A real number characterising the real-value model prediction quality (goodness of fit):

R^2 = 1 - \frac{MSD(x, \hat x)}{MSD(x, \bar x)} = 1 - \frac{\sum_i (x_i -\hat x_i)^2}{\sum_i (x_i -\bar x)^2}

where

	observed variable represented by a discrete dataset of numerical samples
	predictor of variable , represented by another discrete dataset of numerical samples, with the same number of samples predicted at the same conditions as the original samples
	mean value of the variable , which can be considered as some sort of extreme predictor with zero variability
	mean square deviation between a variable and its predictor
	mean square deviation between a variable and its mean value

It is similar to Mean Square Deviation (MSD) but quantifies the model prediction efficiency in normalized way which is normally more suitable for assessment goodness of fit.

The coefficient of determination normally ranges between :

0, indicating that prediction error is within the variance of the observed variable around its mean value

and

1, indicating a fine fit, fairly reproducing the variability of the

The values falling outside the above range indicate a substantial mismatch between variable and model prediction and have a meaning that gap between predicted and actual values is higher than the variance of the actual data.

See also