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# Simple/Linear Regression Tutorial

Regression Definition:
A regression is a statistical analysis assessing the association between two variables. It is used to find the relationship between two variables.

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
= 0.19

Step 5: Now, again substitute in the above intercept formula given.
Intercept(a) = (ΣY - b(ΣX)) / N
= (18.6 - 0.19(311))/5
= (18.6 - 59.09)/5
= -40.49/5
= -8.098

Step 6: Then substitute these values in regression equation formula
Regression Equation(y) = a + bx
= -8.098 + 0.19x.

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
= -8.098 + 0.19(64).
= -8.098 + 12.16
= 4.06

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.