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Wize University Statistics Textbook > Inference for Linear Regression

Standard Error and R-Square

Simple Linear Regression AnalysisPrevious Section
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R Squared and Standard Error Formulas


Coefficient of Determination (R Squared)


R2=SSRSST=1− SSESST\displaystyle\boxed{R^2=\frac{SSR}{SST}=1-\ \frac{SSE}{SST}}


R2R^2 measures the how much of the variation in Y is explained by X.

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Residual Standard Deviation Se\colorOne{S_e}

Other names:
  • Standard deviation of the estimate
  • Standard error
  • Overall standard deviation

Se=SSEn−k−1=MSE\displaystyle\boxed{S_e=\sqrt{\frac{SSE}{n-k-1}}=\sqrt{MSE}}
where,
  • k=k= # of explanatory variables, xi′sx_i^{'}s
  • SeS_e measures the overall spread of the residuals (or errors).
  • It is the vertical distances between the actual points and the predicted points obtained using the regression line.



Watch Out!
For simple linear regression where k=1k=1, the denominator is n−k−1=n−1−1=n−2n-k-1=n-1-1=n-2.

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Sum of Squared Errors (SSE)

SSE=SSyy−SSxy2SSxx\boxed{SSE=SS_{yy}-\frac{SS_{xy}^2}{SS_{xx}}}

SSE=∑y2−b0∑y−b1∑xy\boxed{SSE=\sum_{ }^{ }y_{ }^2-b_0\sum_{ }^{ }y_{ }-b_1\sum_{ }^{ }xy_{ }}

This is where SST, SSM, and SSE are located in the ANOVA table:


R.E.T. "Really Exciting Table"



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Example: Residual Standard Deviation and R-Square

What is the residual standard deviation?

Se=SSEn−k−1=\displaystyle{S_e=\sqrt{\frac{SSE}{n-k-1}}=}

Se=1422.515610−1−1=477.81445=13.3347\displaystyle{Se=\sqrt{\frac{1422.5156}{10-1-1}}=\sqrt{477.81445}=13.3347}
As you can see, this is the Standard Error in the above output.

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What is the coefficient of determination?

R2=SSRSST=\displaystyle{R^2=\frac{SSR}{SST}=}

R2=417.48441840=0.2269\displaystyle{R^2=\frac{417.4844}{1840}=0.2269}

23
% of the variation in yy can be attributed to its linear relationship with xx.

As you can see, this is the R Square in the above output.


A restaurateur wants to see if there is a relationship between the number of chefs working in the kitchen and profit. Here is the limited output:


(i) Find the coefficient of determination R2R^2 .