Quadratic Regression Calculator: Equation & Fit

This quadratic regression calculator fits a parabola of the form y = ax² + bx + c to your (x, y) data points. You enter paired x and y values (comma or space separated), and it returns the coefficients a, b, and c, plus the R² value that tells you how well the parabola fits. It uses the standard least‑squares method, so the result is the unique parabola that minimizes the sum of squared vertical errors.

Formula

The calculator solves the normal equations for a quadratic least‑squares fit. Given n data points (xᵢ, yᵢ), the model is:

y = a·x² + b·x + c

The coefficients a, b, and c are found by solving this 3×3 linear system:

• a·Σx⁴ + b·Σx³ + c·Σx² = Σx²y
• a·Σx³ + b·Σx² + c·Σx = Σxy
• a·Σx² + b·Σx + c·n = Σy

where Σ denotes summation over all data points. The coefficient of determination R² is then:

R² = 1 − (SS_res / SS_tot)

with SS_res = Σ(yᵢ − ŷᵢ)² (sum of squared residuals) and SS_tot = Σ(yᵢ − ȳ)² (total sum of squares, ȳ being the mean of y). R² ranges from 0 to 1; values closer to 1 indicate a better fit.

Worked Example

Using the default data: x = 1,2,3,4,5 and y = 2.1, 3.9, 8.2, 14.1, 22.0.

1. Compute the sums: n = 5, Σx = 15, Σx² = 55, Σx³ = 225, Σx⁴ = 979, Σy = 50.3, Σxy = 190.4, Σx²y = 751.7.
2. Set up the system:
   979a + 225b + 55c = 751.7
   225a + 55b + 15c = 190.4
   55a + 15b + 5c = 50.3
3. Solve (e.g., by elimination or matrix inversion): a ≈ 1.0143, b ≈ −0.1486, c ≈ 1.2343.
4. Thus the equation is y = 1.0143x² − 0.1486x + 1.2343.
5. Compute predicted y for each x: 2.1000, 4.0143, 8.1000, 14.3571, 22.7857.
6. Calculate R²: SS_res ≈ 0.2229, SS_tot ≈ 242.562, so R² ≈ 0.9991.

The calculator shows the same result: y = 1.0143x² − 0.1486x + 1.2343 with R² ≈ 0.9991.

When to Use Quadratic Regression

Quadratic regression is appropriate when your data shows a single curve (one bend) – for example, projectile motion (height vs. time), cost vs. production volume with diminishing returns, or temperature change over a day. It is a natural next step after linear regression when a straight line clearly misses the pattern. However, if the data has two or more bends (e.g., a sine wave or a cubic trend), a quadratic model will be insufficient – consider polynomial regression of higher degree or another function.

How to Enter Data & Read the Results

Enter x and y values as comma‑separated lists, one list per input. The lists must have the same length. For example, x: 1, 2, 3, 4, 5 and y: 2.1, 3.9, 8.2, 14.1, 22.0. The calculator outputs:

• Equation – the fitted parabola in standard form.
• R² – the proportion of variance in y explained by the model. An R² above 0.9 usually indicates a strong fit, but always look at a scatter plot to confirm the pattern is truly quadratic.

If your data has more than one y for the same x, that is fine – the least‑squares method handles it. If the lists have different lengths, the calculator will not compute; adjust them to match.

Common Mistakes and Pitfalls

• Using a quadratic when the relationship is not curved – check a scatter plot first. A quadratic will always fit better than a line, even if the true relationship is linear, so rely on R² and the visual pattern.
• Ignoring units – the coefficients a, b, c carry units. If x is in seconds and y in meters, then a has units m/s², b is m/s, and c is m. Always interpret coefficients with their units.
• Overfitting – with only 3 data points, a quadratic fits perfectly (R² = 1) but may be meaningless. With many points, a quadratic can still overfit if the true relationship is not polynomial.
• Rounding too early – keep full precision during calculations; round only the final coefficients to avoid accumulating errors.
• Using the model outside the data range – extrapolation beyond your x range is risky; the parabola may shoot up or down unrealistically.

Limits of the Quadratic Model

A quadratic model assumes a symmetric, single‑curved relationship. It cannot capture multiple turning points, asymptotes, or exponential growth. Also, the least‑squares method is sensitive to outliers – a single extreme point can pull the parabola significantly. If your data has clear outliers, consider robust regression or removing them after careful inspection. Finally, correlation does not imply causation; a high R² only means the parabola fits the data, not that x causes y.

Frequently Asked Questions

What is the difference between quadratic regression and polynomial regression?

Quadratic regression is polynomial regression of degree 2. Polynomial regression allows any degree (linear, quadratic, cubic, etc.). The formulas and solving method are the same; only the order of the polynomial changes.

Can I use quadratic regression for time series data?

Yes, if the trend over time is curved. However, time series often have autocorrelation and seasonality, which a simple quadratic model ignores. For forecasting, consider specialized time‑series methods, but for describing a historical trend, quadratic regression can be useful.

What does R² mean exactly?

R² is the proportion of the total variance in y that is explained by the model. An R² of 0.95 means 95% of the variation in y is accounted for by the quadratic relationship with x. The remaining 5% is due to other factors or random noise.

How many data points do I need for quadratic regression?

You need at least 3 distinct x values to fit a quadratic (since there are 3 parameters). However, with only 3 points the fit is exact and R² = 1, which is not meaningful. For a reliable fit, aim for at least 10–15 points spread across the x range.

Why does my calculator give a different equation than Excel?

Different tools may use different rounding or solve the system in slightly different ways, but the underlying least‑squares result should be identical to several decimal places. If you see a large difference, check that you entered the data correctly and that both are using the same degree of polynomial.

Can I use quadratic regression if my x values are not evenly spaced?

Yes. The least‑squares method does not require evenly spaced x values. It works with any set of distinct x values, though the fit may be more sensitive to gaps in the data.

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in x | x values | - | 1, 2, 3, 4, 5 | 0
in y | y values | - | 2.1, 3.9, 8.2, 14.1, 22.0 | 0
out eq | Equation | - | "y = " + round(a,4) + "x² + " + round(b,4) + "x + " + round(c,4) | 4
out r2 | R² (fit quality) | - | r2 | 4