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.

Formula :
least-square-regression Another formula for Slope: Slope = (N∑XY - (∑X)(∑Y)) / (N∑X2 - (∑X)2)
Where,
b = The slope of the regression line a = The intercept point of the regression line and the y axis. X = Mean of x values Y = Mean of y values SDx = Standard Deviation of x SDy = Standard Deviation of y
Example

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

X ValuesY Values
60 3.1
613.6
623.8
634
654.1
To Find,

Least Square Regression Line Equation

Solution :
Step 1 :

Count the number of given x values.
N = 5

Step 2 :

Find XY, X2 for the given values. See the below table

X ValueY ValueX*YX*X
603.160 * 3.1 =18660 * 60 = 3600
613.661 * 3.6 = 219.661 * 61 = 3721
623.862 * 3.8 = 235.662 * 62 = 3844
63463 * 4 = 25263 * 63 = 3969
654.165 * 4.1 = 266.565 * 65 = 4225
Step 3 :

Now, Find ∑X, ∑Y, ∑XY, ∑X2 for the values ∑X = 311 ∑Y = 18.6 ∑XY = 1159.7 ∑X2 = 19359

Step 4 :

Substitute the values in the above slope formula given. Slope(b) = (N∑XY - (∑X)(∑Y)) / (N∑X2 - (∑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.

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