6
CHAPTER
SUPPLEMENT
Statistical Process
Control
DISCUSSION QUESTIONS
1.
The Central Limit Theorem provides the basis for the calculation of required sample size.
2.
The ultimate goal of the X and R-charts is to ascertain, by a sampling procedure, that the relevant
parameter is kept within specific upper and lower bounds. The X bar chart alone tells us only that the
average or variable values are within the appropriate limits. The combination of X and the R-charts
allows one to determine that both the average and the deviations are within the limits.
3.
X -charts: depict the variation in average value of a variable (weight or diameter, for example)
R-charts: depict the average range or deviation of a variable
p-charts: depict the average value of an attribute (percent defective, for example)
c-charts: depict the number of times an attribute (defect, for example) occurs
4.
A process can be out of control because of assignable variation, which can be traced to specific
causes. Examples include such factors as:

Tool wear

A change in raw materials

A change in working environment (temperature or humidity, for example)

Tired or poorly trained labor
5.
Walter Shewhart brought his knowledge of statistics to bear on the problems of statistical sampling
and quality control, providing the foundations of statistical quality control as we know it.
6.
Natural variations are those variations that are inherent in the process and for which there is no
identifiable cause. These variations fall in a natural pattern.
Assignable causes are variations beyond those that can be expected to occur because of natural
variation. These variations can be traced to a specific cause.
7.
Occurrences (items) can fall outside of the expected range of the process and still be in control
because we expect that to happen in the tails of the distribution.
Also, patterns such as trends, too many points on one side of the mean, or strong fluctuations in
readings can cause a process to be out of control.
8.
Producer’s risk: the risk of rejecting a good lot
Consumer’s risk: the risk of accepting a defective lot
9.
Type I error: when one rejects a hypothesis that is in fact true (i.e., conclude that a batch is
unacceptable when it is not)
Type II error: when one accepts a hypothesis that is in fact false (i.e., conclude that a batch is
acceptable when it is not)
10.
Cpk is the way we express process capability. It measures the difference between desired and an
actual dimension of products made in a process.
A Cpk of 1.0 means that the process variation is centered within the desired upper and lower
specifications.
Chapter 6 Supplement: Statistical Process Control
87
11.
12.
A “run of 5” implies that assignable variation is present.
The 5 steps used to establish X and R charts are:
Collect 20–25 samples of n  4 or n  5 from a stable process.

Compute X and R , set limits (usually at 3 sigma). If the current process is not stable, use a
desired mean instead of X .

Graph the sample means and ranges and see if they fall outside the limits.

Look for points or patterns indicating the process is out of control.

Collect more samples, and revalidate limits if needed.

13.
The desired mean is used when the mean of a process being observed is unknown or out of control.
14.
A run test is used to help spot abnormalities in a control chart process. It is used if points are not
individually out of control, but form a pattern above or below the nominal line.
15.
Managerial issues include:
Selecting places in their process that need SPC
Deciding which type of control charts best fit
Setting rules for workers to follow if certain points or patterns emerge



