# Experimental error

#### green-fresh

hi everyone !

when we say
(for example)a,b and c are the dimension of Parallelogram

a=2.267+- 0.002 cm
b=3.376+-0.001 cm
and
c=0.207+-0.001 cm

dose that means that we used equipment to calculat a and another to calculate b and c ?

because i know that delta a or b or c is half of the smallest Grad of the measure equipment

and if they asked for the volume
v=a*b*c (2.267*3.376*0.207)
and the delta v what would bee???

this is not a homework it is for sake of knowledge

thank you :}

#### kote

hi everyone !

when we say
(for example)a,b and c are the dimension of Parallelogram

a=2.267+- 0.002 cm
b=3.376+-0.001 cm
and
c=0.207+-0.001 cm

dose that means that we used equipment to calculat a and another to calculate b and c ?

because i know that delta a or b or c is half of the smallest Grad of the measure equipment

and if they asked for the volume
v=a*b*c (2.267*3.376*0.207)
and the delta v what would bee???

this is not a homework it is for sake of knowledge

thank you :}
Complicated question . There are different methodologies that can be used for calculating error. To get the maximum error range in your volume you can assume that all of your measurements were on the large side and subtract all of your errors from your measurements before multiplying. You can then do the opposite and add all of the errors before multiplying. This will give you maximum and minimum bounds on your volume, assuming that the errors in your measurements represent the absolute maximum and minimum possibilities.

But DON'T do it this way. This is NOT how it is typically done. It might seem logical, but it's bad statistics.

We usually assume that the measurement values are randomly (normally) distributed around the actual value somewhere, with the error bounds being some number of standard deviations out. Using this assumption, when we multiply measurements together it is more likely that their errors will cancel each other out than magnify each other. We don't do it how it's described above for this reason.

http://www.physics.uc.edu/~bortner/labs/Physics Lab web site/Appendix 2/Appendix 2 Error Propagation htm.htm looks like it has a pretty good description of what is most often the right way to do it. See especially the "Rules for the Propagation of Error" section. I just found this on google, so better resources probably exist, but it looks pretty good.

If you can't assume a normal distributions in your measurements then some other method would be better suited, and you'd probably have to take some advanced stats to get into those .

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#### kote

By the way, the error given by taking half of the measurement interval is only one aspect of the error involved. It only tells you how much you as a human might have been off when you took the measurement. What if your measuring device itself is calibrated incorrectly? There are other sources of error you have to account for as well, and there's another method for adding each of those errors.

#### Bob S

If the three dimensions a, b, and c are uncorrelated, and the uncertainties are Gaussian distributed, The fractional variancies are additive. In this case

(δv/v)2=(δa/a)2+(δb/b)2+(δc/c)2

Bob S

#### green-fresh

kote!! thank you very much so better resources probably exist, but it looks pretty good.
i think this resource is much better than what i found and i will try to get a book about number and error analysis .

Bob S sorry but what is this case is different from:
$$\Delta$$v/v =$$\Delta$$a/a + $$\Delta$$b/b + $$\Delta$$c/c
where
delta a is known
delta b is known
delta c is known
and then we calculate $$\Delta$$v from the last equation because we know v
from a*b*c.

i am sorry again Bob S but i didn't understand why we would write as you said in this case. :)

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#### Bob S

kote!! thank you very much i think this resource is much better than what i found and i will try to get a book about number and error analysis .

Bob S sorry but what is this case is different from:
$$\Delta$$v/v =$$\Delta$$a/a + $$\Delta$$b/b + $$\Delta$$c/c
where
delta a is known
delta b is known
delta c is known
and then we calculate $$\Delta$$v from the last equation because we know v
from a*b*c.

i am sorry again Bob S but i didn't understand why we would write as you said in this case. :)
You are adding errors linearly. Statistical (Gaussian) fractional errors add quadratically. This is the same as adding fractional variances. Example: If you have two multiplicable factors A and B each with an uncorrelated 10 ppm uncertainty, the product C=A*B has a 14 ppm uncertainty.
Bob S

[added] See post #2 in
by Astronuc.

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#### gmax137

If you made the three measurements yourself and you used the same ruler for each measurement, then you should probably add the errors as described by kote in Post 2 (in the DO NOT DO IT THIS WAY way). There are several error terms that will not be independent (the ruler is too short, like maybe it starts at 1/16 inch instead of zero; it is stretched, like maybe it was marked at 68degF and you are measuring at 200F; etc etc. If you can't prove the errors are independent and 'gaussian' then you shouldn't assume they are.

Calculate the error that way, then calculate it the second way (adding in quadrature) - the first way gives a higher error, but so what? what are you trying to gain by reducing the error?

#### Bob S

If you made the three measurements yourself and you used the same ruler for each measurement, then you should probably add the errors as described by kote in Post 2 (in the DO NOT DO IT THIS WAY way). There are several error terms that will not be independent (the ruler is too short, like maybe it starts at 1/16 inch instead of zero; it is stretched, like maybe it was marked at 68degF and you are measuring at 200F; etc etc. If you can't prove the errors are independent and 'gaussian' then you shouldn't assume they are.

Calculate the error that way, then calculate it the second way (adding in quadrature) - the first way gives a higher error, but so what? what are you trying to gain by reducing the error?
If you have uncertainties that include both correlated and uncorrelated uncertainties, then the proper analysis uses variances with a correlation matrix.
Bob S

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