Propagating Uncertainty for sets of data?

[NewtonsHead]

I'm writing the paper on this experiment I just did. Basically I took sets of data for two variables (x,y) and I fit the points to a line in Origin to extract the value that I was trying to measure (J).
*Using generic variables here*
I found a value for J where

J = [8*∏*x*a(b+c)] / y

I did this by graphing the line

y(x) = [8*∏*a(b+c) / J] * x
and extracting J from the slope

The problem is reporting my value of J with an uncertainty. a, b, and c are distances that I measured only 1 time, so I know the uncertainty on those. However, x and y were both measured 10 times in 3 trials each. I fit the average of those 3 trials (10 data points) to a line to obtain J.

Anyone have experience with fitting a lot of data to a line and reporting uncertainties?

 

[Stephen Tashi]

However, x and y were both measured 10 times in 3 trials each.

What errors are involved in measuring x and y? In some measurement scenarios, one sets the value of x and then measures the value of y. So the error in y depends not only on the error of the instrument that measures y, it also depends on the error made in setting x by the instrument that sets x. In other scenarios, you attempt to measure x and y at a time t and use some instrument that measures t.

 

[FactChecker]

My two cents:
When you do a linear regression of x versus y and get the slope, you can calculate the confidence interval of the slope (see http://stattrek.com/regression/slope-confidence-interval.aspx ). Suppose S is the slope from the linear regression, and [L1, L2] is the 95% confidence interval for S.
So X = S*Y, where L1 < S < L2 with 95% confidence.

Now the question is whether you can say that L1 < [8*∏*a(b+c) / J] < L2 with 95% confidence, where a, b, c are your measurements and J has some physical meaning.

That is, can we say that (assuming L1 & L2 are positive), 8*∏*a(b+c)/L2 < J < 8*∏*a(b+c)/L1 with 95% confidence?

Your measurement of a, b, c were not part of a known statistical process. You either have to make repeated measurements and get statistics on those measurements, or you can make some worst-case estimates of the measurement errors and adjust the confidence interval for J accordingly. Most experiments have some small measurement errors and people adjust their conclusions accordingly. Hopefully, they are small enough to still get useful results.

 

[saveGuard]

mayby you should have a look into "Data Fitting and Uncertainty - A practical introduction to weighted least squares and beyond", ISBN 978-3-8348-1022-9, This textbook explains the determination of uncertainties of model parameters quite well and also tells you something about error propagation.

 

[blue_raver22]

As a scientist, it is important to acknowledge and address uncertainties in data analysis. In this case, it seems like there are two main sources of uncertainty: the measurements of a, b, and c, and the measurements of x and y. To accurately report the value of J with its associated uncertainty, it would be helpful to first determine the uncertainty in each of these variables separately. This could involve calculating the standard deviation or error for each set of measurements.

Next, it would be important to consider how these uncertainties propagate through the equation used to calculate J. This can be done using techniques such as error propagation or Monte Carlo simulations. By propagating the uncertainties through the equation, a range of possible values for J can be determined, along with an associated uncertainty.

Another approach could be to use a regression analysis method, such as linear regression, to fit the data and obtain a more precise value for J. This method takes into account the uncertainties in both the dependent and independent variables and provides a statistical measure of the fit, which can be used to determine the uncertainty in the calculated value of J.

In conclusion, when reporting values with uncertainties, it is important to consider all sources of uncertainty and use appropriate methods to determine and propagate them through the calculations. This will provide a more accurate and comprehensive representation of the data and its associated uncertainties.