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PHYSICS 331 (revised 28 March 2005) p.2 of 3
1
x2=
1
n2
1
N
In general, the quantity reported for a measurement will be x x
. If the
parent distribution is a normal(Gaussian BR pp. 27-30)
distribution, the mostcommon case, then the true mean will be
within the range x x 68% of thetime. Stated in other terms, if the
set of Nmeasurements is repeated, 68% of the
time the mean obtained in the second set will lie within the
range x x .
Propagation of Errors
When the quantity being measured, let's call it u, is some
combination of
independent quantities which we will call x, y,
z,...,u=u(x,y,z,...),there is a simple
general rule for calculating the uncertainty in u,u , given
the
uncertainitiesx ,y ,z ,...in x, y, z,.... This is
u2=
u
x
x ,y ,z ,
2
x2+ u
y
x,y ,z ,
2
y2+ u
z
x,y ,z ,
2
z2+...
Two examples of this rule are of particular interest. The first
is the
situation in which uis the sum or difference of the quantities
x, y, z,..., forexample, u= x+ y z. The partial derivatives of
uwith respect to x, yand z areeither +1 or 1 so the expression for
the uncertainty of u reduces to (BR p.42)
u2 = x
2+ y
2+ z
2
The square of the uncertainty of uis the sum of the squares of
the uncertainties ofx, y,and z.
The second example is that in which u can be expressed as the
product
and/or quotient of x, y, and z, for example, u=xy
z. It is a simple matter to show
that the general expression reduces in this case to (BR p.
43)
u
u
2
=x
x
2
+y
y
2
+z
z
2
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PHYSICS 331 (revised 28 March 2005) p.3 of 3
Thus the square of thefractional(or relative) uncertainly in
u,u
u, is the sum of
the squares of thefractional(or relative) uncertainties in x, y,
and z.
For both of the examples above, if the uncertainty in one of the
quantitiesx, y, or zis several times that of the uncertainties in
the other quantities itdominates the uncertainty in u. For example,
if the uncertainty is xis twice that
in yand z, x = 2y = 2z , the uncertainty in xcontributes 82% of
the
uncertainty in u.
