Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Propagation of uncertainty in the slope

  1. Nov 4, 2018 #1
    I'm doing a lab report from electronic spectrum of iodine. I did Birge-Sponer plot from my data. Excel gave it to me a slope y = -2,0698x+133,34. From regression analysis I get uncertainties for slope and intercept.

    Slope: ##-2,069761731 \pm 0,075075941##
    Intercept: ##133,3385857 \pm 2,396753622##



    But the real equation of the slope is $$\Delta E_{v'} = \tilde{\nu}_e'-2\tilde{\nu}_e'x_e'(v'+1)$$, where
    $$\tilde{\nu}_e' = 133,3385857 \pm 2,396753622 $$
    $$-2\tilde{\nu}_e'x_e' = -2,069761731 \pm 0,075075941 $$
    $$(v'+1)=x $$.

    Now the question: How I can calculate uncertainty of $$\tilde{\nu}_e'x_e'$$ and $$x_e'$$?
     
    Last edited by a moderator: Nov 4, 2018
  2. jcsd
  3. Nov 4, 2018 #2

    Dale

    Staff: Mentor

    I cleaned up your equations a bit. To do what you want you have to write your desired parameters in terms of your measured parameters (the slope and the intercept).
     
  4. Nov 4, 2018 #3
    So, you mean like this:
    $$\tilde{\nu}_e' - 2\tilde{\nu}_e'x_e' = 0$$
    $$ x_e' = \frac{\tilde{\nu}_e'}{2\tilde{\nu}_e'}$$

    and the uncertainty is
    $$\delta x_e' = \sqrt{ \left( \frac{\partial \frac{ \tilde{\nu}_e' }{2 \tilde{\nu}_e' }}{\partial \tilde{\nu}_e' } \times \delta \tilde{\nu}_e' \right)^2} $$
    $$\delta x_e' = \sqrt{ \left( \frac{1/2}{\partial \tilde{\nu}_e'} \times 2,396753622 \right)^2}$$.

    There for $$\delta x_e' =0.$$ Is this right?
     
  5. Nov 4, 2018 #4

    Vanadium 50

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2017 Award

    Maybe. You haven't told us what your measured parameters are.

    And by the way 133.3385857 +/- 2.396753622 should be 133.3 +/- 2.4. Do you understand why?
     
  6. Nov 4, 2018 #5
    Yes, I understand. I just copy/paste the values from excel. :) What do you mean by measured parameters?

    I think I figured out this problem...

    $$Lets~calculate~x_e':$$
    \begin{align*}
    && k= - 2\tilde{\nu}_e'x_e' &= -2,069761731\\ \\
    \iff && x_e' &= -\frac{k}{2\tilde{\nu}_e'}\\ \\
    \iff && x_e' &= \frac{2,069761731}{2\tilde{\nu}_e'} = 0,007761301
    \end{align*}
    and the uncertainty is:
    \begin{align*}
    &&\delta x_e' &= \sqrt{ \left( \frac{\partial x_e'}{\partial \tilde{\nu}_e' } \times \delta \tilde{\nu}_e' \right)^2 + \left(\frac{\partial x_e'}{\partial k} \times \delta k\right)^2} \\ \\
    \iff && \delta x_e' &= \sqrt{\left(\frac{k}{2\times (\tilde{\nu}_e')^2} \times \delta \tilde{\nu}_e'\right)^2 + \left(-\frac{1}{2 \times \tilde{\nu}_e'}\times \delta k \right)^2} \\ \\
    \iff && \delta x_e' &= \sqrt{ \left( \frac{-2,069761731}{2\times (133,3385857)^2} \times 2,396753622 \right)^2 + \left(-\frac{1}{2\times133,3385857}\times 0,075075941\right)^2} \\ \\
    && &= 0,000314195 \approx 0,001
    \end{align*}
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted