Cubic Regression Calculator

Fit the cubic model y = ax³ + bx² + cx + d to your data by least squares, then read off the coefficients, R², and a prediction for any x.

Quick Facts

Method
Ordinary least squares fit of y = ax³ + bx² + cx + d
Solves the 4×4 normal equations; needs at least 4 points.

Your Results

Calculated
Fitted equation
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y = ax³ + bx² + cx + d
R² (goodness of fit)
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Fraction of variance explained
Predicted y
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Model value at your x
Data points used
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Paired (x, y) observations

Ready

Enter your x and y values, then run the fit.

What cubic regression does

Cubic regression finds the third-degree polynomial that best fits a set of data points. The model is

y = ax³ + bx² + cx + d

The four coefficients a, b, c, and d are chosen so that the sum of the squared vertical distances between each observed point and the curve — the residual sum of squares — is as small as possible. This is the same least-squares principle used in straight-line regression, extended to a cubic. Because a cubic can bend twice (it has up to two turning points and one inflection point), it fits data that rises then falls then rises again, or curves that flatten and steepen, which a line (degree 1) or a parabola (degree 2) cannot capture.

How the coefficients are computed

For n data points, least squares reduces to a system of four linear equations in the four unknowns a, b, c, d — the "normal equations." In matrix form the design matrix X has rows [xᵢ³, xᵢ², xᵢ, 1], and the coefficients solve XᵀX·β = Xᵀy. The calculator above builds the 4×4 matrix XᵀX (which contains the sums of x, x², x³, up to x⁶) and the vector Xᵀy, then solves the system by Gaussian elimination with partial pivoting. No iteration or guessing is involved: for a given data set the answer is exact and unique whenever the x values are distinct and there are at least four of them.

Reading R²

R² (the coefficient of determination) reports the fraction of the variance in y that the fitted cubic explains, computed as 1 − SS_res/SS_tot, where SS_res is the residual sum of squares and SS_tot is the total sum of squares about the mean of y. R² = 1 means the curve passes exactly through every point; R² near 0 means the cubic is no better than simply predicting the average y. With exactly four distinct points a cubic always achieves R² = 1 because it interpolates them, so a high R² is only meaningful when you have comfortably more points than coefficients.

Where cubic regression is used

  • Growth and dose-response curves: biological and chemical responses often accelerate, level off, then accelerate again — an S-shape a cubic can approximate over a limited range.
  • Engineering calibration: sensor and instrument calibration curves are frequently modeled with low-degree polynomials, and a cubic captures mild curvature without over-fitting.
  • Economics and forecasting: cost, yield, and trend data with a single hump or dip are common candidates for a cubic term.

Frequently Asked Questions

How is cubic regression different from linear or quadratic regression?
All three use least squares; they differ in the highest power of x. Linear regression fits y = cx + d (a straight line), quadratic fits y = bx² + cx + d (a parabola with one turning point), and cubic fits y = ax³ + bx² + cx + d (up to two turning points and an inflection). A higher degree always fits the sample at least as well, but a degree higher than the true relationship over-fits noise and predicts poorly outside the data.
How many data points do I need?
You need at least four points because a cubic has four coefficients. With exactly four distinct x values the curve passes through every point and R² = 1, but that is interpolation, not a fit you can trust. To estimate the relationship rather than just connect the dots, use appreciably more points — ten or more is a sensible minimum — so residuals and R² actually mean something.
Can I trust predictions outside my data range?
Be very cautious. Cubics grow without bound: as x gets large the ax³ term dominates and the curve shoots off toward ±∞. Extrapolating even a little past the smallest or largest x you measured can produce wildly wrong values. Cubic regression is reliable for interpolation within the observed range, not for forecasting far beyond it.
What does a negative leading coefficient (a) mean?
The sign of a sets the curve's overall direction. When a is positive the cubic falls on the far left and rises on the far right; when a is negative it rises on the left and falls on the right. The middle terms bx² and cx bend the curve and position its turning points, but a governs the long-run behavior at the extremes.

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