How this instrument works
Quartic regression fits a degree-4 polynomial, y = a4x⁴ + a3x³ + a2x² + a1x + a0, to a set of (x, y) data points using the method of least squares — finding the five coefficients that make the sum of squared differences between the curve and the actual data as small as possible. A quartic curve can bend through up to three direction changes, giving it far more flexibility than a straight line (which can't bend at all) or a quadratic parabola (which can only bend once).
Under the hood, the fit is found by solving a system of linear equations built from the data (the 'normal equations'), the same general approach used for linear and quadratic regression, just extended to five unknown coefficients instead of two or three. With exactly 5 data points, a quartic curve can, in principle, pass through every point exactly (a perfect fit, R² = 1.0); with more than 5 points, the fit becomes a genuine best-approximation compromise, and R² tells you how much of the data's variation the curve actually captures.
Because the constant term in y = ax⁴+bx³+cx²+dx+e would naturally be labeled 'e' — which collides with Euler's number as a reserved constant in this site's calculation engine — this calculator labels its five outputs a4, a3, a2, a1, and a0 instead, in descending order of the power of x each one multiplies. The coefficients still mean exactly the same thing as the more familiar a/b/c/d/e notation; only the labels differ.
- Enter your x-values into X values — separated by commas, spaces or new lines.
- Enter the paired y-values, in the same order, into Y values — separated by commas, spaces or new lines — you need at least 5 pairs to fit a quartic.
- Read Coefficient of x⁴ through Constant term (a4, a3, a2, a1, a0) — together they define the fitted curve y = a4x⁴+a3x³+a2x²+a1x+a0.
- Read R² — goodness of fit — closer to 1.0 means the quartic curve closely tracks your data; a value noticeably below 1.0 suggests the data doesn't follow a clean quartic pattern.
Worked example — six points on the curve y = x⁴ − 2x³ + 3x² − x + 5
Enter 0, 1, 2, 3, 4, 5 into X values and 5, 6, 15, 56, 177, 450 into Y values — six points generated exactly from the polynomial y = x⁴ − 2x³ + 3x² − x + 5 (for instance, at x=3: 81 − 54 + 27 − 3 + 5 = 56).
With one more point than the minimum 5 needed to fit a quartic, the least-squares fit still recovers the true underlying coefficients essentially exactly: Coefficient of x⁴ (a4) reads 1.000000, Coefficient of x³ (a3) reads -2.000000, Coefficient of x² (a2) reads 3.000000, Coefficient of x (a1) reads -1.000000, and Constant term (a0) reads 5.000000, with R² — goodness of fit reading 1.000000 — a perfect fit, since the data really did come from exactly this quartic with no added noise.
Questions
How many data points do I need for quartic regression?
At minimum 5 points, since a degree-4 polynomial has 5 coefficients to determine (a4, a3, a2, a1, a0), and 5 points give exactly enough information to solve for them uniquely — with exactly 5 points, if they truly lie on some quartic curve, the fit will pass through every one of them exactly (R²=1.0). With more than 5 points, the fit becomes a genuine least-squares approximation rather than an exact pass-through, and R² will typically be less than 1.0 unless the data is a perfect quartic.
Why are the coefficients labeled a4, a3, a2, a1, a0 instead of a, b, c, d, e?
Purely a naming choice to avoid a collision: the constant term in the familiar y=ax⁴+bx³+cx²+dx+e notation would be labeled 'e', but 'e' is already reserved in this site's calculation engine for Euler's number (≈2.71828). Using descending-power subscripts (a4 for the x⁴ coefficient down to a0 for the constant) sidesteps that collision entirely while keeping the same meaning — a4 is simply what you'd otherwise call 'a', and a0 is what you'd otherwise call 'e'.
What does R² tell me about a quartic fit?
R² reports the fraction of the variation in your y-values that the fitted quartic curve explains, on a scale from 0 to 1. An R² of 1.0 means the curve passes through the data perfectly (or as close to it as floating-point arithmetic allows); an R² noticeably below 1.0 means real scatter remains that the quartic shape doesn't capture — which could mean the data is genuinely noisy, or that a quartic simply isn't the right shape for this particular relationship.
How is this different from the quadratic-regression or polynomial-regression calculators on this site?
All three use the same least-squares fitting approach, just for different polynomial degrees. quadratic-regression fits a fixed degree-2 curve (3 coefficients); this quartic-regression calculator fits a fixed degree-4 curve (5 coefficients) for data with more bends than a parabola can follow; and polynomial-regression is the general-purpose version, letting you pick any degree from 1 to 5 yourself. If you specifically need a degree-4 fit, this dedicated page is a faster path than setting the degree manually on the general one.
Can a higher-degree fit always get R² closer to 1.0?
In general, yes — a higher-degree polynomial has more coefficients and more flexibility to bend through scattered points, so R² tends to rise (or at worst stay the same) as you increase the degree, especially once the number of coefficients approaches the number of data points. That flexibility is a double-edged sword: a high-degree curve can start fitting the noise in your specific dataset rather than the true underlying pattern (a problem known as overfitting), so a higher R² alone doesn't always mean a better or more trustworthy model.