How to use the Least Squares Curve Fitting Calculator
- 1. Enter the dataType the x values and the matching y values, separated by commas.
- 2. Pick the curveStraight line, parabola, y = a e^(bx), y = a bˣ or y = a xᵇ.
- 3. Read the workingSee the table of sums, the normal equations, the constants and each residual.
The method of least squares
The best-fitting curve is the one that makes the sum of squared errors Σ(y − ŷ)² as small as possible. Setting the partial derivatives to zero gives the normal equations. For a straight line y = a + bx they are Σy = na + bΣx and Σxy = aΣx + bΣx². For a parabola y = a + bx + cx² there is a third unknown and a third equation, using Σx³, Σx⁴ and Σx²y.
Curves like y = a e^(bx), y = a bˣ and y = a xᵇ are fitted by taking logarithms, which turns them into straight lines in ln y (and ln x). This is the textbook approach; it minimises the error in ln y rather than in y.
Worked example
Fit y = a + bx + cx² to x = 0, 1, 2, 3, 4 and y = 1, 1.8, 1.3, 2.5, 6.3. The sums are n = 5, Σx = 10, Σx² = 30, Σx³ = 100, Σx⁴ = 354, Σy = 12.9, Σxy = 37.1 and Σx²y = 130.3.
The normal equations are 12.9 = 5a + 10b + 30c, 37.1 = 10a + 30b + 100c and 130.3 = 30a + 100b + 354c. Solving them gives a = 1.42, b = −1.07, c = 0.55, so the best parabola is y = 1.42 − 1.07x + 0.55x².
Which curve to choose
Plot the points first. A steady rise suggests a line, one bend suggests a parabola, and growth by a constant factor suggests an exponential. Compare the sums of squared errors. For correlation and both regression lines, use the regression line calculator.
Frequently asked questions
What are normal equations?
They are the equations you get by setting the derivative of the squared error to zero for each unknown. Solving them gives the least squares constants.
Why take logarithms for exponential curves?
ln(a e^(bx)) = ln a + bx is a straight line in x, so the line formulas apply. The tool then converts back to a and b.
How many points do I need?
At least one more than the number of unknowns: 3 points for a line and 4 for a parabola. More points give a more reliable fit.
Is this the same as linear regression?
The straight-line fit is the regression line of y on x. The other curves extend the same idea.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.