r^2

the corelation squared

measures how well the y hat is than just using y bar or the other way around...

I don't remeber what r^2 measures I have what it means on my calculator but I still don't understand it

My teacher told me it's basically somehting like how much better y hat is to predict y values than y bar or somethiing like that

can you please explain
thanks

this is on the mark, it is not particularily easy reading. Read it carefully.

http://en.wikipedia.org/wiki/Coefficient_of_determination

Certainly! The term r^2 refers to the coefficient of determination, which is a statistical measure that represents the proportion of the variance in the dependent variable (usually denoted as y) that can be explained by the independent variable(s) (usually denoted as x). It quantifies the goodness-of-fit of a regression model.

To explain how to calculate r^2 and its interpretation, let's assume you have a set of paired data points (x, y). Here's how you can calculate r^2:

1. Calculate the average of the y values, denoted as ȳ.
2. Calculate the predicted y values, denoted as ȳ, using a regression model or a simple average (ȳ).
3. Calculate the sum of squares total (SST), which measures the total variability in the observed y values. SST is calculated by summing up the squared differences between each observed y value and the mean y value (ȳ).
4. Calculate the sum of squares regression (SSR), which measures the variability in the predicted y values. SSR is calculated by summing up the squared differences between each predicted y value (ȳ) and the mean y value (ȳ).
5. Calculate the sum of squares error (SSE), which measures the unexplained variability in the observed y values. SSE is calculated by summing up the squared differences between each observed y value and its corresponding predicted y value (ȳ).
6. The coefficient of determination (r^2) is then given by the equation: r^2 = SSR / SST, or equivalently, r^2 = 1 - (SSE / SST).

Now, let's interpret the value of r^2. It ranges from 0 to 1, with 0 indicating that none of the variance in y is explained by x, and 1 indicating that all the variance in y is explained by x. In other words, r^2 tells us the proportion of the total variation in y that can be accounted for by the regression model or the independent variable x.

When r^2 is closer to 1, it suggests that the model can predict y values with higher accuracy compared to simply using the mean y value (ȳ). On the contrary, when r^2 is closer to 0, it indicates that the model has limited predictive power and using the mean y value would be just as good.

So, in summary, r^2 is a measure of how well the regression model fits the data, and it tells us the proportion of the variation in the dependent variable (y) that can be explained by the independent variable(s) (x).