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247Chapter 15 Quality Assurance
. .
R
3 77
10
1 67
2.00 ppm
ppm ppm
100 5%#=
-
=
 Because this recovery is within the limit of ±10%, the laboratory 
spike’s recovery is acceptable, suggesting a time-dependent change in 
the analyte’s concentration.
7. he mean and the standard deviation for the 25 samples are 34.01 
ppm and 1.828 ppm, respectively, which gives us the following warn-
ing limits and control limits
. ( ) ( . ) .UCL 34 01 3 1 828 39 5= + =
. ( ) ( . ) .U LW 34 01 1 828 37 72= + =
. ( ) ( . ) .LLW 34 01 1 828 30 42= - =
. ( ) ( . ) .L LC 34 01 1 8283 28 5= - =
 Figure SM15.1 shows the property control chart. Note that the high-
lighted region contains 14 consecutive cycles (15 samples) in which 
the results oscillate up and down, indicating that the system is not in 
a state of statistical control.
8. he mean and the standard deviation for the 25 samples are 99.84% 
and 14.08%, respectively, which gives us the following warning limits 
and control limits
( ) ( ). . .UCL 399 84 14 08 142 1= + =
. ( ) ( . ) .UWL 99 84 14 082 128 0= + =
. ( ) ( . ) .LWL 99 84 14 082 71 7= =-
. ( ) ( . ) .LCL 99 84 3 14 08 57 6= - =
 Figure SM15.2 shows the property control chart, which has no fea-
tures to suggest that the system is not in a state of statistical control.
9. he 25 range values are 4, 1, 3, 3, 2, 0, 2, 4, 3, 1, 4, 1, 2, 0, 2, 4, 3, 
4, 1, 1, 2, 1, 2, 3, 3, with a mean of 2.24. he control and warning 
limits, therefore, are
( . ) ( . ) .UCL 3 267 2 24 7 3= =
( ) ( . ). .UWL 2 242 512 5 6= =
 Figure SM15.2 shows the precision control chart, which has no fea-
tures to suggest that the system is not in a state of statistical control.
Figure SM15.1 Property control chart for 
the data in Problem 7. he highlighted re-
gion shows 14 consecutive cycles in which 
the results oscillate up and down, a sign that 
system is not in a state of statistical control.
Figure SM15.2 Property control chart for 
the data in Problem 8. 
Figure SM15.3 Precision control chart for 
the data in Problem 9. 
5 10 15 20 25
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