PLTL session on Error Propagation

Prof. A. Cucchiara, UVI

GOAL: this PLTL workshop is designed to teach students the rules of error propagation in simple arithmetic operations and, at the end, allows the teams to discuss further aspects of the importance of proper treatment of uncertainties in estimating quantities and derive them from complex functions.

The function that the students will work on is:

\begin{equation} f(a,b,c)=\sqrt[]{\frac{a^{3}(b-c)}{3.4\times bc}}\nonumber \\ \end{equation}

where \(a,b\), and \(c\) are the following quantities:
\(a=3.1\pm 0.1\) m
\(b=14.7\pm 0.2\) m
\(c=10.5\pm 0.3\) m

Procedure: the workshop is divided in multiple parts and follows a round robin approach. Ideally the students should know already from the class the rules of error propagation. The Team leader may aid them, providing them, in case the team gets stuck at any point.

Step 1: The team leader draws from the deck of cards up to 7 cards and let the team members pick one card per member until each member has at least one card (max of three). At this point the leader shows the team that each card corresponds one propagation. Here is an example, assuming we use the cards values of ”Ace, King, Queen, Jack, 10, 9,8 , and 7” (or other sequence):

  • 8: \(b-c\)

  • 9: \(a^{3}\) (or \(a\times a\times a\))

  • 10: \(bc\)

  • Jack: \(3.4\times bc\)

  • Queen: \(a^{3}(b-c)\)

  • King: \(\frac{a^{3}(b-c)}{3.4\times bc}\)

  • Ace: \(\sqrt[]{\frac{a^{3}(b-c)}{3.4\times bc}}\)

Step 2: Starting from the lower value card (the 8) each member of the team work on propagating the error and solving the piece of the function until the final result. The team leader may help the students suggesting them similar calculations done in class on in physics labs.


Step 3: At this point the solution should be reached and every students had his/her chance to apply the rules of error propagation. Now, assuming that the results is the number \(NN\pm ee\). The question is: Does the actual order in which these have been calculated matter? If so, which ones are interchangeable?


Step 4: Question (if there is time left after 50 minutes): How about the following expression:

  • \({\rm log}_{10}(NN\pm ee)\)

  • \({\rm cos}(NN\pm ee)\)

  • \(3^{NN\pm ee}\)

These can be worked in pair or discussed altogether.

Review of the role of propagation:
Given two measured quantities \(x\pm\delta x\) and \(y\pm\delta y\), the resulting uncertainty of the sum/difference of these quantities is:

\begin{equation} \delta_{x+y}=\sqrt{(\delta x)^{2}+(\delta y^{2})}\nonumber \\ \end{equation}

and

\begin{equation} \delta_{x-y}=\sqrt{(\delta x)^{2}+(\delta y^{2})}\\ \nonumber \\ \end{equation}

The resulting uncertainty of the product/ratio of these quantities is:

\begin{equation} \%\delta_{xy}=\sqrt{(\%\delta x)^{2}+(\%\delta y^{2})}\nonumber \\ \end{equation}

and

\begin{equation} \%\delta_{x/y}=\sqrt{(\%\delta x)^{2}+(\%\delta y^{2})}\\ \nonumber \\ \end{equation}

where:
\(\%\delta_{xy}=\frac{\delta_{xy}}{xy}\times 100\%\)
\(\%\delta_{x/y}=\frac{\delta_{x/y}}{x/y}\times 100\%\)
 \(\%\delta x=\frac{\delta x}{x}\times 100\%\)
\(\%\delta y=\frac{\delta y}{y}\times 100\%\)