Relation between the slope in linear regression and the covariance between x and y

by Kurious Fox

The relationship between the slope in linear regression and the covariance between x and y is an important concept in statistics and data analysis. In simple linear regression, the slope of the regression line represents the change in the dependent variable (y) for a one-unit change in the independent variable (x). This slope is related to the covariance between x and y through the formula for the slope in linear regression, which involves the covariance and variance of the variables. Understanding this relationship can provide insights into how changes in the independent variable impact the dependent variable and the strength and direction of their association. Additionally, it can help in interpreting the significance of the slope and making meaningful conclusions about the data being analyzed. This post will show the exact relations between the two quantities in terms of formulas.

Now, recall that the formula for the slope in linear regression is

b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}

and the formula for the sample covariance s_{xy} is:

s_{xy} = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})

The sample covariance s_{xy} captures the direction and strength of the linear relationship between X and Y . If X and Y tend to increase together, s_{xy} will be positive. If one variable tends to increase while the other decreases, s_{xy} will be negative. If there is no linear relationship, s_{xy} will be close to zero.

The slope (b ) of the regression line can be expressed using s_{xy} and the variance of X (s_x^2 ) as follows:

b = \frac{s_{xy}}{s_x^2}

Where: s_x^2 is the sample variance of X , calculated as:
s_x^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2

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