Least squares method calculator: linear approximation
Calculator finds out coefficient of linear function that fits best into series of (x, y) points.

Beta version#

This online calculator is currently under heavy development. It may or it may NOT work correctly.
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However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
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Calculation data - measurement points#

Format of input data

Results - approximation of your dataset#

Regression typeApproximation formulaCoefficient of determination R2
Linear regressionShow sourcey=2 x+1y=2~x+11

Summary - function best fitting to your data#

Measurement points
Number of points4
Points you entered(1, 3), (2, 5), (3, 7), (4, 9)
Regression typeLinear regression
Function formulaShow sourcey=2 x+1y=2~x+1
Coefficient of determination R21
Line slope a2
Free term b1
Helper values
Sum of x-values x\sum x10
Sum of y-values y\sum y24
Sum of x squares x2\sum x^230
Sum of multiplies xy\sum xy70

Some facts#

  • ⓘ Hint: If you're not sure what type of regression this is, let us do the hard work for you and visit another calculator: Regression types.
  • Approximation of a function consists in finding a function formula that best matches to a set of points e.g. obtained as measurement data.
  • The least squares method is one of the methods for finding such a function.
  • The least squares method is the optimization method. As a result we get function that the sum of squares of deviations from the measured data is the smallest. Mathematically, we can write it as follows:
    i=1n[yif(xi)]2=min.\sum_{i=1}^{n} \left[y_i - f(x_i)\right]^2 = min.
    • (xi,yi)(x_i, y_i) - coordinations of the i-th measurement point, these are points that we know,
    • f(x)f(x) - the function we are searching for, we want this function to best match to the measurement points,
    • nn - number of measurement points.
  • If we limit the search to linear function only, then we say about linear regression or linear approximation.
  • If we set a condition that we are only looking for a linear function:
    f(x)=ax+bf(x) = ax + b
    we get following solution:
    a=n SxySx Syn Sxx(Sx)2a = \dfrac{n~S_{xy} - S_x~S_y}{n~S_{xx} - \left(S_x\right)^2}
    b=Sya Sxnb = \dfrac{S_y - a~S_x}{n}
    • SxS_{x} - sum of x-values xi\sum x_i,
    • SyS_{y} - sum of y-values yi\sum y_i,
    • SxxS_{xx} - sum of squares xi2\sum x_i^2,
    • SxyS_{xy} - sum of multiplies xi yi\sum x_i~y_i.

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