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The Equation of the Regression Line for Y on X

When making regression predictions for Y observations based on X observations,
one uses the regression line for Y on X. While all such estimates can be
achieved using only the slope of the regression line

some people prefer to use the equation of the regression line. To obtain
this equation, you need to note the slope of the line [as above] and remember
that the line passes through the center of averages (A_{X} , A_{Y}).
Using the most common form of a linear equation Y = (slope) X + (Y-intercept)
we immediately see that

To find the Y-intercept, we use the fact that the regression line passes
through (A_{X} , A_{Y}). This tells us

Solving this expression for the Y-intercept, the equation of the regression
line for Y on X becomes

Using some basic algebra, this equation has an equivalent expression:

With a bit more algebra, we see:

Recalling the conversions to standard units for both X and Y observations,
we see from this expression that the equation of the regression line for
Y on X __in terms of standard units__ is

Last revised: February 2000. Questions to: kvalente@mail.colgate.edu

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