Uncertainty analysis is a vital part of any experimental program or
measurement system design. Common sources of experimental
uncertainty were defined in Chapters 2 through 4. In this chapter,
methods are provided to combine uncertainties from all
sources to estimate the uncertainty of the final results of an
experiment.
INTRODUCTION
Any experimental measurement will involve some level of
uncertainty that may originate from causes such as inaccuracy in
measurement equipment, random variation in the quantities
measured, and approximations in data-reduction relations. All
these uncertainties in individual measurements eventually
translate into uncertainty in the final results. This "propagation of
uncertainty" is an important aspect of any engineering
experiment. Uncertainty analysis is performed during the design
stage of an experiment to assist in the selection of measurement
techniques and devices. It is also performed after data have been
gathered, in order to demonstrate the validity of the
results.
we can estimate the maximum uncertainty in R by forcing all terms
on the right-hand side of Eq to be positive.
In mathematical form, this becomes
(7.3)
It is not very probable that all terms in Eq. (7.2) will become positive simultaneously
or that errors in the individual x's will all be at the extreme value of the uncertainty
interval. Consequently, Eq. (7.3) will produce an unreasonably high estimate for up.
A better estimate for the uncertainty is given by
Comment: The maximum
uncertainty of 40 W is 4% of
the power (P = VI = 100 x 10
=1000 W), whereas the
uncertainty estimate of 28.3
W is 2.8% of the power. The
maximum error
estimate is too high in most
circumstances.
Solution: The relationship between the pressure
and the measured column height for a manometer is:
So, The maximum
allowable uncertainty in
the gage fluid density:
CONSIDERATION OF SYSTEMATIC AND RANDOM COMPONENTS OF UNCERTAINTY
In the early phases of the design of an experiment, it is often not practical to separate
the effects of systematic and random uncertainties. For example, manufacturer-specified
instrument accuracies inherently combine random and systematic uncertainties.
The Eq.
will then be used with these combined uncertainties in measured variables to
estimate the uncertainty of the result.
In more detailed uncertainty analyses, it is usually desirable to keep separate track of the
systematic uncertainty, denoted B, and the random uncertainty, denoted P, associated with the
measurement. Estimating random uncertainty depends on the sample size. A systematic error
does not vary during repeated readings and is independent of the sample size. It is this
distinction that makes it desirable to handle the terms separately in uncertainty analysis.
The random uncertainty is estimated using the t-distribution, introduced in
Chapter 6. If the variable x is measured n times, then the standard deviation and mean of the
sample can be determined from
In case the mean value is obtained from a different set of tests, the
final value of the mean of x, denoted by
is determined from M
measurements of x and is given by
In this case, M = 1, random uncertainty in this single measurement is
given by
Usually, S, is determined in tests with a large sample size (n > 30). In
addition, a confidence level of 95% is commonly used to determine the
random uncertainty interval.
For these two conditions, the appropriate value of t is 2.0. A large sample
size simplifies an uncertainty analysis considerably, and for this reason, the
ASME Standard [ASME, (1998)] suggests the use of large sample sizes in
most uncertainty analyses.
the systematic uncertainty in the measured variable x, remains constant
if the test is repeated under the same conditions. Systematic uncertainties
include those errors which are known, but have not been eliminated through
calibration, as well as other fixed errors that can be estimated, but not
eliminated from the measurement process.
Systematic and random uncertainties are combined to obtain the total
uncertainty, using RSS. For the mean of .
for a single measurement of x,
Example 7.3
In a chemical-manufacturing plant, load cells are used to measure the mass of a
chemical mixture during a batch process. From 10 measurements, the average of the
mass is measured to be 750 kg.
From a large number of previous measurements, it is known that the standard deviation
of the measurements is 15 kg (which implies that t : 2.0 for 95% confidence level).
Assuming that the load cells do not introduce any random uncertainty into the
measurement, calculate for 95% confidence level
(a) the standard deviation and random uncertainty of each measurement.
(b) the standard deviation and random uncertainty of the mean value of the ten
measurements.
Example 7.4
In estimating the heating value of natural gas from a gas field, 10 samples are
taken and the heating value of each sample is measured by a calorimeter. The
measured values of the heating value, in kJ/kg, are as follows:
48530, 48980, 50210, 49860, 48560, 49540, 49270, 48850, 49320, 48680
Assuming that the calorimeter itself does not introduce any random uncertainty
into the measurement, calculate (for 95% confidence level)
(a) the random uncertainty of each measurement.
(b) the random uncertainty of the mean of the measurements.
(c) the random uncertainty of the mean of the measurements, assuming that S
was calculated on the basis of a large sample (n > 30); but has the same value
as computed inparts (a) and (b).
Example 7.5
The manufacturer's specification for the calorimeter in Example 7.4 states that
the device has an accuracy of l.5% full range from 0 to 100,000 kJ/kg. Estimate
the total uncertainty of
(a) the mean value of the measurement in Example 7.4 and
(b) a single measurement of heating value of 49,500 kJ/kg, obtained later than
the measurements in Example 7.4.
95% confidence
assumed)
We have assumed that the "accuracy" is entirely a systematic uncertainty.
(a)
which is 3.1% of the mean value.
Problem
error type
hysteresis
Lineariz.error
Resolution error
zero off set
repeatability
 0.1C
0.2% of reading
0.10C
0.1C
0.1C
systematic
systematic
random
systematic
random
Assuming that the random errors have been determined with
samples>30,