Residual estimate of slope uncertainty through the origin ($s_{\hat{\beta }}^2=\mathrm{RSS}/[(n-1){\sum }_i{x}_i^2]$)

For independent errors of common unknown variance in $Y_i=\beta x_i+{\epsilon }_i$, the ordinary least squares estimate is $\hat{\beta }={\sum }_i{x}_i{Y}_i/{\sum }_i{x}_i^2$. Fitting one parameter leaves $n-1$ residual degrees of freedom. Therefore $s^2={\sum }_i(Y_i-\hat{\beta }{x}_i{)}^2/(n-1)$ 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.