Cubic Regression Calculator

Cubic regression fits a third-degree polynomial to paired observations. The model has the form y = ax³ + bx² + cx + d, where a controls the strongest curvature, b contributes quadratic bending, c describes a linear trend, and d is the predicted value when x equals zero. Unlike interpolation, regression does not normally force the curve through every point. It chooses the coefficients that minimize the sum of squared vertical differences between observed and predicted y values. This cubic regression calculator uses the ordinary least-squares method. From the entered x values it builds the four normal equations associated with the basis terms 1, x, x², and x³. It then solves that four-by-four system with pivoted Gaussian elimination. Pivoting improves numerical reliability by selecting the strongest available pivot at each step. At least four paired observations and four distinct x values are required because four independent coefficients must be estimated. The reported coefficient of determination, R², compares the model's remaining squared error with the total variation in the observed y values. An R² of 1 means the fitted curve reproduces every observation exactly. A value near zero means the model explains little more variation than simply using the mean. A high R² is useful evidence of fit, but it does not prove that a cubic relationship is scientifically meaningful or that predictions outside the observed range are reliable. Cubic models are useful when data change direction up to twice, such as an accelerating trend followed by a slowdown and another rise. They can describe calibration curves, physical trajectories over limited intervals, growth measurements, demand patterns, and smooth experimental responses. Always plot the data when possible. One unusual observation can have substantial leverage, and large or tightly clustered x values can make polynomial coefficients sensitive to small changes. Use matching positions in the two lists: the first x is paired with the first y, the second x with the second y, and so on. More observations generally produce a more informative fit than the minimum four. For forecasting, remain cautious beyond the data range because the x³ term can grow rapidly. Compare the cubic model with a simpler linear or quadratic model and prefer the simpler form when extra curvature does not provide a meaningful improvement.