Quadratic Regression Calculator
This quadratic regression calculator fits the best parabola through your x and y values. Example: x = 0 to 5 and y = 1, 2.8, 6.1, 10.8, 17.2, 24.5 give ŷ = 0.7125 · x² + 1.16321 · x + 0.960714 with R² = 0.999876.
Source: Wikipedia – Polynomial regression. Updated: .
How it is calculated
Quadratic regression fits a parabola y = a · x² + b · x + c to data points when the relationship is curved and a straight line is not enough.
Method and working
The task is to find a, b and c so that the sum of squared distances between the measured y and the parabola is as small as possible (least squares). The sums Σx, Σx², Σx³, Σx⁴, Σy, Σxy and Σx²y give a system of three equations in three unknowns, which the calculator solves. The coefficient of determination R² = 1 − SSE / SST shows what share of the variation in y the parabola explains. The vertex lies at x = −b / (2a).
Worked example
For x = 0, 1, 2, 3, 4, 5 and y = 1, 2.8, 6.1, 10.8, 17.2, 24.5 the sums are n = 6, Σx = 15, Σx² = 55, Σx³ = 225 and Σx⁴ = 979. The system gives a = 0.7125, b = 1.16321 and c = 0.960714, so ŷ = 0.7125 · x² + 1.16321 · x + 0.960714 with R² = 0.999876. The prediction at x = 6 is 0.7125 · 36 + 1.16321 · 6 + 0.960714 = 33.59. The vertex is at x = −0.816291 with y = 0.485954; because a > 0 it is a minimum.
- If a is positive the parabola opens upwards, if negative downwards.
- With n = 3 the parabola passes exactly through all points (R² = 1).
- For a straight trend use linear regression; the residual calculator checks the residuals.
Frequently asked questions
How do you calculate a quadratic regression?
Find a, b and c so that the sum of squared deviations is minimal. This means solving the system built from the sums Σx to Σx⁴, Σy, Σxy and Σx²y.
When is quadratic better than linear regression?
When the residuals of the line form a curve or the data are visibly curved, for example braking distances or projectile paths.
What does the vertex mean?
It is the lowest or highest point of the parabola at x = −b / (2a). In our example it lies at x = −0.816291, outside the measured values.
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