R-Squared Explained The Coefficient Of Determination

What is r 2 ? The coefficient of determination (r-sq, r 2 ) Mathematically it is the r-value squared (r 2 = r * r)

The LSRL (least-squares regression line or y-hat) is just one way to model data. Remember…since our goal is to predict the y-values with the least amount of error, the LSRL minimizes the vertical (y) distance from each point to the line.

What if we didn’t have the LSRL?

Because a modeling equation is used to predict y-values, maybe we could try predicting all y- values will be the average (mean) y. Without fancy tools (like a calculator), we might do something simple.

Would a horizontal line be a good fit here? Why? Why not?

Not a good fit! The error (difference between “best fit” line and data) is shown by the red lines.

The y-values are not all the same. (Can you see that the y-values are 2, 4, 6, 8, 15? The y-values vary!) We need a line that slants!!

Look at how small the total error is now. That is, the y-hat equation gets closer to all y-values of the data.

In a way, r-Squared tells us how much BETTER the y-hat equation is to predict y-values than using the mean (y-bar).

Another way to think about it: If we used the mean to make predictions, the variation in y would not be taken into account (all predicted y-values would be the same). R-squared tells us what percent of the variation (differences) in y-values is explained by using a slanted line (y-hat) to make predictions instead (so that we get variety in predicted y-values).

R-squared is… _____ percent of the variation in the _______ that is due to (or explained by) the linear relationship with between _______ and ____. Write r-sq as a % y-variable name x-variable name

Example: From the other day, we found that the r-value for coffee price and deforestation was r = Therefore, r 2 = =.9123 So…91.23% of the variation in the deforestation can be explained by the linear relationship between coffee price and deforestation percent. So, our y-hat equation is a very good fit and predictor.

r 2 can be any number 0 to 1. When r close to 0 … then the y-hat equation is not good for making predictions. When r close to 1 … then the y-hat equation makes reliable predictions.

EXAMPLE Using the data for height and hand-span… (Let’s say for this example that I am using the data to try to predict heights (y) using hand span (x).) – For my Room 12 Stats class, the r-value is.7927 – For my Room 17 Stats class, the r-value is.4859 Determine r 2 for each class and interpret each. (Interpretation sentence starter: “___ % of the variation…”) Should I use the corresponding y-hat equations to predict for each class?

EXAMPLE ANSWERS - For my Room 12 Stats class, the r-value is.7927 r 2 =.6284 So 62.84% of the variation in heights can be explained by the linear relationship between hand span and height. – For my Room 17 Stats class, the r-value is.4859 r 2 =.2361 So 23.61% of the variation in heights can be explained by the linear relationship between hand span and height. Should I use the corresponding y-hat equations to predict for each class? We definitely should not use the Room 17 y-hat equation to predict (it is a very bad fit). We might use the Room 12 y- hat equation to predict, but honestly, I wouldn’t. 62% is still too low for good predictions.