END-OF-SUPPLEMENT PROBLEMS
. , z3
S6.1 n  10 , X  75 ,   195
195
.
F
H10 IK 76.85
195
.
LCL  75  3F I  7315
H K .
UCL  75  3
10
S6.2 n  5 , X  50 ,   0.18 , z  3
1.72
F
H5 IK 52.30
1.72
LCL  50  3F I  47.70
HK
UCL  50  3
5
. , D3  0
S6.3 n  5 . From Table S6.1, A2  0.577 , D4  2115
(a)
UCL  X  A  R  50  0.577  4  52.308
X
(b)
2
LCLX  X  A2  R  50  0.577  4  47.692
UCLR  D4  R  2.115  4  8.456
LCLR  D3  R  0  4  0
S6.4 n  6 . From Table S6.1, A2  0.483 , D4  2.004 , D3  0
UCLX  X  A2  R  46  0.483  2  46.966
LCLX  X  A2  R  46  0.483  2  45.034
UCLR  D4  R  2.004  2  4.008
LCLR  D3  R  0  2  0
88
Instructor’s Solutions Manual t/a Operations Management
S6.5 n  10 . From Table S6.1, A2 = 0.308, D4  1777
, D3  0.233
.
UCLX  X  A2  R  60  0.308  3  60.924
LCLX  X  A2  R  60  0.308  3  59.076
UCLR  D4  R  1.777  3  5.331
LCLR  D3  R  0.223  3  0.669
S6.6
Hour
1
2
3
4
5
6
7
8
X
3.25
3.10
3.22
3.39
3.07
2.86
3.05
2.65
Hour
9
10
11
12
13
14
15
16
R
0.71
1.18
1.43
1.26
1.17
0.32
0.53
1.13
X
3.02
2.85
2.83
2.97
3.11
2.83
3.12
2.84
R
0.71
1.33
1.17
0.40
0.85
1.31
1.06
0.50
Hour
17
18
19
20
21
22
23
24
X
2.86
2.74
3.41
2.89
2.65
3.28
2.94
2.64
R
1.43
1.29
1.61
1.09
1.08
0.46
1.58
0.97
Average X  2.982, Average R = 1.02375, n  4 . From Table S6.1, A2  0.729 , D4  2.282 ,
D3  0.0 .
UCLX  X  A2  R  2.982  0.729  1.024  3.728
LCLX  X  A2  R  2.982  0.729  1.024  2.236
UCLR  D4  R  2.282  1.024  2.336
LCLR  D3  R  0  1.024  0
The smallest sample mean is 2.64, the largest 3.39. Both are well within the control limits. Similarly,
the largest sample range is 1.61, also well within the control limits. We can conclude that the process
is presently within control. However, the first five values for the mean are above the expected mean;
this may be the indication of a problem in the early stages of the process.
Control Chart X
4.00
3.50
LCL
UCL
3.00
2.50
2.00
0
5
10
15
20
25
Sample
Chapter 6 Supplement: Statistical Process Control
89
Control Chart R
2.50
2.00
1.50
LCL
UCL
1.00
0.50
0.00
0
5
10
15
20
R
1.3
1.8
1.0
1.8
1.5
1.7
1.4
1.1
1.8
Sample
19
20
21
22
23
24
25
25
Sample
S6.7
Sample
1
2
3
4
5
6
7
8
9
X
63.5
63.6
63.7
63.9
63.4
63.0
63.2
63.3
63.7
R
2.0
1.0
1.7
0.9
1.2
1.6
1.8
1.3
1.6
Sample
10
11
12
13
14
15
16
17
18
X
63.5
63.3
63.2
63.6
63.3
63.4
63.4
63.5
63.6
X
63.8
63.5
63.9
63.2
63.3
64.0
63.4
R
1.3
1.6
1.0
1.8
1.7
2.0
1.5
X  63.49 , R  1.5 , n  4 . From Table S6.1, A2  0.729 , D4  2.282 , D3  0.0 .
UCLX  X  A2  R  63.49  0.729  1.5  64.58
LCLX  X  A2  R  63.49  0.729  1.5  62.40
UCLR  D4  R  2.282  1.5  3.423
LCLR  D3  R  0  1.5  0
The process is in control.
Control Chart X
65.00
64.00
63.00
LCL
UCL
62.00
61.00
60.00
0
90
5
10
15
Sample
20
25
30
Instructor’s Solutions Manual t/a Operations Management
Control Chart R
4.00
3.00
LCL
UCL
2.00
1.00
0.00
0
S6.8
Time
9 AM
10 AM
11 AM
12 PM
1 PM
5
10
Box 1
9.8
10.1
9.9
9.7
9.7
15
Sample
Box 2
10.4
10.2
10.5
9.8
10.1
Box 3
9.9
9.9
10.3
10.3
9.9
20
25
Box 4
10.3
9.8
10.1
10.2
9.9
Average
30
Average
10.10
10.00
10.20
10.00
9.90
10.04
Range
0.60
0.40
0.60
0.60
0.40
0.52
n  4 . From Table S6.1, A2  0.729 , D4  2.282 , D3  0.0 .
UCLX  X  A2  R  10.04  0.729  0.52  10.42
LCLX  X  A2  R  10.04  0.729  0.52  9.66
UCLR  D4  R  2.282  0.52  1187
.
LCLR  D3  R  0  0.52  0
The smallest sample mean is 9.9, the largest 10.2. Both are well within the control limits. Similarly,
the largest sample range is 0.6, also well within the control limits. Hence, we can conclude that the
process is presently within control.
One step the QC department might take would be to increase the sample size to provide a clearer
indication as to both control limits and whether or not the process is in control.
S6.9
X  19.90 , R  0.34 , n  4 , A2  0.729 , D4  2.282
UCL  19.90  0.7290.34  2015
.
(a)


