When R-squared is zero, a model that explains nothing

Question

A regression of lottery winnings on the buyer's shoe size reports 	ext{SST} = 250, \qquad 	ext{SSE} = 250. Find R-squared and interpret the result.

Accepted Answer

Apply the formula: R^2 = 1 - \frac{	ext{SSE}}{	ext{SST}} = 1 - \frac{250}{250} = 1 - 1 = 0. An R-squared of 0 means the model explains none of the variation in y: all of the spread is still sitting in the errors. This is exactly what you would expect for a nonsense predictor like shoe size for lottery winnings. When R^2 = 0, the best-fit line is essentially flat, and predicting with it is no better than just using the average of y for everyone. This is the floor of the scale: R-squared runs from 0 (explains nothing, error equals total) up to 1 (explains everything, error is zero). Here it s…

When R-squared is zero, a model that explains nothing

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