Uncertainty Analysis
In many, if not most, experimental situations, the final desired result is not measured directly.
Instead, measurements of several variables are substituted into a data reduction equation to
obtain the desired quantity. As an example, suppose the density p of a flowing gas stream is
desired. Direct measurements of gas density are difficult, so instead the pressure P and
temperature T are measured and, assuming the gas can be treated as ideal, the density can be
found from the ideal gas equation of state,
P=pRT
The question that naturally arises is: How do the uncertainties in the individual measured
variables P and T propagate through the data reduction equation into the final result for p?
The purpose of uncertainty analysis is to answer this important question.
As discussed in previous sections, bias is a fixed error that can be reduced by calibration.
On the other hand, precision error is a random error that can be reduced by obtaining
multiple measurements. Because of the differing nature of these two components of
uncertainty, it is desirable to investigate their propagation into the experimental result
separately. This approach is consistent with that recommended in the ANSI/ASME Standard
on Measurement Uncertainty.
A. Propagation of Bias Errors
The steps involved in determining the bias limit WR for the experimental result are
sketched below. Each of the individual measurement variables (X1, X2,…,XK) is subject to
several, say M, elemental bias errors. The bias limits for each of these elemental sources are
combined in some manner to obtain the overall bias limit (W1, W2,…, WK) for each variable.
In turn, the techniques of uncertainty analysis are used to combine the bias limits for the
measured variables into a bias limit WR for the experimental result. The following material
will describe the details of the method required to carry out this procedure.
Perhaps the most difficult step in this process is the identification and quantification of
the elemental bias limits that affect each of the measurement variables. As discussed in an
earlier section, elemental bias error sources can generally be placed in three categories:
calibration, data acquisition, and data reduction. However, assigning magnitudes to these
sources is not a straightforward task. Unlike estimating the precision limit from computation
of the precision index S, there is no statistical calculation that can be done to estimate the
bias limit. Since it is a fixed error, the bias is the same for each measurement. However, its
magnitude, being the difference between the sample mean and the true value, is unknown
because the true value is always unknown in any realistic experiment. Thus, the elemental
bias limits must always be estimated. In addition, the bias limit estimates are made at a
95% confidence level for consistency with the precision limit determinations. This can be
interpreted to mean that the magnitude of the bias β is less than or equal to the bias limit W
at a 95% level of confidence. Information on bias errors can be inferred from
comparison of independent measurements that depend on different physical principles
or that have been independently calibrated. Bias limit estimates can also be made
based on previous experience of the experimenter or other individuals, instrument
manufacturer’s information and specifications, and comparison of measurements with
known values.
Once the bias limits for the elemental error sources are estimated, they must be combined
in some manner to obtain the bias limit for each measured variable. The preferred method
for doing this is the root-sum-square (RSS) technique. For measurement variable XK this is
given by:
[
WK = (W
)
2
K 1
+ (W
)
2
K 2
]
1
2 2
K M
+ ... + (W
)
where WK is the 95% confidence estimate of the bias limit for the measurement.
The next step in the procedure is to apply uncertainty analysis to determine how the bias
limits (W1, W2,…, WK) for the individual variables propagate through the data reduction
equation to form the bias limit WR for the experimental result. The data reduction equation is
taken to be of the form
R=R(X1, X2,…, XK)
where it is assumed that this relation is continuous and has continuous derivatives in the
domain of interest and that the bias limits Wi for the measurement variables are independent of
one another. Under these conditions, the bias limit for the result is given by the uncertainty
analysis expression (see Coleman and Steele, Experimentation and Uncertainty Analysis for
Engineers, Wiley, Appendix B).
2
2
⎛ 1 ∂R
⎞
⎞
⎛ 1 ∂R
⎞ ⎛ 1 ∂R
⎛ WR ⎞
W1 ⎟⎟ + ⎜⎜
W2 ⎟⎟ + ... + ⎜⎜
WK ⎟⎟
⎜
⎟ = ⎜⎜
⎝ R ⎠
⎝ R ∂X K
⎠
⎠
⎝ R ∂X1
⎠ ⎝ R ∂X 2
2
2
Use of this expression will be demonstrated in an example at the end of this section.
B. Propagation of Precision Errors
The procedure for determining the precision limit PR for an experimental result is
generally similar to that for determining the bias limit WR. In fact, a sketch of the procedure
would look identical to the one given previously for the bias limit with the substitution of “P” for
“W” and “precision” for “bias.” The measurement of each variable (X1, X2,…, XK) is influenced
by precision errors from a number of elemental error sources. These random errors combine to
cause the precision error in the measurement of each variable; the latter is quantified by
determining the precision limit (P1, P2,…, PK) of each measured variable. These individual
measurement precision limits are then propagated through the uncertainty analysis to obtain the
precision limit PR for the result.
