Cubic Regression Calculator

Find the cubic polynomial (degree 3) that best fits your data using least squares method

Data Input (Enter x,y coordinates)

Number of decimal places for coefficients (1-10)

Point
X Value
Y Value
Predicted Y
Residual
Actions
#1
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#2
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#3
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#4
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At least 4 data points are required for cubic regression

At least 4 data points are required for cubic regression

Currently you have 0 valid data points.

Example Calculation

Sample Data

Points: (0,1), (2,0), (3,3), (4,5), (5,4)

Equation: y = a + bx + cx² + dx³

Result

y = 0.997 - 5.076x + 3.069x² - 0.387x³

R² ≈ 0.99 (excellent fit)

Interpretation

The cubic model explains 99% of the variance in the data, indicating an excellent fit.

Model Fit Guide (R²)

0.9 - 1.0: Excellent
Model fits data very well
0.7 - 0.9: Good
Model fits data well
0.5 - 0.7: Moderate
Decent model fit
0.3 - 0.5: Weak
Poor model fit
0.0 - 0.3: Very Weak
Very poor fit

Coefficient Meanings

a (Constant)
Y-intercept when x = 0
b (Linear)
Rate of change (slope)
c (Quadratic)
Curvature coefficient
d (Cubic)
Controls cubic behavior

Understanding Cubic Regression

What is Cubic Regression?

Cubic regression is a type of polynomial regression that fits a cubic polynomial (degree 3) to your data points. The resulting equation has the form y = a + bx + cx² + dx³, where a, b, c, and d are coefficients determined using the least squares method.

When to Use Cubic Regression

  • Data shows S-shaped or complex curved patterns
  • Linear or quadratic models don't fit well
  • Data has multiple inflection points
  • You need to model complex relationships

Mathematical Method

Least Squares Method

β = (X^T X)^(-1) X^T y

Uses matrix operations to minimize squared residuals

Design Matrix X

[1, x, x², x³] for each data point

Contains powers of x values

Minimum Requirements

At least 4 data points needed for cubic fitting

Advantages

  • Models complex curved relationships
  • Can capture multiple turning points
  • More flexible than linear/quadratic
  • Good for S-shaped data patterns

Limitations

  • Can overfit with small datasets
  • Extrapolation can be unreliable
  • More complex than simpler models
  • Sensitive to outliers

Applications

  • Growth curve modeling
  • Economic trend analysis
  • Engineering curve fitting
  • Scientific data modeling