Meaning of regression analysis
Regression analysis refers to statistical technique for estimating the relationship among variables. Which is concerned with estimating the value of one variable when the value of the other variable is known.
• Regression therefore, is a measure of the average relationship between two or more variables in terms of the original units of data.
REGRESSION LINES
. These refer to graphical devices that describe the average relationship between two variables.
. There are two regression lines namely:
-Regression line of Y on X
-Regression line of X on Y.
REGRESSION EQUATIONS
• Regression equation refers to algebraic expressions of regression lines.
• Since there are two regression lines, there are two regression equations namely:
1. Regression equation of Y on X.
This equation is expressed as Y= a+bx.
Where Y= is the dependent variable to be estimated.
X= is the independent variable
a = is the interception of Y axis
b = slope on the Y axis,
• To determine the values of a and b the following normal equations are to be solved.
∑Y = Na + b∑x
∑XY= a∑X + b∑X²
Or
NB – There is an alternative formula to find the value of a and b
– This formula is only applicable on the regression equation of y on x.
a = Σy – bΣx
n
b = nΣxy – ΣxΣy
nΣx² – (Σx)² 
2. Regression equation of X on Y
• This equation is expressed by x = a + by
• The value of a. and b in the equation are obtained by solving the following normal equations simultaneously.
ASSUMPTIONS OF REGRESSION ANALYSIS🙋
1. The variance of the error terms is constant across all value of independent variables. This is called homoscedasticity.
2. The values of items are normally distributed.
3. There is no correlation between the independent variables in a linear regression equation.
4. Residual errors have a mean value of Zero.
5. There is a linear relationship between the independent variables and dependent variables.
ILLUSTRATION
Use the following values of X and Y to find the regression equation of Y on X and X on Y.
X Y
1 80
2 96
3 83
4 94
5 99
6 92
SOLUTION
For the equation of Y on X
Y = a + bx
Normal equations
∑Y = Na + b∑x
∑XY= a∑X + b∑X²
X Y X² Y² XY
1 80 1 6400 80
2 96 4 9216 192
3 92 9 8464 276
4 83 16 6889 332
5 94 25 8836 470
6 99 36 9801 594
7 92 49 8464 644
∑x= ∑y= ∑x²= ∑y² = ∑xy=
28 636 140 50,070 2588
Hence 636 = 7a + 28b
N= 7 2588 = 28a + 140b
Use elimination or substitution method to find a and b
a = 84.58
b = 1.57 therefore Y = 84.58
Use the alternative method to find a and b i.e
a =636 – b28 = 84.58
7
b = 7 * 2588 – 28*636 = 1.57
7 * 140 – 28²
ii) Regression equation of x on y
x = a +by
Normal equations
∑X = Na + b∑Y
∑XY= a∑Y + b∑Y²
Hence 28 = 7a + 636b
2588 = 636a +58070b
Thus a = -9.63
b = 0.15
Therefore x = 0.15x -9.63
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