# Propagating Uncertainty for sets of data?

1. Jan 29, 2014

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?

2. Jan 31, 2014

### Stephen Tashi

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.

3. Jan 31, 2014

### 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.

4. May 10, 2014

### 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.