Learning measurement uncertainty; could not find any online resources

Question

I'm currently writing a lab report on Atwood's machine, and the gist of it is that $a=a_{g}\tfrac{\left ( m_{2}-m_{1} \right )}{\left ( m_{2}+m_{1} \right )}$.

We're holding $\left (m_{2}+m_{1} \right )$ and $a_{g}$ constant, where $a$ is our dependent variable and $\left (m_{2}-m_{1} \right )$ is our independent variable.

Since we ran multiple trials with the same $\left (m_{2}-m_{1} \right )$, I think we can say that our measurement uncertainty for $a$ is $\sigma_{a}$ (right? ... not entirely sure). However, we don't know the measurement uncertainty for either $m_{2}$ or $m_{1}$ as we were not measuring them ourselves (the weights were given, without to uncertainty). I'm not sure how to deal with this...

Also, as a hypothetical question, if I had a data set $x$ with some measurement uncertainty, what is the uncertainty of $f\left (x \right )$?

Thanks in advance! I tried Google-ing the issue, but so far there's been nothing.

Answer 1

I know this question is relatively old but I ran across it while looking for answers to my own questions and hopefully I have some insight that can help the next person.

First, to address the question of finding online resources to help with measurement uncertainty, there is an international guidance document called the Guide to the expression of uncertainty in measurement (GUM) that comprehensively covers this topic. My further answers are based on that guidance.

Your measurement uncertainty, or Type A uncertainty as defined in GUM 4.2, is not the standard deviation but rather the standard deviation of the mean, which is $\sigma_{a}/\sqrt{n}$ (see GUM 4.2.3).

Regarding the incorporation of the uncertainty for $m_1$ and $m_2$, these would be classified as Type B uncertainty derived from an analysis of information. Typically the uncertainty of a mass would come from the uncertainty reported by the manufacturer of the scale used to measure the mass. See GUM 4.3.7 for an example of this.

This segues nicely into the last question, regarding the uncertainty of $f(x)$. To translate the uncertainty of a quantity $x$ into the uncertainty of the function $f(x)$, you need to take the partial derivative of $f(x)$ with respect to $x$. This is presented in GUM section 5, "Determining combined standard uncertainty".

Answer 2

From your equation, it appears this is simple linear regression through the origin, which you can fit by ordinary least squares in any statistics software and get an estimate for the slope of the regression line, i.e. $a_g$, and its standard error. I'd recommend drawing the corresponding plot, i.e. of $a$ vs. $m_2-m_1$, to check that a straight line through the origin appears to be a good description of your data.

Answer 3

I don't know the physical system at all so I'll focus more on the statistical issues.

Measurement error estimation tends to depend on the (not very testable) assumptions you are happy to make about it, e.g. whether it is 'classical' or 'Berkson', etc.

However, you might find you can dispense with the analytic approach to use Cook and Stefanski's SIMEX approach to back out the true values and the measurement error variance by adding noise and extrapolating to the noise free state. This is implemented in the R package simex.

An interesting aspect of your problem seems to be that two noisy ingredients go into the independent variable. I can't see how you could back out their variances separately, but the general SIMEX logic might be helpful nevertheless.

The answer to your hypothetical question can be found by googling 'change of variables'. Unsurprisingly it will generally depend on the nature of $f$, e.g. whether it has a friendly inverse, whether and how it is non-linear, etc. Continuing the simulation theme, it is always reasonable to generate from some assumption about $p(x)$, transform each variate, and summarize the results to get an idea of the what you might expect.