Section 4.4 Control Charts

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```					Section 4.4 Control Charts

These are the data from the 9:00 lecture from the paper cutting process (the “machine”
was supposed to cut strips 50 mm. in length). The data from the 10:00 lecture are further
down.

Descriptive Statistics

SAMPLE                            N             Mean     Std. Deviation
1          LENGTH                     3          60.33            1.53
2          LENGTH                     3          65.33            3.51
3          LENGTH                     3          63.00            7.21
4          LENGTH                     3          65.00            3.61
5          LENGTH                     3          59.67            5.03
6          LENGTH                     3          61.67            7.64
7          LENGTH                     3          60.33             .58
8          LENGTH                     3          56.33            2.31
9          LENGTH                     3          60.33            5.03
10         LENGTH                     3          57.67            2.52
11         LENGTH                     3          53.67            2.31
12         LENGTH                     3          54.00            1.73
13         LENGTH                     3          57.33            6.81
14         LENGTH                     3          57.00            5.29
15         LENGTH                     3          52.33            3.51

x = 58.93        s = 3.907

We first construct the s chart to see if the process variation is in control. The control
limits for the s chart are s ± Bs = 3.907 ± 1.568 (3.907) = 3.907 ± 6.126 = -2.22 and
10.03. The value of B comes from Table 4.3 on page 8 of Section 4.4. Since standard
deviations can’t be negative, there will obviously be no values below the lower control
limit. There are also no standard deviations above the upper control limit and no run of 9
values either all above or all below s .
Control Chart: LENGTH
12

10

8

6
Standard Deviation

LENGTH
4
UCL = 10.04

2                                                                      Average = 3.91

0                                                                      LCL = .00
1       3       5       7       9        11        13        15
2       4       6       8       10        12        14

Sigma level: 3

Since the process variation is in control, we check to see if the process mean is in control.
The control limits for the x chart are x ± As = 58.93 ± 2.954 (3.907) = 58.93 ± 7.63 =
51.30 and 66.57. There are no sample means outside of the control limits and no run of 9
consecutive means either all above or all below 58.93 (although, it looks like if we had
continued the process, this might have happened).
Control Chart: LENGTH
67.182

63.057

58.933
LENGTH

UCL = 66.57
54.809
Average = 58.93
Mean

50.685                                                                   LCL = 51.30
1        3       5       7       9        11        13        15
2       4       6       8       10        12        14

Sigma level: 3

In practice, we would run this process a little longer (at least 20 samples). If it were in
control at this point, then we would adjust the process if the average length were still
higher than the target of 50 mm. If the process were not in control after 20 samples, then
we would attempt to find out what is making the process unstable and correct it. Once it
was in control, then we could adjust it to achieve the target mean of 50 mm.
These are the data from the 10:00 lecture from the paper cutting process (the “machine”
was supposed to cut strips 50 mm. in length). The analysis proceeds as for the 9:00 data
above. We first do an s chart to see if the process variation is in control. If it’s in control,
then we do an x chart to see if the process s mean is in control.

Descriptive Statistics

SAMPLE                             N             Mean     Std. Deviation
1          LENGTH                      3         53.667           4.041
2          LENGTH                      3         53.333           9.866
3          LENGTH                      3         54.000           4.583
4          LENGTH                      3         52.000           8.185
5          LENGTH                      3         46.000           6.245
6          LENGTH                      3         47.667           7.506
7          LENGTH                      3         49.667           3.786
8          LENGTH                      3         55.667           7.506
9          LENGTH                      3         52.667           3.055
10         LENGTH                      3         55.333           7.572
11         LENGTH                      3         49.667            .577
12         LENGTH                      3         59.333           5.132
13         LENGTH                      3         50.333            .577
14         LENGTH                      3         53.333           6.807
15         LENGTH                      3         56.000           1.732

x = 52.578                 s = 5.145

The control limits for the s chart are s ± Bs = 5.145 ± 1.568 (5.145) = 5.145 ± 8.067 =
-2.92 and 13.21. There are also no standard deviations above the upper control limit and
no run of 9 values either all above or all below s .
Control Chart: LENGTH
14

12

10

8
Standard Deviation

6                                                                      LENGTH

4                                                                      UCL = 13.212

2                                                                      Average = 5.145

0                                                                      LCL = .000
1       3       5       7       9        11        13        15
2       4       6       8       10        12        14

Sigma level: 3

Since the process variation is in control, we check to see if the process mean is in control.
The control limits for the x chart are x ± As = 52.578 ± 1.954 (5.145) = 52.578 ± 10.053
= 42.52 and 62.63. There are no sample means outside of the control limits and no run of
9 consecutive means either all above or all below 52.578.
Control Chart: LENGTH
63.437

58.007

52.578
LENGTH

UCL = 62.632
47.148
Average = 52.578
Mean

41.719                                                                   LCL = 42.523
1        3       5       7       9        11        13        15
2       4       6       8       10        12        14

Sigma level: 3

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