E1C05 09/14/2010
14:36:26 Page 171
But let us think about this. The insidious aspect of systematic error has been revealed. Why
doubt a measurement indication and suspect a systematic error? The mean value of the data set may
be offset from some true value that we do not know. Figuratively speaking, there will be no shoe
heels staring at us. Experience teaches us to think through each measurement carefully because
systematic error is always present at some magnitude. We see that it is difficult to estimate
systematic error without comparison, so a good design should include some means to estimate it.
Various methodologies can be utilized: (1) calibration, (2) concomitant methodology, (3) interlaboratory comparisons, or (4) judgment/experience. When available, calibration using a suitable
standard and method can reduce instrument systematic error to predictable intervals and estimate
its associated uncertainty. A quality instrument may come with a certified calibration certificate.
Concomitant methodology, which is using different methods of estimating the same thing, allows
for comparing the results. Concomitant methods that depend on different physical measurement
principles are preferable, as are methods that rely on calibrations that are independent of each other.
In this regard, analytical methods could be used for comparison
3 or at least to estimate the range of
systematic error due to influential sources such as environmental conditions, instrument response
errors, and loading errors. Lastly, an elaborate but good approach is through interlaboratory
comparisons of similar measurements, an excellent replication method. This approach introduces
different instruments, facilities, and personnel into an otherwise similar measurement procedure.
The variations in the results between facilities provide a statistical estimate of the systematic
uncertainty (2).
In lieu of the above, a judgment value based on past experience may have to be assigned; these
values are usually understood to be made at the 95% confidence level. For example, the value that
first came to mind to you in the bathroom scale example above likely covered a 95% interval.
Note that calibration cannot eliminate systematic error, but it may reduce uncertainty. Consider
the calibration of a temperature transducer against a National Institute of Standards and Technology
(NIST) standard certified to be correct to within 0.01
C. If the calibration data show that the
transducer output has a systematic offset of 0.2
C relative to the standard, then we would just correct
all the data obtained with this transducer by 0.2
C. Simple enough, we correct it! But the standard
itself still has an intrinsic systematic uncertainty of 0.01
C, and this uncertainty remains in the
calibrated transducer. We would include any uncertainty in the correction value applied.
Random Error
When repeated measurements are made under fixed operating conditions, random errors manifest
themselves as scatter of the measured data. Random error
4 is introduced through the repeatability
and resolution of the measurement system components, calibration, and measurement procedure
and technique; by the measured variable’s own temporal and spatial variations; and by the variations
in the process operating and environmental conditions from which measurements are taken.
The estimate of the probable range of a random error is given by its random uncertainty. The
random standard uncertainty,s x , is defined by the interval given by Æs x , where
s x ¼ s x =
ffiffiffiffi
N
p
ð5:5Þ
3 Smith and Wenhofer (3) provide examples for determining jet engine thrust, and several complementary measurements are
used with an energy balance to estimate the uncertainty assigned to the systematic error.
4 This error was called a ‘‘precision’’ error in engineering documents prior to the 1990s.
5.5 Systematic and Random Errors 171
14:36:26 Page 171
But let us think about this. The insidious aspect of systematic error has been revealed. Why
doubt a measurement indication and suspect a systematic error? The mean value of the data set may
be offset from some true value that we do not know. Figuratively speaking, there will be no shoe
heels staring at us. Experience teaches us to think through each measurement carefully because
systematic error is always present at some magnitude. We see that it is difficult to estimate
systematic error without comparison, so a good design should include some means to estimate it.
Various methodologies can be utilized: (1) calibration, (2) concomitant methodology, (3) interlaboratory comparisons, or (4) judgment/experience. When available, calibration using a suitable
standard and method can reduce instrument systematic error to predictable intervals and estimate
its associated uncertainty. A quality instrument may come with a certified calibration certificate.
Concomitant methodology, which is using different methods of estimating the same thing, allows
for comparing the results. Concomitant methods that depend on different physical measurement
principles are preferable, as are methods that rely on calibrations that are independent of each other.
In this regard, analytical methods could be used for comparison
3 or at least to estimate the range of
systematic error due to influential sources such as environmental conditions, instrument response
errors, and loading errors. Lastly, an elaborate but good approach is through interlaboratory
comparisons of similar measurements, an excellent replication method. This approach introduces
different instruments, facilities, and personnel into an otherwise similar measurement procedure.
The variations in the results between facilities provide a statistical estimate of the systematic
uncertainty (2).
In lieu of the above, a judgment value based on past experience may have to be assigned; these
values are usually understood to be made at the 95% confidence level. For example, the value that
first came to mind to you in the bathroom scale example above likely covered a 95% interval.
Note that calibration cannot eliminate systematic error, but it may reduce uncertainty. Consider
the calibration of a temperature transducer against a National Institute of Standards and Technology
(NIST) standard certified to be correct to within 0.01
C. If the calibration data show that the
transducer output has a systematic offset of 0.2
C relative to the standard, then we would just correct
all the data obtained with this transducer by 0.2
C. Simple enough, we correct it! But the standard
itself still has an intrinsic systematic uncertainty of 0.01
C, and this uncertainty remains in the
calibrated transducer. We would include any uncertainty in the correction value applied.
Random Error
When repeated measurements are made under fixed operating conditions, random errors manifest
themselves as scatter of the measured data. Random error
4 is introduced through the repeatability
and resolution of the measurement system components, calibration, and measurement procedure
and technique; by the measured variable’s own temporal and spatial variations; and by the variations
in the process operating and environmental conditions from which measurements are taken.
The estimate of the probable range of a random error is given by its random uncertainty. The
random standard uncertainty,s x , is defined by the interval given by Æs x , where
s x ¼ s x =
ffiffiffiffi
N
p
ð5:5Þ
3 Smith and Wenhofer (3) provide examples for determining jet engine thrust, and several complementary measurements are
used with an energy balance to estimate the uncertainty assigned to the systematic error.
4 This error was called a ‘‘precision’’ error in engineering documents prior to the 1990s.
5.5 Systematic and Random Errors 171