I'm surprised this hasn't been asked before, but I cannot find the question on stats.stackexchange.
This is the formula to calculate the variance of a normally distributed sample:
$$\frac{\sum(X - \bar{X}) ^2}{n-1}$$
This is the formula to calculate the mean squared error of observations in a simple linear regression:
$$\frac{\sum(y_i - \hat{y}_i) ^2}{n-2}$$
What's the difference between these two formulas? The only difference I can see is that MSE uses $n-2$. So if that's the only difference, why not refer to them as both the variance, but with different degrees of freedom?