Power Regression Calculator

Calculator

Enter positive x and y pairs. One pair per line. Use comma or space separators.

Example Data Table

Replace these values with your own measurements or observations.

Formula Used

Power regression fits the model y = a × x^b.

• Take logs: log(y) = log(a) + b·log(x).
• Run linear regression on (log(x), log(y)).
• Intercept gives log(a), slope gives b.
• Convert back: a = exp(intercept) (or 10^intercept for base 10).

This calculator reports accuracy on both original and log scales.

How to Use This Calculator

1. Enter at least two positive x,y pairs in the textarea.
2. Select a log base if you prefer base 10 reporting.
3. Optionally add forecast x values to predict y.
4. Click Calculate to view coefficients and metrics.
5. Use the download buttons to export CSV or PDF.

If any x or y is zero or negative, it is skipped.

Practical Notes for Power Regression

When a power model is the right choice

Power regression is useful when output scales by a constant factor as input changes, such as biological allometry, learning curves, and surface-area relationships. If doubling x tends to multiply y by a near-constant ratio across the range, a power curve often summarizes the trend better than a straight line on the original scale. If your data represent physical laws, validate units and dimensional consistency, then compare fitted a and b against theory to catch transcription mistakes early in practice.

Data preparation that improves stability

Because the fit is performed on log-transformed values, every x and y must be positive. Remove zeros, replace invalid readings, and keep units consistent. Use enough spread in x to avoid a near-vertical or near-flat log relationship. Outliers can dominate the fit, so review residuals and percent errors before trusting forecasts. For best results, collect at least 8–12 pairs spanning the operating range.

How the coefficients behave

The coefficient a sets the scale, while b controls curvature. If b is near 1, the relationship is close to proportional. If b is greater than 1, y accelerates as x increases; if b is between 0 and 1, y grows with diminishing returns. Comparing b across datasets can reveal structural differences, even when absolute values differ.

Interpreting accuracy metrics

R² on the original scale reflects variance explained in y, while log-space R² reflects consistency of multiplicative errors. RMSE and MAE are in y units and are sensitive to large values. MAPE expresses average percent error, which is often easier to communicate when y varies widely. Use several metrics together for reliable decisions. Check for systematic residual patterns; persistent bias at high x can indicate underfitting. If you export the CSV, plot y versus ŷ to confirm errors are roughly centered.

Using weights, forecasts, and exports

Weighted fitting can help when measurement variance grows with magnitude. For example, inverse-y weights can reduce the influence of large y values, while y weights can emphasize high-end accuracy. Forecasts use the fitted model directly, so avoid extrapolating far beyond your observed x range. Exporting CSV supports audits and plotting, while PDF suits quick reporting.

FAQs

1) Why are zero or negative values rejected?

The method uses logarithms to linearize the model. Logs are undefined for zero and negative numbers, so those rows cannot be transformed and are skipped for correctness.

2) Should I use natural log or base 10?

Either choice gives the same b and predictions. Only the reported intercept representation changes. Choose base 10 if you prefer decimal log interpretation in reports.

3) What does an exponent b of 0.5 mean?

It indicates diminishing returns: y increases with x, but at a slowing rate. Doubling x increases y by about 2^0.5, which is roughly 1.414 times.

4) Why do I see two R² values?

One R² is computed on the original y scale, while the other is computed in log space. Log-space R² reflects multiplicative fit quality, which is central to this model.

5) When should I enable weighted fitting?

Use weights when errors are not uniform across the range. If larger measurements are noisier, inverse-y weighting can balance the influence of high values and stabilize the fit.

6) Can I trust forecasts outside my data range?

Forecasts are safest within the observed x range. Extrapolation can be misleading if the true process changes regime, saturates, or follows different physics beyond your samples.