# Quartic Regression Calculator

Free quartic regression calculator. Fit a 4th-degree polynomial to data points using least squares. Get coefficients, R-squared, and predictions.

## What this calculates

Fit a fourth-degree polynomial to your data using the least squares method. Enter summary statistics and get the quartic equation, R-squared value, and predicted values.

## Inputs

- **Number of Data Points (n)** — min 6 — Total number of (x, y) data points. Minimum 6 for quartic regression.
- **Σx** — Sum of all x values.
- **Σx²** — Sum of each x value squared.
- **Σx³** — Sum of each x value cubed.
- **Σx⁴** — Sum of each x to the 4th power.
- **Σx⁵** — Sum of each x to the 5th power.
- **Σx⁶** — Sum of each x to the 6th power.
- **Σx⁷** — Sum of each x to the 7th power.
- **Σx⁸** — Sum of each x to the 8th power.
- **Σy** — Sum of all y values.
- **Σxy** — Sum of x times y for each pair.
- **Σx²y** — Sum of x-squared times y for each pair.
- **Σx³y** — Sum of x-cubed times y for each pair.
- **Σx⁴y** — Sum of x-to-the-fourth times y for each pair.
- **Σy²** — Sum of each y value squared (used for R²).
- **Predict Y for X =** — Enter an x value to predict the corresponding y value.

## Outputs

- **Regression Equation** — formatted as text — The quartic regression equation.
- **a (x⁴ coefficient)** — Coefficient of the x⁴ term.
- **b (x³ coefficient)** — Coefficient of the x³ term.
- **c (x² coefficient)** — Coefficient of the x² term.
- **d (x coefficient)** — Coefficient of the x term.
- **e (constant)** — The constant term.
- **R²** — Coefficient of determination for the quartic fit.
- **Predicted Y** — Predicted y for the given x value.

## Details

Quartic regression fits a fourth-degree polynomial to your data, capturing complex curves with up to three turning points.

**The Model:**
y = ax⁴ + bx³ + cx² + dx + e

The five coefficients are found by solving a 5x5 system of normal equations derived from minimizing the sum of squared residuals.

**When to Use Quartic Regression:**

- Your data shows a complex curve with multiple bends
- Lower-degree polynomials (linear, quadratic, cubic) give poor fits
- The relationship has up to three turning points

**Cautions:**

- Higher-degree polynomials can overfit, especially with few data points
- Quartic models can behave unpredictably outside the range of your data (extrapolation risk)
- Always check if a simpler model (quadratic or cubic) fits nearly as well
- You need at least 6 data points, but more is better for a reliable fit

**Required Summary Statistics:**
You need n, Σx through Σx⁸, Σy, Σxy through Σx⁴y, and Σy². Compute these from raw data in a spreadsheet.

## Frequently Asked Questions

**Q: How many data points do I need for quartic regression?**

A: You need at least 6 data points. Five points would give a perfect fit (R² = 1) regardless of the actual relationship. More data points give a more meaningful and reliable fit. For practical use, aim for at least 10 to 15 points.

**Q: When is quartic regression appropriate?**

A: Use quartic regression when your data clearly shows a complex curve that simpler models cannot capture. Examples include phenomena with multiple peaks and valleys, certain physical processes, or economic cycles. Always start with simpler models first and increase complexity only when justified.

**Q: What is overfitting and how do I avoid it?**

A: Overfitting happens when the model fits the noise in your data rather than the true underlying relationship. Signs include a high R² on training data but poor predictions on new data, and wild behavior between or beyond data points. Use fewer polynomial terms, collect more data, and compare adjusted R² across models.

---

Source: https://vastcalc.com/calculators/statistics/quartic-regression
Category: Statistics
Last updated: 2026-04-08