The way in which the precision limits of the individual measurement variables are
determined depends both on the type of experiment and the phase under consideration. For
example, in the design phase of a new experiment, before any equipment has been specified or
data obtained, estimates for the precision limits are made based on all available information: the
experimenter’s experience, that of others, manufacturer’s specifications, etc. At this stage in the
experiment, the precision limit associated with the measurement system may be the only
precision error source considered. As a general rule-of-thumb, the precision limit resulting
from the readability of an analog instrument can be taken as one-half of the smallest scale
division. Likewise, for a digital output, the precision limit associated with the readability is
one-half of the least digit in the output. For cases in which the precision limit Pi is estimated,
the estimate should be that band which will contain the mean value of the variable with 95%
confidence.
During the execution phase of an experiment, previous measurements may be available
with which to determine each of the Pi values. In other cases, multiple measurements of the
variable may be made during the actual experiment, from which the precision index Si and
precision limit Pi can be calculated from a sample of N readings. Several comments are required
here to clarify this procedure. From the discussion in preceding sections, recall that the
appropriate precision limit to use with a variable Xi that is determined from a single reading is
the precision index of the sample population times the t factor taken from the t-distribution table
for N < 31 or t=2.0 for N ≥ 31,
PXi=tSXi
Of course, for a single reading PXi must be estimated or must be available from previous
measurements. As discussed previously, the ±PXi band around the measurement Xi contains the
mean value of the measure variable with 95% confidence. Therefore, in the uncertainty analysis
equations given below, Xi and Pi should be interpreted as
Xi and PXi
when the value of Xi used in the data reduction equation is determined from a single reading.
When the value of the measurement is determined as the mean X i of N separate readings,
then the precision limit of the sample mean
PXi = tS X =
tS Xi
N
i
should be used. In this case, the ± PX band around the sample mean X i contains the mean value
i
of the measured variable with 95% confidence. Therefore, if the value of the variable that is
used in the data reduction equation is determined as the mean of N separate measurements, the
values
Xi
and
PX
i
should be used in the uncertainty analysis equations.
When several separate factors can be identified as causing the precision error in a
measured variable, it may sometimes be desirable to determine the precision limit by considering
the contributions of the elemental error sources. The procedure is similar to that discussed
previously for the determination of the bias limit. For the XK measurement variable, suppose
that M elemental precision error sources are identified and their 95% confidence precision limits
are determined as (PK)1, (PK) 2 ,…, (PK) M . Then, the overall precision limit PK is given by the
RSS expression
[
PK = (PK )1 + (PK )2 + ... + (PK )M
2
2
2
]
1
2
at the 95% confidence level.
Another factor that must be considered in estimating precision limits is the time period
over which the sample population is obtained. This is summarized in the following rule: Data
sets for determining estimates of precision indices should be acquired over a time period
that is long relative to the time scales of the factors that have a significant influence on the
data and that contribute to the precision errors. If this rule is not followed, the precision
limit estimations would not include long time (low frequency) variations that affect the
measurement, and therefore these limit estimates would be inaccurate.
Once the 95% precision limit Pi for each of the measured variables Xi in the data
reduction equation
R=R(X1, X2, …, XK)
Is determined, the 95% precision limit for the experimental result PR is found from the
uncertainty analysis expression
2
2
⎞
⎛ 1 ∂R
⎞
⎛ 1 ∂R ⎞ ⎛ 1 ∂R
⎛ PR ⎞
P1 ⎟⎟ + ⎜⎜
P2 ⎟⎟ + ... + ⎜⎜
PK ⎟⎟
⎜ ⎟ = ⎜⎜
⎝R⎠
⎠
⎝ R ∂X K
⎝ R ∂X1 ⎠ ⎝ R ∂X 2 ⎠
2
2
(Coleman and Steele, Experimentation and Uncertainty Analysis for Engineers, Wiley, appendix
B). Note that this expression is identical in form to the one used for determination of the bias
limit WR of the result. It is assumed that the data reduction equation is continuous and has
continuous derivatives in the domain of interest and that the precision limits Pi for the measured
variables are independent of one another. In addition, it is assumed that the result R is
determined from the reduction equation only once at a given experimental condition using either
a single measurement Xi or the mean value X i of N repeated measurements.