LCL  19.90  0.729 0.34  19.65
UCL  2.282 0.34   0.78
(b)
LCL  0
(c)
The ranges are ok; the means are not in control.
.
S6.10 X  10 , R  33
.
(a)
standard deviation = 1.36,  x  136
Chapter 6 Supplement: Statistical Process Control
5  0.61
91
(b)
Using  x
UCL  10  30.61  1183
.
LCL  10  3 0.61  817
.
Using A2  0.577
UCL  10  3.30.577  1190
.
LCL  10  3.30.577  810
.
(c)
UCL  2115
. 3.3  6.98
(d)
LCL  0 3.3  0
Yes, both mean and range charts indicate process is in control.
S6.11 RDesired  35
. , XDesired  50 , n  6
UCLX  X  A2  R  50  0.483  3.5  5169
.
LCLX  X  A2  R  50  0.483  3.5  48.31
UCLR  D3  R  2.004  3.5  7.014
LCLR  D4  R  0  3.5  0
The smallest sample range is 1, the largest 6. Both are well within the control limits.
The smallest average is 47, the largest 57. Both are outside the proper control limits.
Therefore, although the range is within limits, the average is outside limits, and apparently
increasing. Immediate action is needed to correct the problem and get the average within the control
limits again.
S6.12 Sample Number Sample Range
1
1.10
2
1.31
3
0.91
4
1.10
5
1.21
6
0.82
7
0.86
8
1.11
9
1.12
10
0.99
11
0.86
12
1.20
Sample Mean
46
45
46
47
48
47
50
49
51
52
50
52
, X  48.583 , n  10 . Use X  47 , R  10
R  1049
.
. , n  10
UCLX  X  A2  R  47  0.308  1  47.308
LCLX  X  A2  R  47  0.308  1  46.692
UCLR  D4  R  1.777  1  1.777
LCLR  D3  R  0.223  1  0.223
The smallest sample range is 0.82, the largest 1.31. Both are well within the control limits.
Almost all the averages are outside the control limits. Therefore the process is out of control.
92
Instructor’s Solutions Manual t/a Operations Management
While the range is within limits, the average is outside limits, and apparently increasing.
Immediate action is needed to correct the problem and get the average within the control limits again.
Control Chart X
54.00
52.00
50.00
LCL
UCL
48.00
46.00
44.00
0
2
4
6
8
Sample
10
12
14
Control Chart R
2.00
1.50
LCL
UCL
1.00
0.50
0.00
0
2
4
6
8
Sample
10
12
14
0.51
0.505 drill bit (largest)
S6.13
0.495 drill bit (smallest)
0.49
0.505  0.49  0.015 , 0.015 0.00017  88 holes within standard
0.495  0.49  0.005 , 0.005 0.00017  29 holes within standard
Any one drill bit should produce at least 29 holes that meet tolerance, but no more than 88 holes
before being replaced.
S6.14 UCLp  p  3
p1  p
n
LCLp  p  3
p1  p
n
Chapter 6 Supplement: Statistical Process Control
93
n = 100
p1  p n
1 p
Percent Defective (p)
0.02
0.04
0.06
0.08
0.10
S6.15 UCLp  p  3
p1  p
n
LCLp  p  3
p1  p
n
LCLp
UCLp
0.0
0.0
0.0
0.0
0.01
0.062
0.100
0.132
0.161
0.190
n = 200
p1  p n
1 p
LCLp
UCLp
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.90
0.0
0.0
0.0
0.0
0.0038
0.0096
0.0160
0.0224
0.0294
0.0364
0.0310
0.0497
0.0663
0.0817
0.0962
0.1104
0.1240
0.1376
0.1506
0.1636
0.98
0.96
0.94
0.92
0.90
Percent Defective (p)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.014
0.020
0.024
0.027
0.030
0.0070
0.0099
0.0121
0.0139
0.0154
0.0168
0.0180
0.0192
0.0202
0.0212
Control Limits for Percent Defective
0.2
0.15
LCL
UCL
0.1
0.05
0.00
0
0.02
S6.16 UCLp  p  3
p1  p
n
LCLp  p  3
p1  p
n
94
0.04
0.06
Percent defective
UCLp  0.015  3
0.015  0.985
 0.0313
500
LCLp  0.015  3
0.015  0.985
 0.0013 or zero
500
0.08
0.1
Instructor’s Solutions Manual t/a Operations Management
S6.17 UCLp  p  3
p1  p
n
LCLp  p  3
p1  p
n
S6.18
UCLp  0.035  3
0.035  0.965
 0.0597
500
LCLp  0.035  3
0.035  0.965
 0.0103
500
0.035 0.965
 0.0184
100
UCL  p  3 p  0.035  3 0.0184   0.0901
p 
LCL  p  3 p  0.035  3 0.0184   0.0201  0
Increased control limits by more than 50%. No, sample size should not be changed, as quality will be
seriously affected.
S6.19 UCLp  p  3
p1  p 
n
LCLp  p  3
p1  p 
n
UCLp  0.011  3
0.011  0.989
 0.0209
1000
LCLp  0.011  3
0.011  0.989
 0.0011
1000
S6.20 n  200 , p  50 10200  0.025
UCLp  p  3
p1  p 
n
LCLp  p  3
p1  p 
n
UCLp  0.025  3
0.025  0.975
 0.0581
200
LCLp  0.025  3
0.025  0.975
 0.0081 or zero
200
The highest percent defective is .04; therefore the process is in control.
S6.21
Day
1
2
3
4
5
6
7
Number
Defective
6
5
6
4
3
4
5
Chapter 6 Supplement: Statistical Process Control
Day
8
9
10
11
12
13
14
Number
Defective
3
6
3
7
5
4
3
Day
15
16
17
18
19
20
21
Number
Defective
4
5
6
5
4
3
7
95
p
 pi 

