Experimental Uncertainty Analysis
Author: John M. Cimbala, Penn State University
Latest revision: 07 February 2007
Experimental Uncertainty Analysis
•
Now that the principles of measurement uncertainty and confidence level have been established, we can
predict the uncertainty of a
calculated
quantity as well.
•
Experimental uncertainty analysis
provides a method for
predicting the uncertainty of a variable based on
its component uncertainties.
•
Some authors call this analysis the
propagation of uncertainty
.
•
Suppose one measures
N
physical quantities (or variables, like voltage, resistance, power, torque,
temperature, etc.),
x
1
,
x
2
, .
..,
x
N
. Also suppose that each of these quantities has a known
experimental
uncertainty
associated with it, which shall be denoted by
i
x
w
, i.e.,
i
ii
x
x
xw
=
±
.
•
Furthermore, unless otherwise specified, each of these uncertainties has a confidence level of 95%. Since the
x
i
variables are components of the calculated quantity, we call the uncertainties
component uncertainties
.
•
Suppose now that some new variable,
R
, is a function of these measured quantities, i.e.,
=
R
(
x
1
,
x
2
, .
..,
x
N
).
The goal in experimental uncertainty analysis is
to estimate the uncertainty in
R
to the same confidence level
as that of the component uncertainties
, i.e., we want to report
R
as
R
R
Rw
=
±
, where
R
w
is the
predicted
uncertainty
on variable
R
. There are two types of uncertainty on variable
R
:
o
Maximum uncertainty
– We define the
maximum uncertainty
on variable
R
as
,max
1
i
iN
Rx
i
i
R
ww
x
=
=
∂
=
∂
∑
.
±
Because of the absolute value signs, this expression assumes that
all
the errors in the component
variable
x
i
measurements are such that the error in
R
is always the
same sign
.
±
Such a case would be highly unlikely, especially for a large number of variables (large
N
), because
some of the errors would be positive and some negative, and the errors would cancel each other out
somewhat. In other words,
this is a
worst case scenario
.
o
Expected uncertainty
– We define the
expected uncertainty
on variable
R
as
2
,RSS
1
i
i
i
R
x
=
=
⎛⎞
∂
=
⎜⎟
∂
⎝⎠
∑
.
±
Expected uncertainty is also called the
root of the sum of the squares uncertainty
, or
RSS
uncertainty
, because of the above equation – the square root of the sum of squared quantities.
±
This uncertainty estimate is more realistic than the maximum uncertainty since it is unlikely that the
maximum error will occur on all component variables simultaneously.
±
RSS uncertainty is the
engineering standard
, and the usual notation is to set
w
R
equal to the RSS
uncertainty, i.e.,
R
=
w
R
,RSS
.
±
It turns out that we can write
R
= calculated
R
±
w
R
, to
the same confidence level
as that of each of
the
x
i
measurements (the same confidence level as that of the individual component measurements)
.