Enter your X values and Y values (one per line or comma-separated) to fit a power regression curve of the form ŷ = a · xᵇ. You get back the coefficients a and b, the R² value, and a scatter plot with the fitted curve — ideal for modeling relationships that follow a power law.
Fitted Equation
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Coefficient a
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Exponent b
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R² (Coefficient of Determination)
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Correlation Coefficient (r)
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Number of Data Points
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Power regression is a type of curve fitting that models the relationship between two variables using a function of the form ŷ = a · xᵇ. It is commonly used when one variable increases or decreases as a power of another — for example, area vs. radius or speed vs. drag force.
The calculator linearizes the power function by taking the natural logarithm of both sides: ln(y) = ln(a) + b·ln(x). This transforms the problem into a simple linear regression on the log-transformed data. The coefficients are then back-transformed to recover a and b.
R² (the coefficient of determination) measures how well the power curve fits your data. A value of 1.0 means a perfect fit, while 0.0 means the model explains none of the variability. Generally, an R² above 0.9 is considered a strong fit.
Power regression works by applying a logarithmic transformation to both X and Y values. Since logarithms are only defined for positive numbers, all X and Y inputs must be greater than zero. If your data includes zeros or negative values, a different regression model should be used.
In power regression (ŷ = a · xᵇ), the independent variable x is raised to a power b. In exponential regression (ŷ = a · eᵇˣ), x appears as an exponent itself. Power regression is better suited for relationships like scaling laws, while exponential regression fits phenomena like population growth or radioactive decay.
You need at least 3 data points to perform a meaningful power regression, since the model has two parameters (a and b). However, more data points generally yield a more reliable and stable fit. A minimum of 5–10 points is recommended for practical use.
Yes. A negative exponent b means that Y decreases as X increases, following an inverse power relationship (e.g., intensity decreasing with distance). The calculator handles both positive and negative exponents automatically.
The coefficient a represents the value of ŷ when x = 1 (the scale factor), and b is the exponent that controls the rate and direction of growth. If b > 1, growth is accelerating; if 0 < b < 1, growth is decelerating; if b < 0, Y decreases as X increases.