R square or coefficient of determination is the percentage variation in y expalined by all the x variables together.

Example, say we are trying to predict Rent based on square feet and number of bedrooms in the apartment. Say the R square for our model is 72% – that means that all the x variables i.e. square feet and number of bedrooms together explain 72% variation in y i.e. Rent.

Now let say we add another x variable, for example age of the building to our model. By addiding this third relevant x variable the R square is expected to go up. Let say the new R square is 95%. This means that square feet, number of bedrooms and age of the building together explain 95% of the variation in the Rent.

Remember, coefficient of determination or R square can only be as high as 1 (it can go down to 0, but not any lower).

If we can predict our y variable (i.e. Rent in this case) then we would have R square (i.e. coefficient of determination) of 1.

Usually the R square of .70 is considered good.

For those cases where we really know nothing much about say the hormones which increase our body’s immunity against Cancer – in such cases if we have a regression model with say R square of .05 or even .02, is also considered very good. It shows that atleast our x variables (what ever they are) are predicting some effect on cancer immunity.

Can a Regression Model with a Small R-square Be Useful

(source of the above url is http://www.theanalysisfactor.com/small-r-squared/)