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Re: [Phys-l] propagation of error

At 13:15 -0400 9/5/06, wrote:

Let a = 40 +/- 5 m
b = 30 +/- 3 m
t = 1.2 +/- 0.1 s

What is a + b and a/t?

Here's how I would approach these two.

a + b:

Method 1 (conservative), simply add the uncertainties, so a + b = (70 +/- 8) m.
Method 2 (more realistic, perhaps) add the uncertainties quadratically, to allow for the likelihood that they won't both always have the same sign, so a + b = (70 +/- 6) m.


Method 1 (conservative), add the relative uncertainties and multiply by the quotient of a/t to get the total uncertainty, so a/t = 40/1.2*(1 +/- 5/40 +/- .1/1.2) = (33.3 +/- 6.9) m/s.
fMethod 2 (more realistic), add the relative uncertainties quadratically, and multiply by the quotient of a/t, giving a/t = (33.3 +/- 5.0) m/s.

As I see it, uncertainties in simple arithmetic operations simply add (either in total, as in addition and subtraction, or relatively, as in multiplication and division--assuming that the quadratic term in the product of (a +/- ea)*(b +/- eb) is small relative to a*eb + b*ea, so it can be neglected. when the uncertainties become large then the rules can change. But one expects that, on average, the uncertainties in the various numbers will be positive about half the time and negative about half the time, so the sum is likely to be less that to total of the absolute values. Adding them quadratically is a way to account for this likelihood.

Of course, some systematic error in the measurements could make the uncertainties all of the same sign, in which case all bets are off, and the only realistic method of finding the total uncertainty is to all the absolute values.

Decisions like that depend on the individual cases and are hard to deal with in general.

Note that the uncertainties, regardless of which method one uses are the same for addition and subtraction, and for multiplication and division, although the percentage uncertainties can vary by a large amount.


Hugh Haskell

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