Calculating error for a specific physics experiment I asked a question for a project I am supposed to do and have changed my topic. Now I will use results from a physics lab I did. It is to estimate the value of $g$, the gravitational acceleration using a ramp and frictionless cart. I want to estimate the error for my calculation using propagation of error, but I am running into issues.
So to begin, we measured five values of the height of the ramp to obtain five measured values of $\theta$ for five different $\theta$ values.$\theta$ is the angle of inclination of the ramp. Then I measured five values for the average acceleration of the cart down the ramp for each of these theta values.
Finally I used the equation $g=\frac{a_x}{\sin \theta}$ and plotted the $\sin \theta$ values on a scatterplot against the acceleration and calculated a line of best fit to estimate the value of $g$ (the slope). Now with this $g$ value I want to calculate an error using propagation of error techniques. I was thinking of calculating the average and standard deviation of $\theta$ values for each different $\theta$ value and doing the same for the acceleration values, then estimating the variance of $\hat{g}=\frac{a_x}{\sin \theta}$ using $\sigma_{\hat{g}}=\sqrt{\frac{\partial \hat{g}}{\partial a_x}\sigma_{a_x}^2+\frac{\partial \hat{g}}{\partial \theta}\sigma_{\theta}^2}$ to obtain a value of $\hat{g} \pm \sigma_{\hat{g}}$, but then I realize there are also errors from using a regression line. Or to be more exact, I am unsure of what error values I need to calculate since I used the equation $\frac{a_x}{\sin\theta}$ indirectly. My question is what can I do to obtain a successful error for my estimated $g$ value?
 A: Well, if you really want to do propagation of error, you'd need to know the uncertainties (or, more precisely, the covariances) of your experimentally measured quantities, right? Those are the errors you'd be propagating. You'd also want them to be small, ideally, since your equations seem to use linear expansion. See, for example, Wikipedia's "propagation of error". If I recall doing intro physics labs correctly, you might have been expected to try to estimate the uncertainties in angles, heights, accelerations, etc.
Alternately, if you don't have the uncertainties, you could just look at your set of measured values for $g$. The experimental standard deviation of this set can be used to estimate the uncertainty. See, for example, Wikipedia's "Unbiased estimation of standard deviation". This does have disadvantages, particularly with small numbers. For example, suppose that you know that your measurement uncertainty is 10, but you make only two measurements and obtain 92.4,91.8. Probably your answer should be something like $92.1 \pm 10$ but if you tried to estimate the uncertainty experimentally you'd get something small, probably just by the coincidence of having only a few data points. But again, to get the more plausible uncertainty of 10, you'd need to have prior knowledge of the uncertainty. Otherwise you'd have to go with the small number derived experimentally, though being aware that the uncertainty in this uncertainty is itself quite high.
