Intermediate Physics Lab Analysis, Uncertainty and Linear Regression

definition

1. The problem statement, all variables and given/known data
"You are asked to do an experiment where you will need to use a rotating blade to measure the wind speed. You measure the number of rotations of the blade at 10 different wind speeds, 10 times each and will make a linear fit to determine the wind speed as a function of rotations."

Justification Question: Suppose the uncertainty of the independent variable is the same for each. Give an upper estimate of the value of that uncertainty that justifies neglecting this uncertainty when doing the linear fit.


2. Relevant equations
-Uncertainty we assign is + or - half of the least count.
-Propagation of Errors? L = L0 + ΔL
-Mean of sample distribution?


3. The attempt at a solution

This is a uncertainty and analysis type problem for my intermediate physics lab course. So I am given a set of data points, which I didn't post because I am more interested in arriving on how to do the calculations. I just want to make it clear that I'm not seeking to get the work done, but would rather learn..

So a set of example data points are however:
10 mph: 12, 13, 14, 17, 14, 14, 15, 13, 14, 14 rotations
15 mph: 18, 19, 20, 18, 18, 18, 17, 19, 21, 19 rotations

Therefore would the uncertainty be just 0.5 rotations per each data point? But then is it really possible to have an error of rotations? So I just apply the fractional uncertainty equation, which is (Uncertainity)/(Value), in which for the value I just take the average for each set, since this is not a combination type uncertainty measurement.

I get a value for each mph data points, example, i get 3.5% for 10mph, and 2.6% for 15mph. Is this approach correct? However, I do not understand why this uncertainty would be neglected in linear regression. Is it because in linear regression, it takes the sample distribution?

Thank you very much!

Simon Bridge

The uncertainty on each data point is well estimated to be below 0.5 rotations if you are always rounding to a whole number of revolutions.
(presumably you are counting number of rotations in a set time frame?)

It is certainly valid to have an uncertainty in the number of rotations - since it is unlikely that an exact whole number of rotations will have occurred. You can see from the statistical variation in the number of rotations that the actual count is uncertain.

For each speed you can find a mean number of rotations, the uncertainty can be estimated from the standard deviation of the distribution of measurements. You can also use the formula for finding the uncertainty on the mean of a gaussian distributed measurement.

Linear regression is itself a statistical method that provides it's own estimates for uncertainties for, say, slope and intercept.

All these are methods for estimating the uncertainties. The best estimate is the smallest value that is certain to be bigger than the "actual" uncertainty. You are going through the process of learning different techniques to make sure the estimate is a good one.