E1C05 09/14/2010
14:36:28 Page 187
Example 5.11
The measurement of x contains three elemental random errors from data acquisition. Each
elemental error is assigned a random standard uncertainty as follows:
s x
ð Þ 1 ¼ 0:60 units; n 1 ¼ 29
s x
ð Þ 2 ¼ 0:80 units; n 2 ¼ 9
s x
ð Þ 3 ¼ 1:10 units; n 3 ¼ 19
Assign the random standard uncertainty due to these data-acquisition errors.
KNOWN s x
ð Þ k with k ¼ 1; 2; 3
FIND s x
SOLUTION The random standard uncertainty due to data-acquisition errors is
s x ¼ s x
ð Þ
2
1 þ s x
ð Þ
2
2 þ s x
ð Þ
2
3
h
i 1=2 ¼ 0:60
2
þ 0:80
2
þ 1:10
2
À
Á 1=2 ¼ 1:49 units
at one standard deviation confidence and with degrees of freedom
v ¼
X 3
k¼1
s x
ð Þ
2
k
! 2
X 3
k¼1
s x
ð Þ
4
k =v k
¼
0:6
2
þ 0:8
2
þ 1:1
2
À
Á 2
0:6
4
=29
À
Á þ 0:8
4
=9
À
Á þ 1:1
4
=19
À
Á¼ 38:8 % 39
Example 5.12
In reporting the results of an experiment to measure stress in a loaded beam, an uncertainty analysis
identifies three elemental errors in the stress measurement having the following values of
uncertainty:
b s
ð Þ 1 ¼ 0:5 N/cm
2
b s
ð Þ 2 ¼ 1:05 N/cm
2
b s
ð Þ 3 ¼ 0 N/cm
2
s s
ð Þ 1 ¼ 4:6 N/cm
2
; n 1 ¼ 14
s s
ð Þ 2 ¼ 10:3 N/cm
2
; n 2 ¼ 37
s s
ð Þ 3 ¼ 1:2 N/cm
2
; n 3 ¼ 8
where the degrees of freedom in the systematic uncertainies are very large. If the mean value of the
stress in the measurement is s ¼ 223:4 N/cm
2 , determine the best estimate of the stress at a 95%
confidence level, assuming all errors are accounted for.
KNOWN Experimental errors with assigned uncertainties
ASSUMPTIONS All elemental errors (K ¼ 3) have been included.
FIND s Æ u s (95%)
SOLUTION We seek values for the statement, s
0
¼ s Æ u s 95%
ð
Þ, given that s ¼ 223:4 N/cm
2 .
The measurement random standard uncertainty is
s s ¼ s s
ð Þ
2
1 þ s s
ð Þ
2
2 þ s s
ð Þ
2
3
h
i 1=2 ¼ 11:3 N/cm
2
5.8 Multiple-Measurement Uncertainty Analysis 187