In regression analysis there are two types of variables. The variable whose value is to be predicted is called dependent variable and the variable which is used for prediction is called independent variable.
Regression analysis is a mathematical measure of the average relationship between two or more variables in terms of the original units of the data.

**Regression Analysis**

**Introduction:**

So far we have studied
correlation analysis which measures the direction and strength of the
relationship between two variables. Here we can estimate or predict the value
of one variable from the given value of the other variable. For instance, price
and supply are correlated. We can find out the expected amount of supply for a
given price or the required price level for attaining the given amount of
supply.

The term
“ regression” literally means “ Stepping back towards the average”. It was
first used by British biometrician Sir Francis Galton (1822 -1911), in
connection with the inheritance of stature. Galton found that the offsprings of
abnormally tall or short parents tend to “regress” or “step back” to the
average population height. But the term “regression” as now used in Statistics
is only a convenient term without having any reference to biometry.

**Definition 9.1**

Regression analysis is a mathematical measure of the average
relationship between two or more variables in terms of the original units of
the data.

**Definition 9.2**

In regression analysis there are two types of variables. The variable
whose value is to be predicted is called dependent variable and the variable
which is used for prediction is called independent variable.

Regression
helps us to estimate the value of one variable, provided the value of the other
variable is given. The statistical method which helps us to estimate the
unknown value of one variable from the known value of the related variable is
called Regression.

Regression
equations are algebraic expressions of the regression lines. Since there are
two regression lines, there are two regression equations. The regression
equation of *X *on* Y *is used to describe the variation in the values of* X *for given changes in* Y *and* *the regression equation of *Y*
on *X* is used to describe the
variation in the values of *Y* for
given changes in *X*. Regression
equations of (i) *X* on *Y* (ii) *Y* on *X* and their
coefficients in different cases are described as follows.

When we
deal with actual values of X and Y variables the two regression equations and
their respective coefficients are written as follows

is known as the regression coefficient of X on *Y*, and* r *is the correlation coefficient between* X *and* Y*, σ_{x}* *and σ_{y}* *are standard deviations* *of
*X* and *Y* respectively.

is known
as the regression coefficient of X on Y,
and *r* is the correlation coefficient
between *X* and Y, σ* _{x}* and σ

The
calculation can very much be simplified instead of dealing with the actual
values of *X* and *Y*, we take the deviations of *X*
and *Y* series from their respective
means. In such a case the two regression equations and their respective
coefficients are written as follows:

Note:
Instead of finding out the values of correlation coefficient σ* _{x}*, σ

When
actual means of *X* and *Y* variables are in fractions the
calculations can be simplified by taking the deviations from the assumed means.
The regression equations and their coefficients are written as follows

**(i) Regression Equation of Y on X**

where *dx*=*X*–A,
*dy*=*Y*–*B*, *A* and *B* are assumed means
or arbitrar*y* values are taken from *X* and *Y* respectively.

(i) Correlation
Coefficient is the geometric mean between the regression coefficients *r* =

(ii) If
one of the regression coefficients is greater than unity, the other must be
less than unity.

(iii) Both
the regression coefficients are of same sign.

Tags : Maths , 11th Business Mathematics and Statistics(EMS) : Chapter 9 : Correlation and Regression analysis

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11th Business Mathematics and Statistics(EMS) : Chapter 9 : Correlation and Regression analysis : Regression Analysis | Maths

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