I am trying to work out the errors of $m$ and $b$ in a linear least-square fit of a straight $y = m \cdot x + b$ to $N$ equally-spaced data points $Y_i$ on the interval $I = [-\Delta x \cdot N/2,\Delta x \cdot N/2]$, where $\Delta x$ is the distance between each data point. I do not assume an error for the $x$-position of that data-points and the error for $Y_i$ is $\sigma_Y$. I know that a linear dependence is a good model for my data-points due to the physical system from which the data was obtained.
My question now is how the fit-errors $\sigma_m$ and $\sigma_b$ of $m$ and $b$ will theoretically depend on the error $\sigma_Y$ and $N$, when correspondingly growing the interval $I$. This seems like a trivial problem and I would expect the their error to decrease as $N$ increases, but I can't work out the exact dependence. Perhaps someone can point me in the right direction, thanks in advance!