APEX AP Stats 3.3.5 The Meaning of r-Squared Study Guide

r^2

The square of the correlation coefficient r in bivariate regressions; how well x predicts y.

When is r^2 useful?

1. When looking at multivariable situations, it lets us get a measure of how precisely all of the independent variables together are predicting the dependent variable 2. find the percent variation of the y variable that can be accounted for with changes in the x variable

What pattern should the residuals follow if the model/line fits well?

follow a pattern of normal distribution

What can r^2 tell us about y?

r^2 tells us how much better y can be predicted when we know x.

How can we use r^2 to examine the relationships on the least-squares regression line?

r^2 measures how well x can predict y.

Explained variation

Variation explained by the least squares regression line.

Residual

Unexplained variation; variation in the y variable that isn't explained.

Unexplained variation

There will always be some variation of y that is unexplained.

Total variance of y

The sum of the explained variance and unexplained variance.

Formula to predict y

y-hat = y-bar.

Total variation

The variance of y.

Residuals squared

Add up the lengths of the residuals squared using the equation y-hat = y-bar and dividing by N.

Example of total variance calculation

In this situation, y(2, 3, 5, 5, 10) is added up and divided by 7.6.

Unexplained variance calculation

Take each residual, square it, and add up the squared residuals before dividing by N.

Percentage of explained variation

Find (total variance - explained variance)/total variance.

Correlation coefficients

r and r^2 both measure how closely correlated x and y are.

Causation

r and r^2 cannot prove causation.

Steps finding r^2

1. Get the total variation (the variance of y) 2. Calculate the unexplained variance (take each residual, square it, and add up the squared residuals before dividing by N, the population) 3. Find (total variance - unexplained variance)/total variance