Section 6.5 Appendix II: Estimating the Standard Error of a Regression Slope 1
This section is adapted from David M. Lane. "Inferential Statistics for b and r." Online Statistics Education: A Multimedia Course of Study. https://onlinestatbook.com/2/regression/inferential.html
1
This section is adapted from David M. Lane. "Inferential Statistics for b and r." Online Statistics Education: A Multimedia Course of Study. https://onlinestatbook.com/2/regression/inferential.html
This appendix shows how to compute the estimated standard error for the slope of a simple linear regression. The estimated standard error of \(\beta\) is computed using the following formula:
\begin{equation*}
s_{\beta} = \frac{s_{est}}{\sqrt{SSX}}
\end{equation*}
where \(s_{\beta}\) is the estimated standard error of \(\beta\text{,}\) \(s_{est}\) is the standard error of the estimate, and \(SSX\) is the sum of squared deviations of \(X\) from the mean of \(X\text{.}\) \(SSX\) is calculated as:
\begin{equation*}
SSX = \sum{(X-\bar{X})^2}
\end{equation*}
where \(\bar{X}\) is the mean of \(X\text{.}\) The standard error of the estimate can be calculated as:
\begin{equation*}
s_{est} = \sqrt{\frac{(1-r^2)SSY}{n-2}}
\end{equation*}
where \(r\) is the correlation between \(X\) and \(Y\text{,}\) and \(SSY\) is the sum of squared deviations of \(Y\) from the mean of \(Y\text{.}\)
These formulas are illustrated with the data shown in Table 6.5.1. These data are reproduced from Section 3.4. The column X has the values of the independent variable and the column Y has the values of the dependent variable. The third column, x, contains the differences between the values of column X and the mean of X. The fourth column, x\(^2\text{,}\) is the square of the x column. The fifth column, y, contains the differences between the values of column Y and the mean of Y. The last column, y\(^2\text{,}\) is simply square of the y column.
| X | Y | x | x\(^2\) | y | y\(^2\) | |
|---|---|---|---|---|---|---|
| 1.00 | 1.00 | -2.00 | 4 | -1.06 | 1.1236 | |
| 2.00 | 2.00 | -1.00 | 1 | -0.06 | 0.0036 | |
| 3.00 | 1.30 | 0.00 | 0 | -0.76 | 0.5776 | |
| 4.00 | 3.75 | 1.00 | 1 | 1.69 | 2.8561 | |
| 5.00 | 2.25 | 2.00 | 4 | 0.19 | 0.0361 | |
| Sum | 15.00 | 10.30 | 0.00 | 10.00 | 0.00 | 4.5970 |
\(SSY\) is the sum of squared deviations from the mean of Y. It is, therefore, equal to the sum of the y\(^2\) column and is equal to 4.597. The correlation (\(r\)) between \(X\) and \(Y\) is 0.6268, and there are 5 observations (n=5). Thus, the standard error of the estimate is:
\begin{equation*}
s_{est} = \sqrt{\frac{(1-(0.6268)^2)(4.597)}{5-2}} = \sqrt{\frac{2.791}{3}} = 0.964
\end{equation*}
We now have all the information to compute the standard error of \(\beta\text{:}\)
\begin{equation*}
s_{\beta} = \frac{0.964}{\sqrt{10}} = 0.305
\end{equation*}
