Correlation Coefficient and Linear Regression

The correlation coefficient measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where -1 indicates a perfect negative linear relationship, 1 indicates a perfect positive linear relationship, and 0 indicates no linear relationship.

Linear regression, on the other hand, is a statistical technique used to model the relationship between two variables by fitting a linear equation to the observed data. The equation takes the form: y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.

Interpreting the Correlation Coefficient

In this case, the correlation coefficient is 0.8, which indicates a strong positive linear relationship between the variables. This means that as one variable increases, the other variable tends to increase as well, and vice versa.

Explained Variation

When we perform linear regression, we aim to explain the variation in the dependent variable (y) using the independent variable (x). The correlation coefficient squared (r^2) represents the proportion of the variation in y that can be explained by the linear regression model.

In this case, r^2 = (0.8)^2 = 0.64, which means that 64% of the variation in y can be explained by the linear regression model.

Unexplained Variation

The percentage of variation not explained by the linear regression model is equal to 100% minus the explained variation. Therefore, the percentage of variation not explained is 100% - 64% = 36%.

Conclusion

In summary, the percentage of variation not explained by linear regression is 36%. This means that there are other factors or variables not captured by the linear regression model that contribute to the remaining variation in the dependent variable.