Quartic Regression Calculator

Fit a fourth-degree polynomial y = ax⁴ + bx³ + cx² + dx + e to your (x, y) data by least squares, and read off all five coefficients plus the R² goodness of fit.

Quick Facts

Method
Least-squares fit of y = ax⁴ + bx³ + cx² + dx + e
Solves the 5×5 normal equations by Gaussian elimination; needs at least 5 points.

Your Results

Calculated
Fitted equation
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y as a fourth-degree polynomial in x
a (x⁴)
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Leading coefficient
b (x³)
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Cubic term
c (x²)
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Quadratic term
d (x)
-
Linear term
e
-
Constant term
R²
-
Coefficient of determination

Ready

Enter your (x, y) data points and press Calculate.

What quartic regression does

Quartic regression fits a fourth-degree polynomial to your data:

y = ax⁴ + bx³ + cx² + dx + e

Given a set of (x, y) observations, the calculator finds the five coefficients a, b, c, d, and e that make the curve pass as close as possible to every point. "As close as possible" has a precise meaning: it minimizes the sum of squared residuals, ∑(yᵢ − ŷᵢ)², where ŷᵢ is the value the fitted curve predicts at xᵢ. This is the ordinary least-squares criterion, the same one used for straight-line and quadratic regression — only the model here has more curvature.

How the coefficients are found

Because the polynomial is linear in its unknown coefficients, the least-squares solution is exact, not iterative. Setting the partial derivatives of the squared-error sum to zero gives a system of five linear equations in a, b, c, d, e — the normal equations. Their coefficients are power sums of x (from ∑x⁰ up to ∑x⁸) and cross sums ∑yxʲ. This calculator builds that 5×5 system and solves it by Gaussian elimination with partial pivoting, then reports the coefficients, the reconstructed equation, and R².

Reading R²

The coefficient of determination R² = 1 − SSres/SStot reports the fraction of the variation in y that the fitted curve explains, from 0 (no better than a horizontal line at the mean) to 1 (the curve hits every point). Note that adding polynomial terms can only raise R², so a high R² alone never proves that a quartic is the right model — five points always give R² = 1 regardless of the underlying process.

When a quartic is the right choice

A quartic can bend up to three times (it has up to three turning points and up to two inflection points), so it is useful when data shows an M- or W-shaped pattern, or a rise-fall-rise trend that a line, parabola, or cubic cannot capture. It is common in curve-fitting for calibration curves, empirical physics and engineering data, and interpolating tabulated values. If your data has only one bend, prefer a quadratic; if it is monotonic, prefer a line — fewer terms generalize better.

Frequently Asked Questions

What is quartic regression?
It is the least-squares fitting of a fourth-degree polynomial, y = ax⁴ + bx³ + cx² + dx + e, to a set of (x, y) points. The method finds the five coefficients that minimize the sum of squared vertical distances between the data and the curve.
How many data points do I need?
At least five, because there are five coefficients to determine. With exactly five distinct x-values the curve passes through every point and R² = 1, which tells you nothing about fit quality. Use more points — roughly 8 to 15 well-spread values — so the system is over-determined and R² becomes informative.
Why is my R² equal to 1 even though the data looks noisy?
You almost certainly entered exactly five points (or five distinct x-values). A quartic has enough freedom to pass through five points exactly, so it interpolates rather than fits. Add more points and R² will drop to a value that reflects the real scatter.
Is quartic regression the same as a quartic interpolation?
Only in the exactly-five-points case. With more than five points the curve generally does not pass through any single point; it balances the errors across all of them. Interpolation forces the curve through every node, which is a different goal and is usually done with splines, not a single global quartic.

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