Editing Coefficient of determination (section)

=== As explained variance ===
A larger value of ''R''<sup>2</sup> implies a more successful regression model.<ref name=Devore/>{{rp|463}}
Suppose {{nowrap|1=''R''<sup>2</sup> = 0.49}}. This implies that 49% of the variability of the dependent variable in the data set has been accounted for, and the remaining 51% of the variability is still unaccounted for. 
For regression models, the regression sum of squares, also called the [[explained sum of squares]], is defined as
: <math>SS_\text{reg}=\sum_i (f_i -\bar{y})^2</math>

In some cases, as in [[simple linear regression]], the [[total sum of squares]] equals the sum of the two other sums of squares defined above:
: <math>SS_\text{res}+SS_\text{reg}=SS_\text{tot}</math>

See [[Explained sum of squares#Partitioning in the general ordinary least squares model|Partitioning in the general OLS model]] for a derivation of this result for one case where the relation holds. When this relation does hold, the above definition of ''R''<sup>2</sup> is equivalent to
: <math>R^2 = \frac{SS_\text{reg}}{SS_\text{tot}} = \frac{SS_\text{reg}/n}{SS_\text{tot}/n}</math>
where ''n'' is the number of observations (cases) on the variables.

In this form ''R''<sup>2</sup> is expressed as the ratio of the [[explained variation|explained variance]] (variance of the model's predictions, which is {{nowrap|''SS''<sub>reg</sub> / ''n''}}) to the total variance (sample variance of the dependent variable, which is {{nowrap|''SS''<sub>tot</sub> / ''n''}}).

This partition of the sum of squares holds for instance when the model values ''ƒ''<sub>''i''</sub> have been obtained by [[linear regression]]. A milder [[sufficient condition]] reads as follows: The model has the form
: <math>f_i=\widehat\alpha+\widehat\beta q_i</math>
where the ''q''<sub>''i''</sub> are arbitrary values that may or may not depend on ''i'' or on other free parameters (the common choice ''q''<sub>''i''</sub>&nbsp;=&nbsp;''x''<sub>''i''</sub> is just one special case), and the coefficient estimates <math>\widehat\alpha</math> and <math>\widehat\beta</math> are obtained by minimizing the residual sum of squares.

This set of conditions is an important one and it has a number of implications for the properties of the fitted [[Errors and residuals in statistics|residuals]] and the modelled values. In particular, under these conditions:
: <math>\bar{f}=\bar{y}.\,</math>