How to Find Regression Equation?

A simple linear regression is a method in statistics which is used to determine the relationship between two continuous variables. A simple linear regression fits a straight line through the set of n points. Learn here the definition, formula and calculation of simple linear regression. Check out this simple/linear regression tutorial and examples here to learn how to find regression equation and relationship between two variables. using the slope and y-intercept.

Simple / Linear Regression Tutorial, Examples

Regression Definition:

A regression is a statistical analysis assessing the association between two variables. In simple linear regression, a single independent variable is used to predict the value of a dependent variable.

Regression Formula:

Regression Equation(y) = a + bx Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX² - (ΣX)²) Intercept(a) = (ΣY - b(ΣX)) / N Where, x and y are the variables. b = The slope of the regression line a = The intercept point of the regression line and the y axis. N = Number of values or elements X = First Score Y = Second Score ΣXY = Sum of the product of first and Second Scores ΣX = Sum of First Scores ΣY = Sum of Second Scores ΣX² = Sum of square First Scores

Regression Example:

To find the Simple/Linear Regression of

X Values	Y Values
60	3.1
61	3.6
62	3.8
63	4
65	4.1

To find regression equation, we will first find slope, intercept and use it to form regression equation.

Step 1:

Count the number of values. N = 5

Step 2:

Find XY, X² 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:

Find ΣX, ΣY, ΣXY, ΣX². ΣX = 311 ΣY = 18.6 ΣXY = 1159.7 ΣX² = 19359

Step 4:

Substitute in the above slope formula given. Slope(b) = (NΣXY - (ΣX)(ΣY)) / (NΣX² - (ΣX)²) = ((5)*(1159.7)-(311)*(18.6))/((5)*(19359)-(311)²) = (5798.5 - 5784.6)/(96795 - 96721) = 13.9/74 = 0.18784

Step 5:

Now, again substitute in the above intercept formula given. Intercept(a) = (ΣY - b(ΣX)) / N = (18.6 - 0.18784(311))/5 = (18.6 - 58.41824)/5 = -39.81824/5 = -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 know 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). = -7.964+12.032. = 4.068 This example will guide you to find the relationship between two variables by calculating the Regression from the above steps.

Related Calculator:

Simple/Linear Regression Calculator.

This tutorial will help you dynamically to find the Simple/Linear Regression problems.