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.

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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

For X = —

Statistics

Statistic	Value
Standard Error
Sample Size (n)
Degrees of Freedom

Chart

Step-by-Step Solution

使い方 Power Regression Calculator

成長と減衰

Fits y = a·x^b to your data using log-transformed least squares.

入力データ

Enter paired X and Y values to find the best-fitting power curve.

統計出力

Get the equation y = a·x^b, R-squared value, and full predictive diagnostic data.

Power regression is ideal for modeling allometric scaling, physics laws, and engineering power curves.

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 · x^b, 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

Power Regression Calculator