Quadratic Regression

Regression and Prediction Models

Enter your data points as (x, y) pairs to find the quadratic equation of best fit.

Examples

Click on an example to load the data into the calculator.

Projectile Motion

physics

Modeling the height of a thrown object over time.

0,0 1,25 2,40 3,45 4,40 5,25

Cost Curve

economics

Analyzing the U-shaped relationship between production units and average cost.

10,50 20,35 30,25 40,20 50,22 60,30

Population Growth

biology

Modeling a population that grows rapidly and then slows down due to limiting factors.

1,100 2,250 3,420 4,550 5,600 6,580

Material Stress

engineering

Examining the stress-strain curve for a particular material under load.

0.1,5 0.2,18 0.3,38 0.4,65 0.5,88
Other Titles
Understanding Quadratic Regression: A Comprehensive Guide
Explore the principles, applications, and mathematics behind finding the 'parabola of best fit' for your data.

What is Quadratic Regression?

  • Defining the Parabola of Best Fit
  • Quadratic vs. Linear Regression
  • The Role of the Least Squares Method
Quadratic regression is a statistical method used to model the relationship between two variables by fitting a second-degree polynomial equation to the observed data. The goal is to find the parabola (y = ax² + bx + c) that best represents the trend in the data points. Unlike linear regression, which models a straight-line relationship, quadratic regression is ideal for datasets that show a curved, U-shaped, or inverted U-shaped pattern.
The Core Equation
The general form of the quadratic equation is y = ax² + bx + c, where 'y' is the dependent variable, 'x' is the independent variable, and 'a', 'b', and 'c' are the coefficients that determine the shape and position of the parabola. The coefficient 'a' dictates how wide or narrow the parabola is and whether it opens upwards (a > 0) or downwards (a < 0).
Why Not Just Use a Straight Line?
Many real-world phenomena do not follow a simple linear trend. For example, the height of a projectile over time, the profit of a company as it scales, or the response of a crop to fertilizer often increase to a maximum point and then decrease. A straight line would fail to capture this peak, leading to inaccurate predictions. Quadratic regression provides a more flexible model that can accurately describe these curved relationships.

Step-by-Step Guide to Using the Calculator

  • Entering Your Data Correctly
  • Interpreting the Results
  • Making Predictions
1. Data Entry

In the 'Data Points (x,y)' text area, enter your coordinate pairs. Each pair must be on a new line, with the x and y values separated by a comma. For instance, if you have the points (1, 5), (2, 11), and (3, 21), you would enter them as: 1,5 2,11 3,21 You must provide at least three distinct points to define a unique parabola.

2. Calculation
Once your data is entered, click the 'Calculate' button. The tool will instantly process the points using the method of least squares to determine the optimal coefficients for the quadratic equation.
3. Analyzing the Output
The results section will display: The final equation (y = ax² + bx + c), the specific values for coefficients a, b, and c, and the Coefficient of Determination (R²). R² is a crucial metric, ranging from 0 to 1, that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable. A higher R² value signifies a better fit.
4. Predicting New Values
To use the model for prediction, enter a new x-value into the 'Predict Y for a given X' field. The calculator will substitute this value into the derived equation to compute the corresponding predicted y-value.

Real-World Applications of Quadratic Regression

  • Physics and Engineering
  • Economics and Finance
  • Biology and Environmental Science
Quadratic regression is not just an abstract mathematical concept; it has numerous practical applications across various fields.
Physics: Projectile Motion
The path of an object thrown into the air, under the influence of gravity, follows a parabolic trajectory. Quadratic regression can be used to model this path, predicting the object's height at any given time and determining its maximum height.
Economics: Cost and Revenue Analysis
Businesses often face U-shaped average cost curves, where costs per unit decrease with economies of scale before increasing again due to inefficiencies. Similarly, revenue might peak at a certain price point. Quadratic models help identify the production level that minimizes cost or the price that maximizes revenue.
Agriculture: Crop Yield
The relationship between the amount of fertilizer used and the resulting crop yield is often quadratic. Too little fertilizer results in low yield, but too much can also damage the crops and decrease yield. Regression helps farmers find the optimal amount of fertilizer to use.

Mathematical Derivation and Formulas

  • The Method of Least Squares
  • Solving the System of Normal Equations
  • Calculating the R-Squared Value
The 'best fit' in quadratic regression is achieved by minimizing the sum of the squared differences between the observed y-values and the y-values predicted by the quadratic model. This is known as the Method of Least Squares.
The Normal Equations

To find the coefficients a, b, and c that minimize this error, we take partial derivatives of the sum of squared errors with respect to a, b, and c, set them to zero, and solve the resulting system of three linear equations. These are called the normal equations:

  1. (Σy) = c(n) + b(Σx) + a(Σx²)
  2. (Σxy) = c(Σx) + b(Σx²) + a(Σx³)
  3. (Σx²y) = c(Σx²) + b(Σx³) + a(Σx⁴) Where 'n' is the number of data points. This system can be solved using matrix algebra.
R-Squared (R²) Formula

The Coefficient of Determination is calculated as: R² = 1 - (SSres / SStot).

  • SSres (Sum of Squared Residuals) is Σ(yᵢ - ŷᵢ)², where ŷᵢ is the value predicted by the regression equation for xᵢ. It represents the error of the model.
  • SStot (Total Sum of Squares) is Σ(yᵢ - ȳ)², where ȳ is the mean of all observed y values. It represents the total variation in the data. A model that perfectly explains the data would have an SSres of 0 and thus an R² of 1.

Common Misconceptions and Best Practices

  • Correlation vs. Causation
  • The Danger of Extrapolation
  • Choosing the Right Regression Model
Assuming Causation
A high R² value indicates a strong correlation and a good fit, but it does not imply that changes in 'x' cause changes in 'y'. There could be a third, unobserved variable influencing both. Always be cautious about claiming causality based solely on a regression result.
Extrapolating Beyond the Data Range
A quadratic model may fit your observed data range very well, but it can give absurd predictions for x-values far outside this range. The parabolic trend is unlikely to continue indefinitely. Use the model for interpolation (predicting within your data's range) but be extremely wary of extrapolation.
Is Quadratic Always Best?
Don't assume a quadratic model is needed just because a linear one isn't perfect. Always visualize your data first. Sometimes, a different non-linear model (like exponential or logarithmic) might be more appropriate, or the data might not have a clear pattern at all. Adding complexity to a model (like moving from linear to quadratic) when it's not justified can lead to overfitting, where the model fits the noise in your data rather than the underlying trend.