Coefficient of Determination Math Example 1

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Example 1

medium
A regression model has SST=500SST = 500 (total variation) and SSE=125SSE = 125 (unexplained variation). Calculate R2R^2 and interpret its meaning.

Solution

  1. 1
    R2=1−SSESST=1−125500=1−0.25=0.75R^2 = 1 - \frac{SSE}{SST} = 1 - \frac{125}{500} = 1 - 0.25 = 0.75
  2. 2
    Alternatively: R2=SSRSST=SST−SSESST=375500=0.75R^2 = \frac{SSR}{SST} = \frac{SST - SSE}{SST} = \frac{375}{500} = 0.75
  3. 3
    Interpretation: the regression model explains 75% of the variation in yy; 25% remains unexplained
  4. 4
    For the square root: r=0.75≈0.866r = \sqrt{0.75} \approx 0.866 (if positive association)

Answer

R2=0.75R^2 = 0.75. The model explains 75% of variation in y.
R2=1−SSE/SSTR^2 = 1 - SSE/SST measures the proportion of total variation in y explained by the model. SST = total variation; SSE = residual (unexplained) variation; SSR = explained variation. R2=r2R^2 = r^2 for simple linear regression.

About Coefficient of Determination

The proportion of the total variation in the response variable yy that is explained by the linear relationship with the explanatory variable xx. It equals the square of the correlation coefficient: r2r^2.

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More Coefficient of Determination Examples

Example 2 hard

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Example 3 easy

The correlation between study hours and test score is [formula]. Calculate [formula] and interpret i

Example 4 hard

A model has [formula]. A researcher concludes 'the model is perfect and ready for deployment.' Ident

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