How to Combine Gradient Uncertainty with other Uncertainty?

[Banker]

Homework Statement

I did an experiment to measure the speed of sound(using two microphones and a hammer). I changed the distance between the two mics and calculated(using a fast timer) the time taken for the sound to reach from the start mic to the end mic. I made a graph(distance on x axis, time on y axis) on excel using my results and added a line of best fit. I need error bars and the uncertainty in the gradient. Also, I need to combine the uncertainty from the gradient with the random uncertainty, calibration and scale reading uncertainty(from meter stick). How can I do this?

Homework Equations

The Attempt at a Solution

I know the formula for random uncertainty and the Pythagoras-like formula for combining uncertainties. I just don't know how to combine all of this with the gradient uncertainty.

 

[mfb]

How can I do this?

In general: include uncorrelated uncertainties in the individual datapoints, make sure your fit takes those uncertainties into account (not sure if excel can do that). Correlated uncertainties need a different approach.

 

[Banker]

@mfb Thanks for the reply I did a little experimenting and I combined the random uncertainty for each of my times with the digital reading uncertainty(using ∆w^2 = ∆x^2 + ∆y^2 + ∆z^2, x = random uncertainty, y = scale/digital reading uncertainty, z= calibration uncertainty ) and also did the same with my distances. I then plotted these in my excel graph as a custom error bar for each of my points. Is this the correct way to go? How would I go about finding the uncertainty of the gradient now, with the vertical and horizontal error bars in my graph too?

 

[mfb]

Excel has a function for the uncertainty of parameters of linear functions, I don't know if you can also directly get them from a trend line, and I don't know if the uncertainties are taking into account properly (change them to see if the result changes).