Question

What is the relationship between the linear correlation coefficient rand the slope\({b_1}\)of a regression line?

Short answer

There is a direct relationship between the correlation coefficient and slope of the regression line.

Step-by-step solution

Define the correlation coefficient r

A correlation coefficient provides a measure for the magnitude and direction of linear association between the variables.

State the slope of the regression line

The slope of the regression line helps in describing the level ofchange in y variable due to the unit change in x variable.

Describe the relationship between the correlation coefficient and the slope

The slope is computed as,

\({b_1} = r\frac{{{s_y}}}{{{s_x}}}\)

where\({b_1}\)represents the slope of regression equation, r represents the correlation coefficient,\({s_y}\)represents the standard deviation of y and\({s_x}\)represents the standard deviation of x.

It can be observed from the above formula that as the value of correlation increases, the value of slope increases. Similarly, as the value of correlation decreases, the value of slope decreases.

Thus, the relationship between the correlation coefficient r and the slope of the regression line\({b_1}\)is a direct relationship.

Also, this implies, if the value of r is positive, the slope value is also positive. And, if the value of r is negative, then the slope value is negative as the measure of ratio for standard deviations must always be positive.