Power Law Regression Calculator

Enter Data for Power Law Regression

Provide one pair per line using comma, space, tab, semicolon, or vertical bar. Only positive x and y values work because the model uses logarithms.

Example Data Table

This sample follows a power-shaped pattern and works well for testing the tool.

Observation	x	y
1	1.0	2.5
2	2.0	6.4
3	3.0	11.0
4	4.0	16.2
5	5.0	22.0
6	6.0	28.3

Formula Used

Power model: y = a × x^b

Linearized form: ln(y) = ln(a) + b ln(x)

Exponent: b = [nΣ(lnx·lny) − Σlnx·Σlny] / [nΣ(lnx)² − (Σlnx)²]

Coefficient: a = e^{(Σlny − bΣlnx)/n}

Predicted value: ŷ = a × x^b

Residual: e = y − ŷ

RMSE: √[Σ(y − ŷ)² / n]

MAPE: [Σ|y − ŷ| / y ÷ n] × 100

How to Use This Calculator

Enter positive x and y pairs, one pair on each line.
Use commas, spaces, tabs, semicolons, or vertical bars as separators.
Add an optional x value if you want a specific prediction.
Choose your decimal precision and fitted curve sample count.
Click Run Regression to estimate the model.
Review the equation, fit statistics, and detailed residual table.
Inspect the Plotly graph to compare actual points and fitted curve.
Download CSV or PDF if you need a saved report.

Frequently Asked Questions

1. What does this calculator estimate?

It fits a power law model, y = a × x^b, to positive data. The tool estimates the coefficient, exponent, predictions, residuals, and fit statistics, then plots the observed points beside the fitted curve.

2. Why must both x and y stay positive?

The regression transforms both variables with natural logarithms. Logarithms are undefined for zero and negative values, so every pair must remain strictly positive before the model can be estimated.

3. How should I interpret the exponent b?

The exponent measures scaling strength. If b = 1, the relationship is proportional. If b > 1, y rises faster than x. If 0 < b < 1, growth slows as x increases.

4. What does the coefficient a represent?

The coefficient sets the curve scale. In this model, it equals the predicted y value when x = 1. Larger values shift the fitted curve upward across the full data range.

5. Why are there two R² values?

The page reports one R² on the original y scale and another on the transformed log scale. The log version reflects the fitted linearized model, while the original version compares actual and predicted values directly.

6. When is a power law model a poor choice?

Avoid it when the pattern changes direction, includes zeros, includes negatives, or follows an additive trend. The model also performs poorly when one exponent cannot describe the full range.

7. Can I forecast beyond the observed range?

Yes, but be careful. Extrapolation assumes the same scaling rule continues outside your data. That assumption may fail quickly, especially in physical, biological, financial, or social datasets.

8. What does MAPE tell me?

MAPE is mean absolute percentage error. It shows the average prediction error relative to actual values, expressed as a percentage, which makes different scales easier to compare.