Say I have a set of measurement values $y_\text{m} = (y_{\text{m},1}, \dots y_{\text{m},N}) $, and compare these with some ground truth $y = (y_1, \dots y_N)$. Then, if I understood correctly, I can estimate the (sample) variance of the error $(y_\text{m} - y)$ in my measurement values as

\begin{equation} \sigma^2 = \frac{\sum_{i=1}^N (y_{\text{m},i} - y_i)^2}{N-1} \end{equation}

(and, if I'm working with a fit instead of ground truth, this changes to the mean square error, with $N-2$ in the denominator).

According to the explanation here, with a sufficiently well-behaved error, I could even compute a confidence interval for $\sigma^2$.

Now, if I'm interested in the relative error, expressed as a percentage, $100 \times (y_\text{m} - y)/y$, how do these estimates for $\sigma^2$ and MSE change? And can I still compute a confidence interval in that case?