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

# Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
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 Values only (just a sequence of numbers)Serie of (x,y) points x-values y-values

# Results - approximation of your dataset#

 Regression type Approximation formula Coefficient of determination R2 Linear regression Show source$y=2~x+1$ 1

# Summary - function best fitting to your data#

 Measurement points Number of points 4 Points you entered (1, 3), (2, 5), (3, 7), (4, 9) Approximation Regression type Linear regression Function formula Show source$y=2~x+1$ Coefficient of determination R2 1 Line slope a 2 Free term b 1 Helper values Sum of x-values $\sum x$ 10 Sum of y-values $\sum y$ 24 Sum of x squares $\sum x^2$ 30 Sum of multiplies $\sum xy$ 70

# 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:
$\sum_{i=1}^{n} \left[y_i - f(x_i)\right]^2 = min.$
where:
• $(x_i, y_i)$ - coordinations of the i-th measurement point, these are points that we know,
• $f(x)$ - the function we are searching for, we want this function to best match to the measurement points,
• $n$ - 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:
$f(x) = ax + b$
we get following solution:
$a = \dfrac{n~S_{xy} - S_x~S_y}{n~S_{xx} - \left(S_x\right)^2}$
$b = \dfrac{S_y - a~S_x}{n}$
where:
• $S_{x}$ - sum of x-values $\sum x_i$,
• $S_{y}$ - sum of y-values $\sum y_i$,
• $S_{xx}$ - sum of squares $\sum x_i^2$,
• $S_{xy}$ - sum of multiplies $\sum x_i~y_i$.

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