N
98
 0.0467
21100 
p1  p

n
0.0467  0.9533
 0.0211
100
For a 3 p-chart, the upper control level is given by:
UCL  p  3  0.0467  3  0.0211  0.11
LCL  0
p Chart
0.20
0.15
LCL
UCL
0.10
0.05
0.00
0
5
10
15
20
25
Sample
The process is in control.
S6.22 Average blemishes/table  2000 100  20 . Using a normal approximation to the Poisson distribution:
c  20
  20  4.472
UCLc  c  3  20  3  4.472  33.4 or 33 blemishes
LCLc  c  3  20  3  4.472  6.6 or 7 blemishes
Yes—42 blemishes is considerably above the upper control limit.
S6.23
c 6
UCL  c  3 c  6  3 6  13.35
LCL  c  3 c  6  3 6  135
. or 0
Ten complaints are within the control limits, so this many complaints would not be considered
unusual.
8135
.
 8.00
L
M


3
0
N .04 ,
O
L
P
M
QN
O
P
Q
8.00  7.865
0135
.
0135
.
. .
. Therefore Cpk  1125
or
 1125
. ,
 1125
.
3 0.04
012
.
012
.
The process is centered and will produce within the specified tolerance.
S6.24 Cpk  min of
96
Instructor’s Solutions Manual t/a Operations Management
S6.25 Cpk  min of
41
. 4
L
M

