Quartic Regression Calculator

Enter your x,y data points (one pair per line) into the Quartic Regression Calculator and get a fitted fourth-order polynomial of the form y = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀. You'll see all five coefficients (a₀ through a₄), the R² goodness-of-fit value, and a curve chart showing how well the polynomial fits your data.

Enter at least 5 x,y pairs separated by a comma or space. One pair per line.

Results

R² (Goodness of Fit)

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a₄ (x⁴ coefficient)

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a₃ (x³ coefficient)

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a₂ (x² coefficient)

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a₁ (x coefficient)

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a₀ (constant)

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Number of Points Used

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Quartic Regression Fit

Results Table

Frequently Asked Questions

What is a quartic function?

A quartic function is a polynomial of degree 4, written as y = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀. It can have up to three turning points, making it suitable for data that changes direction more than once. The name 'quartic' comes from the Latin word for fourth power.

What is the difference between linear and polynomial regression?

Linear regression fits a straight line (y = a₁x + a₀) to data and assumes a constant rate of change. Polynomial regression fits a curved line by adding higher-degree terms (x², x³, x⁴, etc.), allowing it to model more complex, non-linear relationships in the data.

How many data points are needed for quartic regression?

You need at least 5 data points to uniquely determine the five coefficients (a₀ through a₄) of a quartic polynomial. In practice, more data points (10 or more) produce a more reliable and stable regression model.

What does R² mean in quartic regression?

R² (R-squared) is the coefficient of determination and measures how well the quartic model fits your data. A value of 1.0 means a perfect fit, while 0.0 means the model explains none of the variance. An R² above 0.95 is generally considered a good fit.

When should I use quartic regression instead of quadratic or cubic?

Use quartic regression when your data has a complex shape with up to three peaks or valleys that simpler polynomial models cannot capture. If a cubic or quadratic model already gives a high R², adding higher-order terms may cause overfitting, so always compare R² values across models.

Can quartic regression be used for prediction (extrapolation)?

Quartic regression can predict y values for new x inputs, but extrapolation beyond the range of your data can be very unreliable. High-degree polynomials tend to behave erratically outside the observed data range, so predictions are most trustworthy within the original data bounds.

What does each coefficient mean in the quartic equation?

In y = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀: a₄ controls the overall shape and end behavior, a₃ affects asymmetry, a₂ influences curvature, a₁ controls the slope, and a₀ is the y-intercept (the value of y when x = 0).

How is the quartic regression calculated?

Quartic regression is solved using the method of least squares, which minimizes the sum of squared residuals (differences between actual and fitted y values). This requires solving a 5×5 system of linear equations, known as the normal equations, derived from the data.

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