Box-Cox Power Transformation y^(λ) = (y^λ − 1)/λ & Inverse Back-Transformation Calculator

Transform skewed positive data via Box-Cox power parameter λ and bias-corrected inverse.

Box-Cox Transformed Value y^(0.5)
8 transformed
Bias-Corrected Back-Transform Mean 25.038 units
Natural Log Baseline (λ = 0) 3.219 ln(y)
Jacobian Weight y^(λ − 1) 0.2 dy^(λ)/dy
Inputs
⚡ Scenarios:
y
Strictly positive original observation y (> 0).
λ
Box-Cox power parameter λ (−2 to +2; 0=log, 0.5=square-root, −1=reciprocal).
σ²
Transformed-scale forecast error variance σ² for Smearing/Taylor bias correction.
Results & Graph
Box-Cox Transformed Value y^(0.5)
8 transformed
Bias-Corrected Back-Transform Mean
25.038 units
Natural Log Baseline (λ = 0)
3.219 ln(y)
Jacobian Weight y^(λ − 1)
0.2 dy^(λ)/dy

Box-Cox Transformed Value y^(0.5)

Graph 1 • Box-Cox Power Transform • Domain Architecture & Live Telemetry Maps Positive Value Y, Box Cox Lambda, Transformed Residual Var through Box-Cox Power Transform Formula.
Box-Cox Transformed Value y^(0: 8 transformed Box-Cox Power Transform • Invariants & Metrics Box-Cox Transformed Value y^ 8 transformed Bias-Corrected Back-Transfor 25.038 units Natural Log Baseline (λ = 0) 3.219 ln(y) Jacobian Weight y^(λ − 1) 0.2 dy^(λ)/dy
Graph 2 • Box-Cox Transformed Value y^(0.5) Across Positive Value Y Sweep (40% → 160%) Evaluates y^(λ) = (y^λ − 1) / λ for λ ≠ 0, else ln(y); BiasCorrectedInverse ≈ across 7 operating points (current input highlighted).
0 transf 5.86 transf 11.71 transf 4.33 10y 5.75 15y 6.94 20y 8 25y 8.95 30y 9.83 35y 10.65 40y
Graph 3 • Joint Sensitivity: Positive Value Y × Box Cox Lambda Compares Box-Cox Transformed Value y^(0.5) when Box Cox Lambda is shifted +25% (solid) vs. −25% (dashed).
0 transf 6.87 transf 13.74 transf --- Box Cox Lambda @ 75% 7.97 -30% 9.21 -15% 10.36 +0% 11.46 +15% 12.49 +30%
Summary: Transforming y = 25 with λ = 0.5 yields y^(λ) = 8 (bias-corrected back-transform with σ²=0.15 is 25.038).
Summary: Transforming y = 25 with λ = 0.5 yields y^(λ) = 8 (bias-corrected back-transform with σ²=0.15 is 25.038).
⚡ Send result to: Scientific →Statistics & SD →Probability & P-Value →
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Formula & Mathematical Methodology: Box-Cox Power Transform Formula

yλ = (y^λ − 1) / λ for λ ≠ 0, else ln(y)
BiasCorrectedInverse ≈ (λ·ŷ + 1)1/λ × [1 + σ²(1−λ)/(2(λ·ŷ+1)²)]
y^(λ) = (y^λ − 1) / λ for λ ≠ 0, else ln(y); BiasCorrectedInverse ≈ (λ·ŷ + 1)^(1/λ) × [1 + σ²(1−λ)/(2(λ·ŷ+1)²)]
  • positiveValueY: Positive Value Y (y)
  • boxCoxLambda: Box Cox Lambda (λ)
  • transformedResidualVar: Transformed Residual Var (σ²)

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: Box-Cox Power Transform

Transform skewed positive data via Box-Cox power parameter λ and bias-corrected inverse.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: y^(λ) = (y^λ − 1) / λ for λ ≠ 0, else ln(y); BiasCorrectedInverse ≈ (λ·ŷ + 1)^(1/λ) × [1 + σ²(1−λ)/(2(λ·ŷ+1)²)].
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Box-Cox Power Transform via y^(λ) = (y^λ − 1) / λ for λ ≠ 0, else ln(y); BiasCorrectedInverse ≈ (λ·ŷ + 1)^(1/λ) × [1 + σ²(1−λ)/(2(λ·ŷ+1)²)].

Frequently Asked Questions

How does the Box-Cox Power Transformation y^(λ) = (y^λ − 1)/λ & Inverse Back-Transformation Calculator compute the solution?

It evaluates y^(λ) = (y^λ − 1) / λ for λ ≠ 0, else ln(y); BiasCorrectedInverse ≈ (λ·ŷ + 1)^(1/λ) × [1 + σ²(1−λ)/(2(λ·ŷ+1)²)] 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 · Econometrics & Regression

Box-Cox Power Transformation y^(λ) = (y^λ − 1)/λ & Inverse Back-Transformation Calculator: inputs, method, and how to read the result

Transform skewed positive data via Box-Cox power parameter λ and bias-corrected inverse.

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.

  • Positive Value Y: Strictly positive original observation y (> 0).
  • Box Cox Lambda: Box-Cox power parameter λ (−2 to +2; 0=log, 0.5=square-root, −1=reciprocal).
  • Transformed Residual Var: Transformed-scale forecast error variance σ² for Smearing/Taylor bias correction.

What the calculation includes

With the default values shown, Box-Cox Transformed Value y^(0.5) is 8 transformed. The calculation follows Box-Cox Power Transform Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Positive Value Y from 25 y to 28 y changes Box-Cox Transformed Value y^(0.5) from 8 transformed to 8.583 transformed. 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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