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Unit 12 : Linear Regression Analysis : Introduction and Lines of Regression

to have tall sons and short fathers short sons, but the average height of the sons of a group of short fathers is Notes
greater than that of fathers. Thus, by regression we mean the average relationship between two or more variables
which can be used for estimating the value of one variable from the given values of one or more variables.
However, in a bivariate distribution, the analysis is restricted to only two variables only.
12.1 Introduction to Linear Regression Analysis

The study of regression has special importance in statistical analysis. We know that the mutual
relationship between two series is measured with the help of correlation. Under correlation, the
direction and magnitude of the relationship between two variables is measured. But it is not possible
to make the best estimate of the value of a dependent variable on the basis of the given value of the
independent variable by correlation analysis. Therefore, to make the best estimates and future
estimation, the study of regression analysis is very important and useful.
Meaning and Definition
According to Oxford English Dictionary, the word ‘regression’ means “Stepping back” or “Returning
to average value”. The term was first of all used by a famous Biological Scientist in 19th century, Sir
Francis Galton relating to a study of hereditary characteristics. He found out an interesting result by
making a study of the height of about one thousand fathers and sons. His conclusion was that (i)
Sons of tall fathers tend to be tall and sons of short fathers tend to be short in height (ii) But mean
height of the tall fathers is greater than the mean height of the sons, whereas mean height of the short
sons is greater than the mean height of the short fathers. The tendency of the entire mankind to twin
back to average height, was termed by Galton ‘Regression towards Mediocricity’ and the line that
shows such type of trend was named as ‘Regression Line’.
In statistical analysis, the term ‘Regression’ is taken in wider sense. Regression is the study of the
nature of relationship between the variables so that one may be able to predict the unknown
value of one variable for a known value of another variable. In regression, one variable is considered
as an independent variable and another variable is taken as dependent variable. With the help of
regression, possible values of the dependent variable are estimated on the basis of the values of the
independent variable. For example, there exists a functional relationship between demand and price,
i.e., D = f (P). Here, demand (D) is a dependent variable, and price (P) is an independent variable. On
the basis of this relationship between demand and price, probable values of demand can be estimated
corresponding to the different values of price.
Definition of Regression

Some important definitions of regression are as follows :
1. Regression is the measure of the average relationship between two or more variables.
— M.M. Blair
2. Regression analysis measures the nature and extent of the relation between two or more variables,
thus enables us to make predictions. — Hirsch
In brief, regression is a statistical method of studying the nature of relationship between two variables
and to make prediction.
Utility of Regression

The study of regression is very useful and important in statistical analysis, which is clear by the
following points :
(1) Nature of Relationship : Regression analysis explains the nature of relationship between two
variables.
(2) Estimation of Relationship : The mutual relationship between two or more variables can be
measured easily by regression analysis.
(3) Prediction : By regression analysis, the value of a dependent variable can be predicated on the
basis of the value of an independent variable. For example, if price of a commodity rises, what
will be the probable fall in demand, this can be predicted by regression.

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