# LSRL Equation - Definition, Formula, Example

Least Square Regression Line method is the accurate way of finding the 'line of best fit'. Line of best fit is the straight line that is best approximation of the given set of data. It helps in finding the relationship between two variable on a two dimensional plane. It can also be defined as 'In the results of every single equation, the overall solution minimizes the sum of the squares of the errors. Learn here on calculating the least square regression line equation.

## Learn Least Square Regression Line Equation - Tutorial

##### Definition

Least square regression is a method for finding a line that summarizes the relationship between the two variables, at least within the domain of the explanatory variable x.

##### Example

This tutorial helps you to calculate the least square regression line equation with the given x and y values. Consider the values

X Values | Y Values |

60 | 3.1 |

61 | 3.6 |

62 | 3.8 |

63 | 4 |

65 | 4.1 |

##### To Find,

Least Square Regression Line Equation

##### Solution :

###### Step 1 :

Count the number of given x values.

N = 5

###### Step 2 :

Find XY, X^{2} for the given values.
See the below table

X Value | Y Value | X*Y | X*X |

60 | 3.1 | 60 * 3.1 =186 | 60 * 60 = 3600 |

61 | 3.6 | 61 * 3.6 = 219.6 | 61 * 61 = 3721 |

62 | 3.8 | 62 * 3.8 = 235.6 | 62 * 62 = 3844 |

63 | 4 | 63 * 4 = 252 | 63 * 63 = 3969 |

65 | 4.1 | 65 * 4.1 = 266.5 | 65 * 65 = 4225 |

###### Step 3 :

Now, Find ∑X, ∑Y, ∑XY, ∑X^{2} for the values
∑X = 311
∑Y = 18.6
∑XY = 1159.7
∑X^{2} = 19359

###### Step 4 :

Substitute the values in the above slope formula given.
Slope(b) = (N∑XY - (∑X)(∑Y)) / (N∑X^{2} - (∑X)^{2})
= ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)^{2})
= (5798.5 - 5784.6)/(96795 - 96721)
= 0.18783783783783292

###### Step 5 :

Now, again substitute in the above intercept formula given.
Intercept(a) = (∑Y - b(∑X)) / N
= (18.6 - 0.18783783783783292(311))/5
= -7.964

###### Step 6 :

Then substitute these values in regression equation formula
Regression Equation(y) = a + bx
= -7.964 + 0.188x
Suppose if we want to calculate the approximate y value for the variable x = 64 then, we can substitute the value in the above equation
Regression Equation(y) = a + bx
= -7.964 + 0.188(64)
= 4.068

Learn how to calculate the Least Square Regression Line Equation in this tutorial, given with the definition, formula and example.