Meaning of Regression

In the Std XI chapter on correlation we learnt how to measure whether two variables are related and how strongly — Karl Pearson's coefficient $r$ tells us the direction and degree of the linear relationship. But knowing that height and weight are correlated does not, by itself, let us predict a person's weight from their height. Regression is the next step: it is the statistical technique used to estimate (predict) the value of one variable from the known value of the other when the two are correlated.

The word regression was introduced by Sir Francis Galton, who noticed that the heights of sons tended to "regress" towards the average height of the population. Today the term simply means the method of estimating one variable from another using the line of best fit through the data.

Dependent and independent variables. In a regression study, the variable whose value we want to estimate is the dependent variable, and the variable used to make the estimate is the independent variable. Because either variable can be estimated from the other, regression gives us two relationships (and hence two lines):

• estimating $Y$ from $X$ — treating $Y$ as dependent — the regression of $Y$ on $X$;
• estimating $X$ from $Y$ — treating $X$ as dependent — the regression of $X$ on $Y$.

This Std XII Mathematics and Statistics course develops the linear form of both relationships. The regression principles studied here are the standard estimation methods common to the wider mathematics-and-statistics curriculum, and they build directly on the correlation work of Std XI.