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# How can i calculate the uncertainty for the division of two arrays having the error for the two arrays?

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Joseph 3 Feb 2019
Edited: John D'Errico 3 Feb 2019
Hi everyone,
i have two arrays A(n*1) and B(n*1) and their errors er_A(n*1) and er_B(n*1). C=(A./B-1)*1000, is calculted from the arrays A and B. how can i calculate the uncertainity for array C which should be another array er_C(n*1)?
thank you

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John D'Errico 3 Feb 2019
John D'Errico 님이 편집함. 3 Feb 2019
As is often the case, there has been far too little information provided to give an intelligent answer.
You have two vectors of numbers, A, B. You have undcertainties for each, also as vectors of the same lengths.
What do you know about the uncertainties? What are you willing to assume? Is that uncertainty in the ofrm of a normal distribution? Or is it simply a set of bounds, and any number in that interval isequally likely? The difference is rather significant in how you would solve the problem!
Some links that describe what you might do if the variables are normally (Gaussian) distributed.
A significant problem arises when the probability of the denominator can be zero. Remember that a normal distribution has infinite tails. So there is always SOME probability the number can be zero. And anything divided by zero is a problem. In fact, the result gets very nasty if B(i) can ever get to zero.
The same thing applies if your uncertainties are simply represented as pure bounds. So here you might need to think about interval arithmetic. Thus, suppose you represent A(i) and B(i) as the intervals:
[A-dA,A+dA] & [B-dB,B+dB]
Assume that both A and B are always positive, so those intervals do not include zero. Then we can infer a new interval on their ratio as:
[(A - dA)/(B + dB), (A + dA)/(B - dB)]
If A is negative, and does not cross zero (with B always positive), then the new interval will be similar:
[(A + dA)/(B - dB), (A - dA)/(B + dB)]
Note that if the bounds on B do include zero, then all hell breaks loose because of the possible divide by zero.
There are interval arithmetic tools to be found, but none that I know of that are freely downloadable for use in MATLAB.
So it very much depends on what you know and assume about those uncertainties.

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