Quadratic Regression Calculator

Quadratic regression fits the parabola that misses the points by the least

Some data rises and then falls, or falls and then rises: the height of a thrown ball, the profit at different prices, the fuel a car burns at different speeds. A straight line through data like that is the wrong shape however well it is placed. Quadratic regression fits a parabola instead, y = ax² + bx + c, and chooses a, b and c so that the squared vertical misses from the points add up to as little as possible. It is the same least squares idea as linear regression, with one more coefficient to choose.

Minimising the squared misses leads to three equations in a, b and c, called the normal equations, built from the sums of x, x², x³, x⁴, y, xy and x²y. A graphing calculator solves them in floating point. This page solves them with exact fractions, so a coefficient of -8/7 is printed as -8/7 rather than -1.142857, and R² is an exact fraction too.

Every point's predicted value and residual is listed under the answer. A good quadratic fit leaves residuals that look like noise, scattered above and below zero with no pattern. If they still curve, or grow steadily across the table, a parabola is not the right shape either, and a different model is worth trying.

What to notice about the parabola

• The sign of a gives the shape: negative a opens downwards with a highest point, positive a opens upwards with a lowest point.
• Three points with different x values always fit exactly. The points (1, 4), (2, 9) and (3, 20) lie on y = 3x² - 4x + 5 with every residual 0, which says nothing about the data.
• R² for a parabola is 1 minus the sum of squared residuals divided by the total variation in y. It can never be lower than the r² of a straight line on the same data, because a line is a parabola with a = 0.
• The vertex of the fitted curve is at x = -b/(2a). For the data 2, 6, 11, 13, 12, 8 at x = 1 to 6 that is x = 33/8, where the curve peaks at about 12.45.
• The fitted curve is only trustworthy between the smallest and largest x. Past the ends a parabola turns and heads off to plus or minus infinity, whatever the data does.

How to do quadratic regression by hand

1. 1

   Build the sums

   For every point compute x, x², x³, x⁴, y, xy and x²y, then add each column and count the points. For x = 0, 1, 2, 3 and y = 1, 2, 5, 11: n = 4, Σx = 6, Σx² = 14, Σx³ = 36, Σx⁴ = 98, Σy = 19, Σxy = 45 and Σx²y = 121.

2. 2

   Write the normal equations

   aΣx⁴ + bΣx³ + cΣx² = Σx²y, aΣx³ + bΣx² + cΣx = Σxy, and aΣx² + bΣx + cn = Σy. With the sums above: 98a + 36b + 14c = 121, 36a + 14b + 6c = 45, and 14a + 6b + 4c = 19.

3. 3

   Solve the three equations

   Eliminate one letter at a time, keeping fractions exact. This system gives a = 1.25, b = -0.45 and c = 1.05, so y = 1.25x² - 0.45x + 1.05.

4. 4

   Check with the residuals

   Put each x into the curve and subtract the prediction from the actual y. Here the residuals are -0.05, 0.15, -0.15 and 0.05, their squares add to 0.05, and R² = 1214/1215, about 0.9992.

Parabolas of best fit for small data sets

Each row is one data set typed into the calculator above, with the exact coefficients and R² it reports.

Worked examples

x: 0, 1, 2, 3; y: 1, 2, 5, 11

x: 1, 2, 3, 4, 5, 6; y: 2, 6, 11, 13, 12, 8

x: 0, 1, 2, 3, 4; y: 2, 17, 22, 17, 2

x: -2, -1, 0, 1, 2, 3; y: 9, 4, 1, 0, 1, 4

Common mistakes

• Using only three points and reading R² = 1 as a great fit. Any three points with different x values lie on some parabola.
• Rounding the sums or the coefficients part way through. The normal equations are sensitive, and a small rounding in Σx⁴ can move c noticeably. Keep fractions until the end.
• Writing the equations with the sums in the wrong places. The top-left entry is Σx⁴ and the bottom-right is n; each row moves the powers down by one.
• Predicting far outside the data. The parabola turns and runs off to infinity past its ends, whether the real quantity does or not.
• Choosing a parabola because R² went up. R² for a parabola is always at least the r² of a line on the same data; the question is whether the residuals lose their pattern.

Quadratic regression FAQ