Quadratic Scatter Plot Calculator

Calculator Form

Example Data Table

Formula Used

This calculator fits a quadratic model: y = ax² + bx + c. It uses least squares regression to minimize the sum of squared residuals between actual values and fitted values.

The coefficients come from the normal equations:
Σx⁴a + Σx³b + Σx²c = Σx²y
Σx³a + Σx²b + Σxc = Σxy
Σx²a + Σxb + nc = Σy

After solving for a, b, and c, the calculator finds predicted values, residuals, SSE, MSE, RMSE, adjusted R², the vertex, the axis of symmetry, and real roots when they exist.

How to Use This Calculator

1. Enter one X and Y pair on each line.
2. Set the prediction X value if you need an estimated Y.
3. Choose decimal places for displayed values.
4. Set the number of curve points for the plotted fit.
5. Press the calculate button to generate the fitted model.
6. Review the equation, metrics, roots, and residual table.
7. Use the CSV and PDF buttons to export your results.

Frequently Asked Questions

1. What does this calculator measure?

It fits a second-degree regression curve to paired data. The tool summarizes the fitted equation, prediction values, model strength, error size, turning point, and real roots when available.

2. When should I use a quadratic model?

Use it when the scatter pattern bends rather than following a straight line. Many growth, motion, yield, and optimization datasets show curved behavior that suits a quadratic trend.

3. What does R² mean here?

R² shows how much variation in Y is explained by the fitted curve. Values closer to 1 suggest a stronger fit, while lower values suggest more unexplained variation.

4. Why are at least three points needed?

A quadratic equation has three coefficients. Fewer than three valid points cannot define a stable least-squares quadratic fit, especially when X values repeat or lack spread.

5. What is the vertex?

The vertex is the turning point of the parabola. It marks the highest or lowest fitted value, depending on whether the curve opens downward or upward.

6. What are residuals?

Residuals are the differences between actual Y values and fitted Y values. Smaller residuals usually indicate a closer fit between the model and the observed data.

7. Can I predict values outside my data range?

Yes, but use caution. Extrapolation can be less reliable because the fitted curve is based only on the observed range of X values.

8. What happens if the system is singular?

The calculator cannot solve the normal equations reliably. This often happens when X values do not vary enough, or the input structure does not support a stable quadratic fit.