula

P(X > x) = (xm / x)^α
E[X] = α·xm / (α − 1) for α > 1
Gini = 1 / (2α − 1)
P(X > x) = (x_m / x)^α; E[X] = α·x_m / (α − 1) for α > 1; Gini = 1 / (2α − 1)
  • scaleMinimumXm: Scale Minimum Xm (x_m)
  • tailExponentAlpha: Tail Exponent Alpha (α)
  • targetThresholdX: Target Threshold X (x)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Pareto Power-Law & 80/20

Evaluate Pareto tail survival (x_m/x)^α, power-law mean, and implied Gini inequality.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: P(X > x) = (x_m / x)^α; E[X] = α·x_m / (α − 1) for α > 1; Gini = 1 / (2α − 1).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Pareto Power-Law & 80/20 via P(X > x) = (x_m / x)^α; E[X] = α·x_m / (α − 1) for α > 1; Gini = 1 / (2α − 1).

Frequently Asked Questions

How does the Pareto Power-Law Distribution (α, x_m), Tail Survival & Implied Gini Index Calculator compute the solution?

It evaluates P(X > x) = (x_m / x)^α; E[X] = α·x_m / (α − 1) for α > 1; Gini = 1 / (2α − 1) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Probability Distributions & Survival Analysis

Pareto Power-Law Distribution (α, x_m), Tail Survival & Implied Gini Index Calculator: inputs, method, and how to read the result

Evaluate Pareto tail survival (x_m/x)^α, power-law mean, and implied Gini inequality.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Scale Minimum Xm: Minimum scale threshold x_m (>0).
  • Tail Exponent Alpha: Pareto tail shape exponent α (α ≈ 1.16 gives the classic 80/20 rule).
  • Target Threshold X: Target threshold value x (≥ x_m) to evaluate P(X > x).

What the calculation includes

With the default values shown, Pareto Tail Probability P(X > 250) is 25.298 %. The calculation follows Pareto Power-Law & 80/20 Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Scale Minimum Xm from 100 x_m to 110 x_m changes Pareto Tail Probability P(X > 250) from 25.298 % to 29.186 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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