# Quadratic Regression Calculator

Free quadratic regression calculator. Fit a second-degree polynomial (y = ax² + bx + c) to your data using least squares. Get coefficients, R-squared, and predictions.

## What this calculates

Fit a quadratic curve to your data points using the least squares method. Enter summary statistics and get the equation y = ax² + bx + c, plus R-squared and predicted values.

## Inputs

- **Number of Data Points (n)** — min 4 — Total number of (x, y) data points. Minimum 4 for quadratic regression.
- **Sum of X (Σx)** — Sum of all x values.
- **Sum of Y (Σy)** — Sum of all y values.
- **Sum of X² (Σx²)** — Sum of each x value squared.
- **Sum of X³ (Σx³)** — Sum of each x value cubed.
- **Sum of X⁴ (Σx⁴)** — Sum of each x value to the fourth power.
- **Sum of XY (Σxy)** — Sum of the products of each x and y pair.
- **Sum of X²Y (Σx²y)** — Sum of x-squared times y for each pair.
- **Sum of Y² (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

- **Coefficient a (x² term)** — The coefficient of the x² term.
- **Coefficient b (x term)** — The coefficient of the x term.
- **Coefficient c (constant)** — The constant (y-intercept) of the equation.
- **Regression Equation** — formatted as text — The quadratic regression equation y = ax² + bx + c.
- **R² (Coefficient of Determination)** — Proportion of variance in y explained by the quadratic model.
- **Predicted Y** — The predicted y value for the given x.

## Details

Quadratic regression fits a second-degree polynomial to your data. It captures curved (parabolic) relationships that linear regression misses.

**The Model:**
y = ax² + bx + c

The coefficients a, b, and c are found by solving the normal equations, which minimize the sum of squared residuals.

**When to Use Quadratic Regression:**

- Your scatter plot shows a curved pattern
- Linear regression gives a poor R² but the data clearly has a trend
- The relationship has a maximum or minimum (like projectile motion, revenue curves, or growth that levels off)

**Interpreting the Coefficients:**

- **a > 0:** Parabola opens upward (U-shaped)
- **a < 0:** Parabola opens downward (inverted U)
- **b:** Affects the tilt and position of the vertex
- **c:** The y-intercept (value of y when x = 0)

**Required Summary Statistics:**
You need n, Σx, Σy, Σx², Σx³, Σx⁴, Σxy, Σx²y, and Σy². These can be computed from raw data in a spreadsheet.

## Frequently Asked Questions

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

A: You need at least 4 data points, but more is better. Three points will always give a perfect fit (R² = 1) which is not useful. With more data points, R² becomes a meaningful measure of how well the quadratic model fits.

**Q: How do I know if quadratic regression is better than linear?**

A: Compare the R² values. If the quadratic R² is substantially higher than the linear R² and the data shows a curved pattern, the quadratic model is likely better. Also check that the coefficient a is significantly different from zero. If a is near zero, the quadratic term adds no value.

**Q: What are the summary statistics and how do I calculate them?**

A: In a spreadsheet: Σx is the sum of all x values, Σx² is the sum of each x squared, Σx³ is x cubed, Σx⁴ is x to the fourth power, Σxy is the sum of x times y, and Σx²y is the sum of x-squared times y. Each can be computed with a SUMPRODUCT or simple column formulas.

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