Power Regression Calculator (y = a x^b)

Enter your (x, y) data points and fit a power model y = a x^b by least squares. Shows the coefficient a, exponent b, R squared and correlation, a scatter plot with the fitted curve, and a table of predictions and residuals.

Input

Enter one (x, y) pair per line. The tool fits a power regression y = a x^b by least squares (both x and y must be positive).

Data points (x, y)

One pair per line. Separate values with a comma, spaces, or a tab.

Result

Regression equation

y = 1.9924 x^1.5023

Coefficient a

1.992432

Exponent b

1.502333

R squared

Correlation r

Number of points

Scatter plot and fitted curve

Data points and predictions

x	y (observed)	Predicted	Residual
1	2	1.9924	0.0076
2	5.6	5.6446	-0.0446
3	10.4	10.3796	0.0204
4	16	15.9911	0.0089
5	22.4	22.3599	0.0401
6	29.4	29.4053	-0.0053

How it works

The power model is y = a x^b. Taking the natural log of both sides gives ln(y) = ln(a) + b ln(x), which turns the problem into a straight-line fit (simple linear regression) in X = ln(x) and Y = ln(y).
Least squares is applied to the transformed (X, Y) pairs to find the slope b and intercept ln(a), then a = exp(intercept) recovers the original coefficient. Every x and y must be positive because logarithms are used.
The R squared and correlation are computed on the original scale, comparing the observed y with the predicted yhat = a x^b rather than the transformed values. An R squared near 1 indicates a good fit.
Enter one (x, y) pair per line. Values may be separated by a comma, spaces, or a tab, and blank lines are ignored.
If x is constant, so that ln(x) is the same for every point, the slope cannot be determined and the fit is not possible. At least two points are required.

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