Correlation | Linear Regression | Variance and Standard Deviation | Normal Distribution

Calculator Linear Regression

Calculates the simple linear regression, i.e. a straight line that predicts the points of a data set with two sizes as well as possible. If you have two connected quantifiable characteristics, such as height and weight of people, and enter many different values of these sizes in a diagram, then the result is a point cloud in which the points are not randomly distributed, but have a direction. This direction can be represented by a straight line. The regression line and the point cloud will be shown as a graphic.

Please enter the values ​​of the two characteristics separately. For each characteristic, the values ​​must be separated from one another with a blank or a line break. The number of values ​​per characteristic must be the same. The n-th value of the first feature belongs to the n-th value of the second feature. Example calculates with the size (in 1000 km²) and population (in millions) of some European countries.

Characteristic x

Characteristic y

Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Result

Number of value pairs:
Mean value x:
Mean value y:
Intercept β0:
Slope β1:
Regression line f:

Graphic

The formulas are:

n: number of value pairs, Σ: sum i=1 to n
x: mean values of all x_i, y: mean values of all y_i

β₁ = Σ[(x_i−x)*(y_i−y)] / Σ(x_i−x)²

β₀ = y − β₁*x

f = β₀ + β₁x_i

Linear regression is the simplest special case of regression analysis, a branch of statistics. It is assumed that there is a linear relationship between two variables. For some contexts this is true or at least a useful simplification. This is certainly the case in the examples with the size and weight of people and the population and size of a country. Other cases cannot be represented in this way and you have to work with a curve instead of the regression line, for example with a parabola for quadratic relationships or with exponential functions. All of these cases are mathematically much more complex than the linear relationship. This calculator cannot decide whether a linear regression is possible and useful or not; this must be checked for each case.
Like correlation, regression says nothing about causality, i.e. what causes what.

Last updated on 01/10/2026. Author: Jürgen Kummer

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