Polynomial regression is a type of regression analysis that models the relationship between a dependent variable y and an independent variable x as an n-th degree polynomial. Unlike linear regression, which fits a straight line, polynomial regression can fit curves, making it ideal for data with non-linear trends.
This method helps capture patterns in data that change direction or have more complex relationships, providing a better fit and more accurate predictions in such cases.
The purpose of this Polynomial Regression Calculator is to make it easy for users to:
This tool is useful for students, researchers, and anyone needing a quick and intuitive way to analyze data with curved trends.
To get started, you'll need to enter your data points into the text area provided on the page. Each point should represent a pair of values: one for x and one for y, separated by a comma. This tells the calculator what values to use when building the polynomial regression model.
Each data point must be entered on a separate line in the following format:
x,y
Make sure:
Here’s an example of how to enter your data correctly:
1,2 2,4 3,8 4,16
This represents the following points: (1,2), (2,4), (3,8), and (4,16). Once you've entered your data, choose the polynomial degree and click "Calculate Regression" to see the results.
The polynomial degree determines the complexity of the curve that the calculator will fit to your data. A higher degree allows the model to fit more bends and changes in direction, while a lower degree results in a simpler curve or a straight line (degree 1).
Choosing the right degree depends on the nature of your data:
Be cautious with very high degrees. While they can fit the training data closely, they may not generalize well and could lead to overfitting.
To calculate a polynomial regression, you need at least one more data point than the degree of the polynomial. This ensures that there’s enough information to compute the curve accurately. For example:
The calculator will alert you if you don't enter enough data for the selected degree.
In addition to fitting a curve to your data, the calculator also allows you to make predictions. You can enter a specific x value, and the tool will estimate the corresponding y value based on the polynomial regression model.
This step is completely optional. If you’re curious about what the model predicts for a certain x value—even one not included in your original data—you can enter that value in the “Predict y for x value” field before clicking the Calculate Regression button.
The predicted y value will appear along with the regression results, giving you a quick estimate based on the fitted curve.
Once the polynomial equation is calculated, the calculator plugs your chosen x value into that equation to solve for y. For example, if the equation is:
y = 1.2 + 0.5x + 0.3x²
And you enter x = 2, the calculator computes:
y = 1.2 + 0.5(2) + 0.3(2²) = 1.2 + 1 + 1.2 = 3.4
This makes it easy to estimate future or missing values from your dataset using a curve that reflects the trend in your existing data.
After clicking the Calculate Regression button, the calculator will display the results of your polynomial regression. These include the full equation of the best-fit curve, an R² value, and any prediction you requested.
The calculator presents the regression equation in the form:
y = a + bx + cx² + dx³ + ...
Each coefficient (a, b, c, etc.) represents the weight or influence of that power of x. These coefficients are calculated to minimize the error between the actual data points and the values predicted by the model.
The higher the degree, the more terms your equation will include, allowing it to follow more complex curves in your data.
The R² value, also called the coefficient of determination, measures how well the regression model fits your data. It ranges from 0 to 1:
Generally, a higher R² indicates that the model does a better job capturing the pattern in your data.
Use the equation and R² value together to understand the strength and reliability of your model’s predictions.
The calculator includes a built-in chart that visually displays your data and the polynomial regression curve. This makes it easy to see how well the model fits your points and to interpret the results at a glance.
Each of your entered data points is shown as a blue dot on the graph. These dots represent the actual values you provided. They help you verify that your input was correct and allow you to compare real data against the regression curve.
The red line on the chart is the regression curve calculated by the model. It is drawn based on the polynomial equation generated from your data. The shape of this curve depends on the degree you selected:
The curve passes through or near the data points in a way that best minimizes the error across all points.
By looking at how closely the red line follows the blue dots, you can visually assess the accuracy and fit of your regression model. A smooth curve that follows the general pattern of the dots is a sign of a good fit.
While the Polynomial Regression Calculator is designed to be easy to use, it performs a series of complex mathematical operations behind the scenes to generate accurate results. This section provides a simplified explanation of how it works.
The calculator uses the method of least squares to find the polynomial curve that best fits your data. This approach minimizes the total squared difference between the actual y-values and the values predicted by the model. The result is a set of polynomial coefficients that define the best-fit equation.
To calculate these coefficients, the tool builds and solves a system of equations using matrix algebra:
This process ensures the curve has the smallest total error across all data points.
Once the system of equations is created, the calculator solves it using a method called Gauss-Jordan elimination. This is a step-by-step process that transforms the matrix into a simpler form to extract the solution easily:
This technique is reliable and efficient for solving systems of linear equations, especially in the context of polynomial regression models.
All of this is done automatically in your browser when you click the “Calculate Regression” button — no need for manual math!
Yes, you can use both whole numbers and decimals for x and y values. Just be sure to separate them with a comma and place each pair on a new line.
The calculator will check for errors and notify you if any data points are missing, malformed, or non-numeric. You'll need to correct these before the regression can be calculated.
The R² value measures how well the regression curve fits your data. A value closer to 1 means the curve closely follows the data points, while a value closer to 0 means a poor fit.
Polynomial regression aims to fit the overall trend of the data, not pass through every single point. Especially with noisy data, the curve will smooth out variations to capture the general pattern.
Yes — but with caution. While it provides accurate curve fitting, ensure your data is reliable and that the chosen degree reflects the true behavior of what you're modeling to avoid overfitting or unrealistic trends.
You can select a polynomial degree up to 10. However, higher degrees require more data points and can lead to overfitting. Choose the simplest degree that fits your data well.
If you didn’t enter a value in the “Predict y for x value” field, the prediction section won’t appear. Simply enter a number and recalculate to get a result.