3
N01.  ,
4  3.9
3 01
.
01
.
O
or L  0.33,
P
M
N
0
.
Q 3
O
P
Q
01
.
 0.33 . Therefore Cpk  0.33 . The process will
0.3
not produce within the specified tolerance.
S6.26 Cpk  min of
S6.27
16.5  16 16  15.5 O L
0.5 0.5
L
M
M3 , 3 O
P. Therefore C
Q
N31 , 31 P
Qor N
Time
9 AM
10 AM
11 AM
12 PM
1 PM
Box 1
9.8
10.1
9.9
9.7
9.7
Box 2
10.4
10.2
10.5
9.8
10.1
Box 3
9.9
9.9
10.3
10.3
9.9
pk
 0.166 .
Box 4
Average
10.3
10.1
9.8
10.0
10.1
10.2
10.2
10.0
9.9
9.9
Average =
10.04
Std. Dev. =
0.11
101
.  10
10  9.9
 0.3 and
 0.3
3 011
3 011
. 
. 
As 0.3 is less than 1, the process will not produce within the specified tolerance. This makes for an
interesting discussion of the control chart for problem S6.8.
S6.28
X
61.13136
49.776
38.42064
Upper Control Limit
Center Line (ave)
Lower Control Limit
Hour
26
27
28
29
30
(a)
(b)
1
48
45
63
47
45
2
52
53
49
70
38
Range
41.6232
19.68
0.00
Recent Data
Sample
3
4
39
57
48
46
50
45
45
52
46
54
5
61
66
53
61
52
X
51.4
51.6
52.0
57.0
47.0
R
22
21
18
25
16
Yes, the process appears to be under control. Samples 26–30 stayed within the boundaries of
the upper and lower control limits for both X and R charts.
The observed lifetimes have a mean of approximately 50 hours, which supports the claim
made by Ward Battery Corp. However, the variance from the mean needs to be controlled and
reduced. Lifetimes should deviate from the mean by no more than 5 hours (10% of the
variance).
CASE STUDIES
BAYFIELD MUD COMPANY
1.
The first thing that must be done is to develop quality control limits for the sample means. This can
be done as follows. Because the process appears to be unstable, we can use the desired mean as the
Chapter 6 Supplement: Statistical Process Control
97
nominal line. Desired x  50.0 , s  1.2 (from past results of Wet-Land
  s n  12
. 6  12
. 2.45  0.489 . At a 99.73% confidence interval Z  3 :
Drilling),
UCLX  X  3    50  3  0.489  50  1.467  51.47
LCLX  X  3  50  1.47  48.53
Now that we have appropriate control limits, these must be applied to the samples taken on the
individual shifts:
Time
6:00
7:00
8:00
9:00
10:00
11:00
12:00
1:00
Ave
49.6
50.2
50.6
50.8
49.9
50.3
48.6
49.0
Low
48.7
49.1
49.6
50.2
49.2
48.6
46.2
46.4
High
50.7
51.2
51.4
51.8
52.3
51.7
50.4
50.0
Day Shift*
Ave
Low
48.6
47.4
50.0
49.2
49.8
49.0
50.3
49.4
50.2
49.6
50.0
49.0
50.0
48.8
50.1
49.4
Time
2:00
3:00
4:00
5:00
6:00
7:00
8:00
9:00
Ave
49.0
49.8
50.3
51.4
51.6
51.8
51.0
50.5
Low
46.0
48.2
49.2
50.0
49.2
50.0
48.6
49.4
High
50.6
50.8
52.7
55.3
54.7
55.6
53.2
52.4
Evening Shift
Ave
Low
49.7
48.6
47.2
48.4
45.3
47.2
44.1
46.8
41.0
46.8
50.0
46.2
44.0
47.4
44.2
47.0
High
51.0
51.7
50.9
49.0
51.2
51.7
48.7
48.9
Ave
49.8
49.8
50.0
47.8
46.4
46.5
47.2
48.4
Low
48.4
48.8
49.1
45.2
44.0
44.4
46.6
47.2
High
51.0
50.8
50.6
51.2
49.7
50.0
48.9
49.5
Time
10:00
11:00
12:00
1:00
2:00
3:00
4:00
5:00
Ave
49.2
49.0
48.4
47.6
47.4
48.2
48.0
48.4
Low
46.1
46.3
45.4
44.3
44.1
45.2
45.5
47.1
High
50.7
50.8
50.2
49.7
49.6
49.0
49.1
49.6
Night Shift
Ave
Low
46.6
47.2
48.6
47.0
49.8
48.2
49.6
48.4
50.0
49.0
50.0
49.2
46.3
47.2
44.1
47.0
High
50.2
50.0
50.4
51.7
52.2
50.0
50.5
49.7
Ave
49.2
48.4
47.2
47.4
48.8
49.6
51.0
50.5
Low
48.1
47.0
46.4
46.8
47.2
49.0
50.5
50.0
High
50.7
50.8
49.2
49.0
51.4
50.6
51.5
51.9
*
(a)
High
52.0
52.2
52.4
51.7
51.8
52.3
52.4
53.6
Ave
48.4
48.8
49.6
50.0
51.0
50.4
50.0
48.9
Low
45.0
44.8
48.0
48.1
48.1
49.5
48.7
47.6
High
49.0
49.7
51.8
52.7
55.2
54.1
50.9
51.2
Bold-faced type indicates a sample outside the quality control limits.
Day shift (6:00 AM–2:00 PM):
Number of means within control limits 23

