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a= 1.3mm \(\pm\) 0.1mm

b= 2.8mm \(\pm\) 0.2mm

c= 0.8mm \(\pm\) 0.1mm


Calculate \(Q=\frac{ab^2}{\sqrt{c}}\) with its uncertainty.

 Jan 4, 2018

a= 1.3mm \(\pm\) 0.1mm    so a is between 1.2 and 1.4mm

b= 2.8mm \(\pm\) 0.2mm         b is between 2.6 and 3.0mm     b^2 is between 6.76 and 9.0 mm^2

c= 0.8mm \(\pm\) 0.1mm         c is between 0.7 and 0.9mm     sqrt(c) is between  0.836 rounded down and 0.987 rounded up


Calculate \(Q=\frac{ab^2}{\sqrt{c}}\) with its uncertainty.


ab^2 is between 1.2*6.76 and 1.4*9 mm^3 ...       ab^2 is between 8.112   and 12.6 mm^3 


ab^2/sqrtc is between      8.112/0.987     and   12.6/0.836    mm^3/mm^0.5....... 

              ab^2/sqrtc is between 8.218 rounded down  and   15.072 rounded up      mm^2.5


(8.218+15.072)/2  =  11.645

so the average is 11.64, the smallest is 11.645-3.427=8.218    and the biggest is 11.645+3.427=15.072


So that is          11.645 pm 3.427   approx    but this degree of accuracy is not valid and I am not sure what would be valid.

I'd say maybe

11.6 \pm 3.5   OR  12 \pm 4  would be reasonable.


I'm going with 12 pm 4   but I expect there is a more defined way of chosing the degree of uncertainty that is scientifically valid. 

 Jan 6, 2018

The following presentation is adapted from conversational dialogues with Lancelot Link for stochastic measurement errors in the physical sciences.  



To calculate the error range for uncorrelated and random errors, use Gauss’s equations for normal distribution of errors. Using the Gauss equations, gives a 68% confidence interval the measurement error is within this range.

Here are some basic derivatives from the Gauss equations:


\(\small \text{If Q is a combination of sums and/or differences, then the sigma (error) of Q is equal to } \\ \sigma Q = \sqrt{(\sigma a)^2 + (\sigma b)^2 . . . + (\sigma x)^2 . . . } \\ \text{If } Q = b^2 \text{ then } \sigma Q = 2(\sigma (b)) \text{ where } \sigma (b) \text{ is the error of (b)}\\ \text{If } Q = \sqrt{(b)} \text{ then }\sigma Q = \frac{1}{2} (\sigma (b)) \text{ where } \sigma (b) \text{ is the error of (b)}\\ \text{Note how the exponent becomes the multiplier of the error.}\\ \small \text {For this equation, the easiest method is to calculate each error as a }\\ \small \text {decimal and then use the sum of the decimals to calculate the error range. }\\ \text {Uncertainties (In decimal) for a, b, and c. }\\ a_1 = \frac{0.1}{1.3} = 0.0769\\ b_1 = (2)\frac{0.2}{2.8} = 0.1428\\ c_1 = (0.5)\frac{0.1}{0.8} = 0.0625\\ Q= \frac{1.3 \times (2.8)^2}{\sqrt {0.8}} = 11.4 \pm ((0.0769 + 0.1428+ 0.0625) * (11.4))\\ Q= 11.4 \pm 3.2 \\ \small \text { note: pay attention to significant figures when recording the final error range. }\\ \\\)


Relating to this question and others you’ve posted on here , another important thing you need to do is Stop being a lazy dolt!

I would have answered your chemistry questions, except I was too lazy. I caught this laziness from you—it’s more contagious than dumbness.indecision




 Jan 6, 2018
edited by GingerAle  Jan 6, 2018

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