Power Regression Calculator
Free power regression calculator. Fit y = a·x^b with step-by-step solutions, R², graphs, and predictions. All calculations in your browser.
Power Regression Calculator
Free power regression calculator. Fit y = a·x^b with step-by-step solutions, R², graphs, and predictions. All calculations in your browser.
Enter your data points
| # | X | Y |
|---|
Results
Equation
Exponent (b)
Coefficient (a)
R²
Predicted Y
Statistics
| Statistic | Value |
|---|---|
| Standard Error | |
| Sample Size (n) | |
| Degrees of Freedom |
Chart
Step-by-Step Solution
How to Use This Power Regression Calculator
Power-Law Modeling
Fits y = a·x^b to your data using log-transformed least squares.
Enter X & Y Data
Input paired values — all X and Y values must be positive for valid results.
Exponent Analysis
Get equation coefficients, R², and prediction forecasts.
Best For
Allometric scaling, physics laws, metabolic rates, and engineering power curves.
Power regression requires all X and Y values to be positive — zero or negative values will produce errors.
What Is Power Regression?
📐 Power regression models relationships where the dependent variable changes as a power of the independent variable. The general formula is y = a · xb, where a is a scaling coefficient and b is the exponent that determines the curvature.
📊 Unlike linear regression, power regression captures proportional scaling laws found throughout science and engineering. Real-world examples include: (1) Physics — gravitational force falls with the square of distance (b = −2), (2) Biology — metabolic rate scales with body mass to the 3/4 power (Kleiber's law), (3) Engineering — pipe flow resistance scales with diameter to the −4.8 power, and (4) Economics — Cobb-Douglas production functions use power-law relationships.
📐 Power regression is a special case of nonlinear regression that can be linearized by taking the logarithm of both variables: ln(y) = ln(a) + b · ln(x). This transformation allows ordinary least squares to estimate the parameters efficiently.
How Power Regression Works
- 1 Transform both variables: Take the natural logarithm of all X and Y values. This converts the power model y = a·x^b into a linear equation ln(y) = ln(a) + b·ln(x).
- 2 Perform linear regression: Apply ordinary least squares to the transformed pairs (ln(x), ln(y)). The slope equals the exponent b and the intercept equals ln(a).
- 3 Recover parameters: Exponentiate the intercept to obtain a = e^(intercept). The slope directly gives b.
- 4 Assess fit on original scale: Compute R² using the original (non-transformed) data to ensure the power curve actually fits the observations well.
- 5 Predict: For any new X value, calculate ŷ = a · X^b using the recovered parameters.
When to Use Power Regression
- Data follows a scaling law or proportional relationship
- Both variables are strictly positive (required for log transformation)
- The relationship appears as a straight line on a log-log plot
- You need to model allometric scaling, physics laws, or engineering power curves
When to Avoid Power Regression
- When X or Y values are zero or negative (log undefined)
- When the relationship is approximately linear on regular axes
- When data shows an S-curve or saturation pattern (use logistic instead)
- When the relationship is exponential rather than power-law
Frequently Asked Questions
What is power regression?
📐 Power regression fits the model y = a · xb to your data by linearizing it through log transformation. It is ideal for modeling scaling laws and proportional relationships where both variables are positive.
How do I calculate power regression?
🧮 To calculate power regression manually: (1) take ln(x) and ln(y) for all points, (2) perform linear regression on (ln(x), ln(y)), (3) the slope is b and a = e^(intercept), (4) the final equation is y = a·x^b.
💻 Our calculator automates these steps instantly.
What is the difference between power and exponential regression?
📊 Power regression uses y = a · xb where x is raised to a power. Exponential regression uses y = a · ebx where the variable is in the exponent. Power regression requires log-transforming both variables; exponential regression only log-transforms y.
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