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2003 University of Arkansas Department of Physics. Supported in part by a grant from the National Science Foundation
1
Appendix A: Propagation of Errors
This document is a compilation of the formulae we use in this laboratory course for
propagating errors.
It begins with firstorder techniques and moves toward the more sophisticated
and general. Don‟t panic. I do not expect you to know how to do all of the calculus you see in
equations 5 and 6. This is so you can see where the relationships come from, but we will continue to
use the values we “guesstimate” from our measuring devices instead.
I. FirstOrder Formulae
When a quantity q is calculated from experimentally determined quantities x, …, z, u, …, w,
where each of these have uncertainties
x, …,
z,
u, …,
w associated with them, then to first order
we may use the following formulae for determining
q.
Adding and Subtracting:
If
w
u
z
x
q
,
then
w
u
z
x
q
.
Eq. (1)
Products and Quotients:
If
w
u
z
x
q
,
then
w
w
u
u
z
z
x
x
q
q
. Eq. (2)
Multiplying by a constant:
If
x
C
q
,
where C is a constant such as 2, 4/3,
, etc.,
then
x
C
q
.
Eq. (3)
Raising a variable to a power, n:
If
n
x
q
,
then
x
x
n
q
q
.
Eq. (4)
All of the above firstorder approximations to the propagated uncertainties can be derived
from two general firstorder formulae.
For functions of one variable, q = q(x):
x
dx
dq
q
, Eq. (5)
and for functions of many variables, q = q(x, …, z):
z
z
q
x
x
q
q
, Eq. (6)
where partial derivatives are used here.
For students who are not yet familiar with partial differentiation, it is easy once you know how
to take the ordinary derivative of a function of one variable.
Suppose we have a function, q = q(w, x,
…, z), of several variables w, x, …, z and we want to calculate its partial derivative with respect to
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2003 University of Arkansas Department of Physics. Supported in part by a grant from the National Science Foundation
2
one of its variables, say x.
To do this, we pretend that all variables other than x are just constants and
we take the ordinary derivative of q with respect to x.
In other words, think of q as being a function
of x only [q = q(x)], which it would be if all other variables were constant.
When taking the partial
derivative of q with respect to another variable, say w, do the same thing but now let all variables
except w be constant and think of q as a function of w only: q = q(w).
II.
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This note was uploaded on 06/04/2009 for the course PHYS 2054 taught by Professor Stewart during the Spring '08 term at Arkansas.
 Spring '08
 Stewart
 Physics

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