Linear Regression

Regression

The word regression was first used by Frances Galton in 1985. It is defined as “The dependence of one variable upon another variable”. For example, weight depends upon height. The yield of wheat depends upon the amount of fertilizer. In regression we can estimate the unknown values of one (dependent) variable from the known values of the other (independent) variable.

Linear Regression

When the dependence of the variable is represented by a straight line then it is called linear regression; otherwise it is said to be non linear or curvilinear regression.

For example, if $$X$$ is the dependent variable and $$Y$$ is the dependent variable, then the relation $$Y = a + bX$$ is linear regression.

Regression Line of Y on X

Regression lines study the average relationship between two variables. In regression line $$Y$$ on $$X$$, we estimate the average value of Y for a given value of $$X$$.

\[Y = a + bX\]

Here, Y is the dependent variable and $$X$$ is the independent variable. An alternate form of regression line $$Y$$ on $$X$$ is:

\[\begin{gathered} Y – \overline Y = b\left( {X – \overline X } \right) \\ Y – \overline Y = r\frac{{{S_Y}}}{{{S_X}}}\left( {X – \overline X } \right) \\ \end{gathered} \]

Regression Line of X on Y

In regression line $$X$$ on $$Y$$ we estimate the average value of $$X$$ for a given value of $$Y$$.

\[X = c + dY\,\,or\,\,X = a + {b_{XY}}Y\]
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Here, $$X$$ is the dependent variable and $$Y$$ is the independent variable. An alternate form of regression line $$X$$ on $$Y$$ is:
\[\begin{gathered} X – \overline X = d\left( {Y – \overline Y } \right) \\ X – \overline X = r\frac{{{S_X}}}{{{S_Y}}}\left( {Y – \overline Y } \right) \\ \end{gathered} \]