How to find +/- uncertainty with a least squares regression I have a set of data points with an uncertainty on each point. From these data points I can fit a line, who's which the slope is a significant value. How do I use the information I have to get an estimate on the $ \sigma $ of my final value? All I can find online is information on $ R^2$ and $ \chi ^2 $. Bonus points if there's an easy way to do it in python/numpy/scipy.
 A: You could treat it like a multiple imputation problem.  Basically you just specify distributions to characterize your uncertainty for each point, then you take several draws of your dataset.  Fit your model to each set of draws.  You then average the coefficients, average the variance-covariance matrices, and add a non-negative correction to the VCV's to reflect how different the models are from one another.  
I find Gelman's treatment of it to be quite readable at the intro-level.  The formulas are at the end in the section on combining multiple imputations:
www.stat.columbia.edu/~gelman/arm/missing.pdf
One wrinkle:  is the noise non-independent?  Does error in one point predict error in another?  If so, you need to specify a joint distribution in order to use MI, and take a draw from the multivariate pdf.  This would be trickier.
Edit:  MI gives you properly-inflated SE's.  I'm not sure how you'd use it to get an inflated $\sigma$
A: Consider a linear estimator $\mathbf{\hat{y}} = \mathbf{X\theta}$ fitted with linear regression $\mathbf{\theta} = (\mathbf{X^\top X})^{-1}\mathbf{X}^\top \mathbf{y}$.
If $\mathbf{C}_y = \mathrm{diag}(\sigma_1^2,\sigma_2^2,\ldots,\sigma_m^2)$ is the covariance for the observations $\mathbf{y}$, the covariance for $\mathbf{\theta}$ is given by (see the lemma):
$$
\mathbf{C}_\theta = (\mathbf{X^\top X})^{-1}\mathbf{X^\top C}_y \mathbf{X}(\mathbf{X^\top X})^{-1}
$$
Lemma
The covariance of a linear mapping $\mathbf{y} = \mathbf{Ax} + \mathbf{b}$ is $\mathbf{C}_y = \mathbf{AC}_x\mathbf{A^\top}$ with $\mathbf{C}_x$ the covariance for $\mathbf{x}$.
