Quadratic Regression Calculator

What the quadratic regression calculator does

Quadratic regression fits a parabola, a curve of the form y equals a plus bx plus cx squared, through your data. This calculator takes your x and y points and works out the three coefficients of the best-fit parabola. It is the tool for data that rises and then falls, or falls and then rises, where a straight line cannot follow the bend.

A parabola has a single curve to it, which lets it capture a peak or a trough. Below is when that shape fits and how it is found.

How to use it

1. Enter your points as x and y values, at least three of them, adding rows as needed.
2. Press Calculate for the three coefficients of the fitted parabola, or Reset to clear it.

When a curve beats a line

A straight line can only go one way: always rising or always falling. Plenty of data does not behave like that. A projectile climbs then drops, profit rises with price then falls once the price puts buyers off, a plant grows fast then levels. These have a turning point, and a line drawn through them fits badly, missing the bend entirely.

A parabola has exactly one bend, so it can rise to a peak and come back down, or dip to a low and climb again. When your data has a single turning point like this, a quadratic captures the shape a line cannot. If the data bends more than once, though, even a parabola is not enough, and a higher-degree curve would be needed.

How the parabola is fitted

The fitting uses the same least squares principle as a straight line: find the coefficients that make the total of the squared vertical distances from the points to the curve as small as possible. The difference is that there are now three coefficients to find rather than two, since the curve has a constant, a linear part, and a squared part.

The calculator solves this by treating x and x squared as two separate ingredients and finding the best combination of them, along with a constant, to match your y values. The result is the single parabola that fits your points most closely in the least squares sense, described by its three coefficients.

Reading the coefficients

The three coefficients each shape the curve. The constant is the value of y when x is zero, where the curve crosses the vertical axis. The linear coefficient tilts the curve, and the squared coefficient controls the bend.

That squared coefficient is the important one for the overall shape. If it is positive, the parabola opens upward, like a valley with a lowest point. If it is negative, it opens downward, like a hill with a highest point. The larger its size, the tighter and steeper the curve, and the closer to zero it is, the flatter the curve and the more the data really behaves like a straight line.

A word on overfitting

A curve with more flexibility will always fit the data at least as closely as a simpler one, but closer is not always better. A parabola can bend to chase the wobble in your points, and if the underlying relationship is really a straight line, that bending is just following noise, not signal.

So a quadratic is the right choice when there is a genuine reason to expect a turning point, or when the data clearly bends. If a line already fits well and the curve barely improves on it, the simpler line is usually the more honest description. Fitting a bendier curve than the situation calls for reads patterns into randomness.

A worked example

Suppose you enter the points (-2, 4), (-1, 1), (0, 0), (1, 1), and (2, 4), which trace a symmetric U shape.

The calculator returns the model y = x², with a constant of 0, a linear coefficient of 0, and a squared coefficient of 1. The positive squared coefficient means the parabola opens upward, with its lowest point at the origin. A straight line through these points would come out flat and fit terribly, explaining almost none of the variation, while the parabola matches them exactly.

Entering your data

Enter at least three points, adding rows for more, up to thirty. Three points fix a parabola exactly, so a trustworthy fit that shows a real pattern wants more than that, spread across the range. The calculator returns the three coefficients that define the best-fit curve.

Questions people ask

What is quadratic regression?

A method for fitting a parabola, y equals a plus bx plus cx squared, to data with a single turning point, using least squares to find the three coefficients.

When should I use it instead of a line?

When the data rises then falls, or falls then rises, so it has one bend. A line can only go one direction and will fit such data poorly.

What does the squared coefficient tell me?

Which way the parabola opens and how sharply. Positive opens upward with a lowest point, negative opens downward with a highest point, and larger size means a tighter curve.

Is a curve always better than a line?

No. A curve fits at least as closely but can chase noise. If a line already fits well, the simpler line is usually the better description of the data.

References

A quick note on where the methods here come from. Polynomial least squares regression is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering regression and least squares.

1. NIST/SEMATECH e-Handbook of Statistical Methods (polynomial and least squares regression). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (regression and least squares). https://openstax.org/details/books/introductory-statistics-2e