E1C05 09/14/2010
14:36:27 Page 179
Note that as a minimum, design-stage analysis includes only the effects found as a result of u 0
and u c . It is those in-between levels that allow measurement procedure and control effects to be
considered in the uncertainty analysis scheme. The Nth-order uncertainty estimate provides the
uncertainty value sought in advanced design stage or in single-measurement analyses. It is an
appropriate value to be used to report the results from single-measurement tests.
Example 5.4
As an exercise, obtain and examine a dial oven thermometer. How would you assess the zero- and
first-order uncertainty in the measurement of the temperature of a kitchen oven using the device?
KNOWN Dial thermometer
ASSUMPTION Negligible systematic error in the instrument
FIND Estimate u 0 and u 1 in oven temperature
SOLUTION The zero-order uncertainty would be that contributed by the resolution error of the
measurement system only. For example, most gauges of this type have a resolution of 10
C. So from
Equation 5.1 we estimate
u 0 ¼ 5
C
At the first order, the uncertainty would be affected by any time variation in the measured value
(temperature) and variations in the operating conditions. If we placed the thermometer in the center
of an oven, set the oven to some relevant temperature, allowed the oven to preheat to a steady
condition, and then proceeded to record the temperature indicated by the thermometer at random
intervals, we could estimate our ability to control the oven temperature. For J measurements, the
first-order uncertainty of the measurement in this oven’s mean temperature using this technique and
this instrument would be, from Equation 5.22,
u 1 ¼ t JÀ1;95 s T
COMMENT We could continue these verification tests. Let’s suppose we are interested in how
accurately we could set the oven mean temperature. Then the idea of setting the operating condition, the
temperature, becomes important. We could estimate our ability to repeatedly set the oven to a desired
mean temperature (time-averaged temperature). This could be done by changing the oven setting and
then resetting the thermostat back to the original operating setting. If we were to repeat this sequence M
times (reset thermostat, measure a set of data, reset thermostat, etc.), we could compute u 1 by
u 1 ¼ t MðJÀ1Þ;95 hs T i
Note that now the variation in oven temperature with time at any setting is included in the pooled
estimate. The effects of instrument calibration would enter at the Nth order through u c . The uncertainty
in oven temperature at some setting would be well approximated by the estimate from Equation 5.23:
u N ¼ u
2
1 þ u
2
c
À
Á 1=2
By inspection of u 0 , u c , and u i , where i ¼ 1; 2; . . . ; N À 1, single-measurement analysis provides
a capability to pinpoint those aspects of a test that contribute most to the overall uncertainty in the
measurement, as well as a reasonable estimate of the uncertainty in a single measurement.
5.7 Advanced-Stage Uncertainty Analysis 179
14:36:27 Page 179
Note that as a minimum, design-stage analysis includes only the effects found as a result of u 0
and u c . It is those in-between levels that allow measurement procedure and control effects to be
considered in the uncertainty analysis scheme. The Nth-order uncertainty estimate provides the
uncertainty value sought in advanced design stage or in single-measurement analyses. It is an
appropriate value to be used to report the results from single-measurement tests.
Example 5.4
As an exercise, obtain and examine a dial oven thermometer. How would you assess the zero- and
first-order uncertainty in the measurement of the temperature of a kitchen oven using the device?
KNOWN Dial thermometer
ASSUMPTION Negligible systematic error in the instrument
FIND Estimate u 0 and u 1 in oven temperature
SOLUTION The zero-order uncertainty would be that contributed by the resolution error of the
measurement system only. For example, most gauges of this type have a resolution of 10
C. So from
Equation 5.1 we estimate
u 0 ¼ 5
C
At the first order, the uncertainty would be affected by any time variation in the measured value
(temperature) and variations in the operating conditions. If we placed the thermometer in the center
of an oven, set the oven to some relevant temperature, allowed the oven to preheat to a steady
condition, and then proceeded to record the temperature indicated by the thermometer at random
intervals, we could estimate our ability to control the oven temperature. For J measurements, the
first-order uncertainty of the measurement in this oven’s mean temperature using this technique and
this instrument would be, from Equation 5.22,
u 1 ¼ t JÀ1;95 s T
COMMENT We could continue these verification tests. Let’s suppose we are interested in how
accurately we could set the oven mean temperature. Then the idea of setting the operating condition, the
temperature, becomes important. We could estimate our ability to repeatedly set the oven to a desired
mean temperature (time-averaged temperature). This could be done by changing the oven setting and
then resetting the thermostat back to the original operating setting. If we were to repeat this sequence M
times (reset thermostat, measure a set of data, reset thermostat, etc.), we could compute u 1 by
u 1 ¼ t MðJÀ1Þ;95 hs T i
Note that now the variation in oven temperature with time at any setting is included in the pooled
estimate. The effects of instrument calibration would enter at the Nth order through u c . The uncertainty
in oven temperature at some setting would be well approximated by the estimate from Equation 5.23:
u N ¼ u
2
1 þ u
2
c
À
Á 1=2
By inspection of u 0 , u c , and u i , where i ¼ 1; 2; . . . ; N À 1, single-measurement analysis provides
a capability to pinpoint those aspects of a test that contribute most to the overall uncertainty in the
measurement, as well as a reasonable estimate of the uncertainty in a single measurement.
5.7 Advanced-Stage Uncertainty Analysis 179