Calculator finds out coefficient of linear function that fits best into series of (x, y) points.

Beta version#

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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 $\sum x_i~y_i$ .

Tags and links to this website#

Tags:

linear_regression · linear_approximation · least_squares_approximation · least_squares_method

Tags to Polish version:

metoda_najmniejszych_kwadratow · przyblizenie_liniowe · regresja_liniowa

What tags this calculator has#

CALCULATOR · MATH · A -> Z (all)

Permalink#

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https://calculla.com/linear_regression?inDataFormat=xy&inValuesX=1%2C%202%2C%203%2C%204&inValuesY=3%2C%205%2C%207%2C%209

Links to external sites (leaving Calculla?)#