C. Uncertainty of the Experimental Result
In order to determine the overall uncertainty UR of the experimental result, the bias and
precision limits, WR and PR, must be combined. This is accomplished using the root-sumsquare (RSS) method
(
U R = WR2 + PR2
)
1
2
thereby providing 95% coverage of the true value.
Example 3
FD
, is to be reported for the flow of water over
2
⎞
⎛1
2 ⎞⎛ πD
⎟⎟
⎜ pV ⎟⎜⎜
⎝2
⎠⎝ 4 ⎠
a strut-mounted sphere. The drag force FD is measured directly with a force transducer, the
freestream velocity V is measured with a pitot-static probe, and the sphere diameter D is
measured with a micrometer. The table below gives nominal values of the measurement
variables and the water density p, as well as estimates for the bias and precision limits of
each variable at the 95% confidence level. The bias limits have been estimated based on
manufacturer’s specifications and previous experience with the instruments during
independent calibrations. The precision limits, on the other hand, have been determined from
multiple measurements of each variable, together with computation of the precision indices,
tS
S Xi , and PX = Xi
. Estimate the overall uncertainty in the reported drag coefficient
i
N
U CD at a confidence level of 95%.
The drag coefficient, CD =
Measured Variable, Xi
Drag force, FD
Water density, p
Freestream velocity, V
Sphere diameter, D
Nominal Value
0.5N
998 kg/m3
5m/s
10mm
Bias Limit, Wi
0.02N
0.2%
0.1m/s
0.1mm
Precision Limit, Pi
0.032N
--------0.18m/s
0.05mm
SOLUTION
Before beginning the details of the solution, discussion of the percentage uncertainty
listed for the water density p is in order. When using tabular or curve-fit reference values for
quantities such as material properties, it is important to remember that these are not true
values. Rather, they are “best estimates” based on experimental data that have uncertainties
associated with them. However, once a table or curve-fit equation has been chosen to
determine a property, the same value will be obtained for a given experimental condition no
matter how many times the table or equation is used. Thus, the precision limit associated
with a property value determined from a table or equation is zero. All of the uncertainty
in experimental property data is combined into a bias limit that is the best estimate of the
overall uncertainty in the data used to generate the table or equation. In the current case, the
95% bias limit estimate for p (expressed here as a percentage) is quite low, 0.2%, since the
density of water is well known and is also relatively insensitive to environmental factors such
as temperature variations.
Writing the expression for the data reduction equation and the uncertainty expression for
the bias limit W CD :
CD =
8
FD p −1V −2 D −2
π
⎛ WCD
⎜⎜
⎝ CD
2
2
2
2
⎞
⎞
⎞ ⎛ 1 ∂C D
⎞ ⎛ 1 ∂C D
⎞ ⎛ 1 ∂C D
⎛ 1 ∂C D
⎟⎟ = ⎜⎜
WFD ⎟⎟ + ⎜⎜
Wp ⎟⎟ + ⎜⎜
WV ⎟⎟ + ⎜⎜
WD ⎟⎟
⎠
⎠ ⎝ C D ∂D
⎠ ⎝ C D ∂V
⎠ ⎝ C D ∂p
⎝ C D ∂FD
⎠
2
Computing the derivatives:
⎛
⎞
⎜
⎟ 8
W
1 ∂C D
1
⎟⎛⎜ p −1V −2 D −2 ⎞⎟ WFD = FD
WFD = ⎜
C D ∂FD
FD
⎜ 8 F p −1V −2 D −2 ⎟⎝ π
⎠
⎜
⎟
D
⎝π
⎠
⎛
⎞
⎜
⎟ 8
− Wp
1 ∂C D
1
⎟⎛⎜ − FD p − 2 V − 2 D − 2 ⎞⎟Wp =
Wp = ⎜
p
C D ∂p
⎜ 8 F p −1V − 2 D − 2 ⎟⎝ π
⎠
⎜
⎟
D
⎝π
⎠
⎛
⎞
⎜
⎟ 16
− 2WV
1
1 ∂C D
⎟⎛⎜ − FD p −1V − 3D − 2 ⎞⎟ WV =
WV = ⎜
C D ∂V
V
⎜ 8 F p −1V − 2 D − 2 ⎟⎝ π
⎠
⎜
⎟
D
⎝π
⎠
⎛
⎞
⎜
⎟ 16
− 2WD
1 ∂C D
1
⎜
⎟⎛⎜ − FD p −1V − 2 D − 3 ⎞⎟ WD =
WD =
8
C D ∂D
D
⎜ F p −1V − 2 D − 2 ⎟⎝ π
⎠
⎜
⎟
D
⎝π
⎠
Substituting into the uncertainty expression for WC D :
2
2
2
2
2
⎛ WC D ⎞
⎛W ⎞
⎛W ⎞
W
W
⎜
⎟ = ⎜ FD ⎟ + ⎜ p ⎟ + 4⎛⎜ V ⎞⎟ + 4⎛⎜ D ⎞⎟
⎜ p ⎟
⎜ C ⎟
⎜ F ⎟
⎝ D ⎠
⎝ V ⎠
⎝
⎠
⎝ D ⎠
⎝ D ⎠
Thus, the bias limit WC D depends most strongly on WV and WD, i.e., the factors in the data
reduction equation with the largest exponents, which is always the case for reduction
expressions of “power law” form.
