Quadratic Regression Calculator

How to Use the Quadratic Regression Calculator

This calculator finds the quadratic regression equation from paired X and Y data. To use the calculator:

1. Enter the independent variable values in the X Values field.
2. Enter the corresponding dependent variable values in the Y Values field.
3. Select the number of decimal places you want to display.
4. Click Calculate.

The calculator returns the quadratic regression equation, coefficients a, b, and c, R², adjusted R², standard error, vertex, axis of symmetry, and the number of data pairs. It also provides a complete step-by-step solution and plots the observed data together with the fitted quadratic curve.

Besides fitting the quadratic regression, the calculator allows you to predict Y from X using the fitted equation. Just open the Predict Y from X button, enter the X value, and click Predict.

Tip. The calculator allows you to enter values manually or copy-paste values directly from Excel, Google Sheets, or text documents. For manual entry, you need to separate the values by commas, spaces, tabs, or line breaks.

What Is Quadratic Regression?

Quadratic regression is used to model a relationship between two variables when the data follows a curved pattern rather than a straight line.

The general form of the fitted quadratic equation is:

ŷ = ax² + bx + c

where:

• ŷ is the predicted value of Y,
• x is the independent variable,
• a is the quadratic coefficient,
• b is the linear coefficient, and
• c is the constant.

The squared term, x², allows the regression line to bend. As a result, a quadratic model can represent a relationship that decreases and then increases, or increases and then decreases.

For example, suppose performance improves as study time increases but begins to level off and eventually decline after very long study periods. A straight line may not represent that pattern well, while a quadratic curve may provide a better fit.

Therefore, if you create a scatter plot using the scatter plot maker and realize that the data points form a curved pattern, you should fit a quadratic regression rather than a linear regression.

Quadratic Regression Formula

The quadratic regression equation is: ŷ = ax² + bx + c

The values of a, b, and c are chosen using the least-squares method. In other words, the fitted parabola is selected so that the sum of the squared differences between the observed Y values and predicted Y values is as small as possible.

To calculate the coefficients by hand, we can use the following three normal equations:

• Σx²a + Σxb + nc = Σy
• Σx³a + Σx²b + Σxc = Σxy
• Σx⁴a + Σx³b + Σx²c = Σx²y

Once the required sums are known, the three equations can be solved simultaneously to obtain the values of a, b, and c.

How to Find Quadratic Regression by Hand

The normal-equation method provides a systematic way to calculate quadratic regression without statistical software. To find the quadratic regression manually, follow these steps:

1. Create a table containing x, y, x², x³, x⁴, xy, and x²y, then find the total of each column.
2. Substitute the totals into the three quadratic regression normal equations.
3. Solve the three equations simultaneously. Eliminate c first, then eliminate b to find a. Substitute back to find b and c.
4. Substitute the three coefficients into ŷ = ax² + bx + c to write the final quadratic regression equation.

Example

Suppose we have the following paired data:

Find the quadratic regression equation.

Solution

Step 1: Create the table to find the required sums

From the quadratic normal equations, we need to find the sum of x, y, x², x³, x⁴, xy, and x²y. The table below shows how to find these totals.

Since there are 5 pairs of observations, n = 5.

Step 2: Set up the normal equations

Recall. The three normal equations are:

• Σx²a + Σxb + nc = Σy
• Σx³a + Σx²b + Σxc = Σxy
• Σx⁴a + Σx³b + Σx²c = Σx²y

Substitute the totals from the table into the normal equations; we get:

(1) 30a + 10b + 5c = 65

(2) 100a + 30b + 10c = 190

(3) 354a + 100b + 30c = 644

Step 3: Solve the equations simultaneously

To solve the 3 equations simultaneously, follow these steps:

i) Eliminate c from equations (1) and (2).

To eliminate c from equations (1) and (2), we multiply equation (1) by 10 and equation (2) by 5:

Therefore:

10 x Equation (1) = 300a + 100b + 50c = 650

5 x Equation (2) = 500a + 150b + 50c = 950

ii) Subtract the first equation from the second

[500a + 150b + 50c = 950]
− [300a + 100b + 50c = 650]

= 200a + 50b = 300 (4)

iii) Eliminate c from equations (1) and (3).

To eliminate c from (1) and (3), we multiply equation (1) by 30 and equation (3) by 5.

Hence,

30 x Equation (1) = 900a + 300b + 150c = 1950

5 x Equation (3) = 1770a + 500b + 150c = 3220

iv) Subtract the first equation from the second equation

[1770a + 500b + 150c = 3220]
− [900a + 300b + 150c = 1950]

= 870a + 200b = 1270 (5)

v) Eliminate b from equations (4) and (5).

To eliminate b from equations (4) and (5), we multiply equation (4) by 4 and subtract it from equation (5):

4 x Equation (4) = 800a + 200b = 1200

Subtracting this from equation (5), we get:

[870a + 200b = 1270]
− [800a + 200b = 1200]

= 70a = 70

Making a the subject, a = 70 / 70

= 1

vi) Substitute a = 1 into equation (4) and solve for b.

200(1) + 50b = 300

50b = 100

b = 100 / 50

= 2

vii) Substitute a = 1 and b = 2 into equation (1) and solve for c

30(1) + 10(2) + 5c = 65

30 + 20 + 5c = 65

5c = 15

c = 15 / 5

= 3

Step 4: Write the quadratic regression equation

From step (3), we know that:

• a = 1
• b = 2
• c = 3

Substituting these values into the general quadratic regression equation, we get: ŷ = x² + 2x + 3. Therefore, the fitted quadratic regression model is: ŷ = x² + 2x + 3.

How to Interpret a Quadratic Regression Equation

Suppose your fitted equation is ŷ = 1.5x² − 6x + 10. The three coefficients describe different parts of the curve. Here’s a quick interpretation of the coefficients:

• Quadratic coefficient, a. It determines the direction and curvature of the parabola. If a > 0, the curve opens upward and has a minimum. However, if a < 0, it opens downward and has a maximum.
• Linear coefficient, b. It contributes to the location of the curve and, together with a, determines where the turning point occurs.
• Constant, c. It is the predicted value of Y when x = 0.

Note. In a quadratic regression model, the effect of increasing X is not constant because the slope changes as X changes.

What Does R² Mean in Quadratic Regression?

R², or the coefficient of determination, measures how closely the fitted quadratic curve follows the observed data. Its value normally ranges from 0 to 1. An R² closer to 1 indicates that the fitted model accounts for more of the variation in Y within the observed data.

For example, R² = 0.92 means that approximately 92% of the variation in Y in the fitted sample is accounted for by the quadratic model.

A high R² alone does not prove that quadratic regression is the correct model. As such, you should also examine the shape of the data and consider whether a quadratic relationship makes sense for the variables being studied.

For more detail on this statistic, use the coefficient of determination calculator.

Using the Quadratic Equation to Make Predictions

After fitting the quadratic regression equation, you can use it to predict Y for a given X value.

Suppose the fitted equation is: ŷ = 2x² + 3x + 5 and you want to predict Y when x = 4. Simply substitute x = 4 into the equation and solve for ŷ.

Therefore, ŷ = 2(4²) + 3(4) + 5

= 2(16) + 12 + 5

= 49

Alternatively, you can use the prediction option in the calculator to make this prediction automatically.

Remember. Predictions are generally more reliable within the range of X values used to fit the model. Be cautious when extrapolating far beyond the observed data because quadratic curves can rise or fall rapidly outside that range.

Frequently Asked Questions