In physics lab class we are learning about uncertainty and propagation of error. Last week we learned about how to find uncertainty of a calculated value using the equation $$\delta_f = \left(\frac{\partial f}{\partial x}\right)\delta_x + \left(\frac{\partial f}{\partial y}\right)\delta_y$$ if $f$ is a function of x and y. My teacher showed us how this equation comes from the tailor series.
This week we learned about how to find the statistical version of uncertainty by using the equation $$\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x}\sigma_x\right)^2 + \left(\frac{\partial f}{\partial y}\sigma_y \right)^2}$$
My teacher tells us that this is the statistical version of uncertainty that gives us 68 percent of the total uncertainty. I am having a hard time with this definition. It seems if this were true we could just multiply the equation given earlier by 0.68.
From what I have learned in my statistics class is that when you add standard deviations, you have to add their squares (variances). I can see how this equation would make sense if we were trying to find the standard deviation of a calculated value, but my teacher tells us we plug in the uncertainty for x in $\sigma_x$ and the uncertainty for y in $\sigma_y$.
Are the two symbols $\delta_x$ and $\sigma_x$ representing the same thing? I am confused how the second equation is valid. Is the second equation used to find the standard deviation or the uncertainty? Do physicists just use the word standard deviation to refer to uncertainty? Why don't we plug in the standard deviations of the distributions of x and y for $\sigma_x$ and $\sigma_y$, which can be found using $\sqrt{\frac{1}{n-1}\Sigma_i (x_i - \bar{x})}$. If $\sigma_f$ truly is the standard deviation of the distribution of calculated $f$, then plugging in the uncertainties for $\sigma_x$ and $\sigma_y$ doesn't make sense. Wouldn't this mean that you could manipulated the standard deviation $\sigma_f$ just by what values you choose for your uncertainties.
Also, In my lab class, we are taught to choose our uncertainties based on what we think the limitations of our instruments are. However, I have seen a few other people use the standard deviation of their measurements and call this the uncertainty. Is this the more common method? I think this would clear up some of the problems I am having.