Substituting in numerical values:
⎛ WCD
⎜⎜
⎝ CD
⎛ WCD
⎜⎜
⎝ CD
2
⎞ ⎛ 0.02 ⎞
⎛ 0.1 ⎞
⎛ 0.1 ⎞
2
⎟⎟ = ⎜
⎟
⎟ + 4⎜
⎟ + (0.002) + 4⎜
⎝ 10 ⎠
⎝ 5 ⎠
⎠ ⎝ 0.5 ⎠
2
2
2
2
⎞
⎟⎟ = (1.6x10 −3 ) + ( 4x10 −6 ) + (1.6x10 −3 ) + ( 4x10 −4 ) = 3.604x10 −3
⎠
⎛ WC D
∴ ⎜⎜
⎝ CD
⎞
⎟⎟ = (6.003x10 − 2 ) ≈ 6.0%
⎠
Note that the bias limit contributions from the drag force and velocity measurements are
equal and dominate the bias uncertainty for CD and that the bias limit on density contributes
negligibly.
Now considering the precision limits, the uncertainty expression for PCD is:
⎛ PC D
⎜
⎜C
⎝ D
2
⎞
⎛ 1 ∂C D
⎟ = ⎜⎜
PFD
⎟
C
F
∂
D
D
⎝
⎠
2
2
⎞
⎛ 1 ∂C D
⎞
⎛ 1 ∂C D
⎞
⎟⎟ + ⎜⎜
PV ⎟⎟ + ⎜⎜
PD ⎟⎟
⎠
⎝ C D ∂V
⎠
⎝ C D ∂D
⎠
2
The derivatives have already been carried out in considering the bias limit, so the following
result can be written immediately:
⎛ PC D
⎜
⎜C
⎝ D
2
⎞
⎛P
⎟ = ⎜ FD
⎟
⎜ F
⎠
⎝ D
2
2
2
⎞
P
P
⎟ + 4⎛⎜ V ⎞⎟ + 4⎛⎜ D ⎞⎟
⎟
⎝ D ⎠
⎝ V ⎠
⎠
Substituting numerical values:
⎛ PC D
⎜
⎜C
⎝ D
⎛ PC D
⎜
⎜C
⎝ D
2
2
2
2
⎞
0.032 ⎞
⎛ 0.05 ⎞
⎛ 0.18 ⎞
⎟ = ⎛⎜
⎟
⎟ + 4⎜
⎟ + 4⎜
⎟
⎝ 10 ⎠
⎝ 5 ⎠
⎝ 0.5 ⎠
⎠
2
⎞
⎟ = ( 4.096x10 − 3 ) + (5.184x10 − 3 ) + (1x10 − 4 ) = 9.38x10 − 3
⎟
⎠
⎛ PC
∴ ⎜⎜ D
⎝ CD
⎞
⎟ = (9.685x10 − 2 ) ≈ 9.7%
⎟
⎠
Combining the bias and precision limits by the RSS method:
U C D ⎡⎛ WC D
= ⎢⎜⎜
CD
⎢⎝ C D
⎣
UCD
CD
[(
= 3.604 x10
−3
)+ (
2
⎞
⎛P
⎟ + ⎜ CD
⎟
⎜C
⎠
⎝ D
)]
1
−3 2
9.38x10
1
2⎤2
⎞
⎟ ⎥
⎟ ⎥
⎠ ⎦
= 0.114 = 11.4%
Thus, under these conditions the total uncertainty in the drag coefficient is 11.4% at a 95%
confidence level. Since the nominal value of CD is
CD =
⎤
0.5
⎛ 8 ⎞⎡
= ⎜ ⎟⎢
⎥ = 0.510
π pV 2 D 2 ⎝ π ⎠ ⎢⎣ (998)(52 )(0.012 ) ⎥⎦
8
FD
This result can also be written as
CD=0.510 ± 0.058
at 95% confidence.