 96%
Total number of means
24
(b)
Evening shift (2:00 PM–10:00 PM):
Number of means within control limits 12

 50%
Total number of means
24
98
Instructor’s Solutions Manual t/a Operations Management
(c)
Night shift (10:00 PM–6:00 AM):
Number of means within control limits 12

 50%
Total number of means
24
As is now evident, none of the shifts meet the control specifications. Bag weight monitoring
needs improvement on all shifts. The problem is much more acute on the evening and night
shifts staffed by the more recent hires.
Note also, that the number of samples indicating a “short weight” is much greater than the
number indicating excess weight.
With regard to the range, 99.73% of the individual bag weights should lie within 3 of
the mean. This would represent a range of 6 , or 7.2. Only one of the ranges defined by the
difference between the highest and lowest bag weights in each sample exceeds this range. It
would appear, then, that the problem is not due to abnormal deviations between the highest
and lowest bag weights, but rather to poor adjustments of the bag weight-feeder causing
assignable variations in average bag weights.
The proper procedure is to establish mean and range charts to guide the bag packers. The
foreman would then be alerted when sample weights deviate from mean and range control
limits. The immediate problem, however, must be corrected by additional bag weight
monitoring and weight-feeder adjustments. Short-run declines in bag output may be necessary
to achieve acceptable bag weights.
Bayfield Case: Control Chart- X
54.0
LCL
UCL
52.0
50.0
48.0
Day
Eve
Night
46.0
44.0
1
3
5
7
9
11 13
Sample
15
17
19
21
23
SPC AT THE GAZETTE
1.
The overall fraction of errors (p) and the control limits are developed as follows:
p
s
Total number of errors
120

 0.04
Number of samples  Sample size 30  100
p1  p

n
0.04  0.96
 0.0196
100
Then the control limits are given (for a 95% confidence interval; 95% = 1.96) by:
UCL  p  196
. s  0.04  196
.  0.0196  0.0784
LCL  p  196
. s  0.04  196
.  0.0196  0.0016
Chapter 6 Supplement: Statistical Process Control
99
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Errors in
Sample
2
4
10
4
1
1
13
9
11
0
3
4
2
2
8
Fraction of
Errors (n/100)*
0.02
0.04
0.10
0.04
0.01
0.01
0.13**
0.09
0.11**
0.00
0.03
0.04
0.02
0.02
0.08
Errors in
Sample
2
3
7
3
2
3
7
4
3
2
2
0
1
3
4
Sample
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Fraction of
Errors (n/100)
0.02
0.03
0.07
0.03
0.02
0.03
0.07
0.04
0.03
0.02
0.02
0.00
0.01
0.03
0.04
* Bold-faced entries indicate sample fractions outside the quality control limits.
** Indicates sample fractions outside the industry standard quality control limits.
Both the table presented above, and the control chart indicate that the quality requirements of the
Gazette are more stringent than those of the industry as a whole. In five instances, the fraction of
errors exceeds the firm’s upper control limit; in two cases, the industry’s upper control limit is
exceeded. An investigation, leading to corrective action, is clearly warranted.
p Chart
0.14
0.12
0.10
UCL
0.08
0.06
0.04
LCL
0.02
0.00
0
5
10
15
20
Sample
Firm LCL
Firm UCL
25
30
Ind LCL
Ind UCL
INTERNET CASE STUDY
GREEN RIVER CHEMICAL CO.
This is a very straightforward case. Running software to analyze the data will generate the X -chart as
UCLX : 6113
.
Nominal: 49.78
LCLX : 38.42
100
Instructor’s Solutions Manual t/a Operations Management
and the range chart as
UCLR : 41.62
Nominal: 19.68
LCLR : 0.00
Next, students need to take the means and ranges for the five additional samples.
Date
April 6
7
8
9
10
Mean
52
57
47
51.4
51.6
Range
14
25
16
22
21
The mean and the ranges are all well within the control limits for this week. There is, however, a noticeable
change in the original data at time 13, where the range suddenly dropped. It then goes back up at time 16.
The data were generated by students in class, and changes in the process were made at the aforementioned
times. The control chart identifies that these changes took place.
Chapter 6 Supplement: Statistical Process Control
101