# Uncertainty measurements done with multiple devices

so I need to calculate the volume of a cylinder. I have measured both the radius and the height with three different devices (ruler, calipers, micrometer), each of them 10 times (total 90 measurements).

I've calculated the average and combined uncertainies for each parameter, for each device (the average and uncertainty of the radius for calipers, the average and uncertainty of the height for calipers,... and so on).

Now I need to calculate the volume, which would be $$\pi r^2 h$$. I know that for this I need to use the formula with partial derivatives to get the uncertainty, and that I need to calculate the average of all the radii and heights, but I'm not sure how to find their uncertainties.

Do I just do find the average of averages of the radius and height ($$(r1 + r2 + r3)/3$$; $$(h1 + h2 + h3)/3$$) and then just add their uncertainties as the square root of the sum of the square of the uncertainties?

## 1 Answer

A better approach will be to do a weighted average. The weights should be the inverse of the variance. This will give greater importance to the more precise measurements.

Once you have determined the weights then you can use the standard propagation of errors formula to find the uncertainty of the length and radius and from that the uncertainty of the volume.

For your own education I would encourage you to do the calculation twice: once using inverse variance weighting as described and once using an unweighted mean. You should get more uncertainty with the unweighted approach.