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

Come usare questo Power Regression Calculator

Crescita e decadimento

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

Dati di input

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

Output statistico

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