= Residual estimate of slope uncertainty through the origin
{title2=$s_{\widehat\beta}^2=\mathrm{RSS}/[(n-1)\sum_i x_i^2]$}
For independent errors of common unknown <variance> in $Y_i=\beta x_i+\varepsilon_i$, the <ordinary least squares> estimate is $\widehat\beta=\sum_i x_iY_i/\sum_i x_i^2$. Fitting one parameter leaves $n-1$ residual <degrees of freedom>. Therefore $s^2=\sum_i(Y_i-\widehat\beta x_i)^2/(n-1)$ estimates the common error <variance>, and $s^2/\sum_i x_i^2$ estimates the slope <variance>. For errors with a <Gaussian distribution>, the slope's standardized deviation follows a <Student's t-distribution> with $n-1$ <degrees of freedom>. A common <calibration> bias does not appear as independent residual scatter.
Back to article page
