Hello,
I need urgent help to calculate absolute values of standard deviations from surface & volume calculations of cylindrical shapes. The tricky part is to calculate according to the law of error propagation:
The 2 formulas are the following:
lenght of mantle m = root[(R-r)2 + h2]
mantle surface M = (R+r)* Pi* m
measurement values are: r=0,89 R=1,43 h=27, as well as Dr=0,79 DR=0,08 Dh=0,5
Can somebody help me out with the exact formula for the standard deviation of the measures?
Thanks for help! Verena Hoelzer
You first calculate the partial derivatives of the two fromulas for each measured variable. Then you calculate the sum of the products of the errors (Dr, DR, and dh) with the squared corresponding partial derivative.
Example for the length of the mantle:
dm/dR = (R-r)/root(w)
dm/dr = -(R-r)/root(w)
dm/dh = h/root(w)
where w = (R-r)²+h². The squared derivatives are
(dm/dR)² = (R-r)²/w
(dm/dr)² = (R-r)²/w
(dm/dh)² = h²/w
The propagated error is
Dm = DR*(dm/dR)² + Dr*(dm/dr)² + Dh*(dm/dh)²
Dm = DR*((R-r)²/w) + Dr*((R-r)²/w) + Dh*(h²/w)
Dm = (DR+Dr)*((R-r)²/w) + Dh*(h²/w)
I hope I did not make some error somewhere.
You will surely be able to do the same for the simpler formula of the surface.
Thank you a lot Jochen for your help and effort!
I contacted some physic experts, and they were able to propagate me al equations. The y had a hard time, though ;)
What Jochen WIlhelm has presented is "error propagation law" at (practical) work. Without going into details almost everybody would follow this line. But: 1) the presented "law" is incomplete, because correlations between variables are arbitrarily set to zero, hence neglected, while in reality they may be non-zero, either negative or positive, thus modifying the final result, maybe substantially, and 2) the "law" applies to "small" deviations (who knows when the deviation is "small"?). In this case r=0.89 and Dr=0.79 - this is certainly not a "small" deviation, therefore the obtained results may (and will!) differ from the true standard deviations. Their order of magnitude, however, should be correct.
Dear Verena
You can read the following
https://www.easycalculation.com/area/learn-cylinder.php
bhsd228.schoolwires.net/cms/.../AP_BIOLOGY_EQUATION_LIST.doc
mathbio.nmsu.edu/documents/ten-equations.pdf
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