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SUPPLEMENT 6

DISCUSSION QUESTIONS

2. The ultimate goal of the X and R-charts is to ascertain, by a sampling procedure, that the relevantparameter is kept within specific upper and lower bounds. The Xbar chart alone tells us only that

the average or variable values are within the appropriate limits. The combination ofX and theR-charts allows one to determine that both the average and the deviations are within the limits.

4. A process can be out of control because of assignable variation, which can be traced to specificcauses. Examples include such factors as:n Tool wearn A change in raw materialsn A change in working environment (temperature or humidity, for example)n Tired or poorly trained labor

6. Natural variations are those variations that are inherent in the process and for which there is noidentifiable cause. These variations fall in a natural pattern.

Assignable causes are variations beyond those that can be expected to occur because of naturalvariation. These variations can be traced to a specific cause.

11. A run of 5 implies that assignable variation is present.13. The desired mean is used when the mean of a process being observed is unknown or out of control.

END-OF-SUPPLEMENT PROBLEMS

S6.4 n = 6 . From Table S6.1, A2 0 483= . , D4 2 004= . , D3 0=

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

46 0 483 2 46 966

46 0 483 2 45 034

2 004 2 4 008

0 2 0

. .

. .

. .

S6.5 n = 10 . From Table S6.1, A2 0308= . , D4 1777= . , D3 0233= .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

60 0 308 3 60 924

60 0 308 3 59 076

1777 3 5 331

0 223 3 0 669

. .

. .

. .

. .

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S6.7 Sample X R Sample X R Sample X R

1 63.5 2.0 10 63.5 1.3 19 63.8 1.32 63.6 1.0 11 63.3 1.8 20 63.5 1.63 63.7 1.7 12 63.2 1.0 21 63.9 1.04 63.9 0.9 13 63.6 1.8 22 63.2 1.85 63.4 1.2 14 63.3 1.5 23 63.3 1.76 63.0 1.6 15 63.4 1.7 24 64.0 2.07 63.2 1.8 16 63.4 1.4 25 63.4 1.58 63.3 1.3 17 63.5 1.19 63.7 1.6 18 63.6 1.8

X= 6349. , R = 15. , n = 4 . From Table S6.1, A2 0 729= . , D4 2282= . , D3 0 0= . .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

63 49 0 729 15 64 58

63 49 0 729 15 62 40

2 282 15 3 423

0 15 0

. . . .

. . . .

. . .

.

The process is in control.

60.000

Sample

5 2510 2015

61.00

62.00

63.00

64.00

Control Chart

LCL

UCL

65.00

30

X

Sample

0.000 5 2510 2015

1.00

2.00

3.00

4.00

Control Chart R

LCL

UCL

30

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S6.8 Time Box 1 Box 2 Box 3 Box 4 Average Range

9 AM 9.8 10.4 9.9 10.3 10.10 0.6010 AM 10.1 10.2 9.9 9.8 10.00 0.4011 AM 9.9 10.5 10.3 10.1 10.20 0.6012 PM 9.7 9.8 10.3 10.2 10.00 0.60

1 PM 9.7 10.1 9.9 9.9 9.90 0.40Average 10.04 0.52

n = 4 . From Table S6.1, A2 0 729= . , D4 2282= . , D3 0 0= . .

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

4

3

10 04 0 729 0 52 10 42

10 04 0 729 0 52 9 66

2 282 0 52 1187

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 canconclude that the process is presently within control.

One step the QC department might take would be to increase the sample size to provide aclearer indication as to both control limits andwhether or not the process is in control.

S6.9 X= 1990. , R = 0 34. , n = 4 , A2 0 729= . , D4 2282= .

(a) UCL

LCL

= + =

= =

( )

( )

19 90 0 729 0 34 2015

19 90 0 729 0 34 19 65

. . . .

. . . .

(b) UCL

LCL

= =

=

( )2 282 0 34 0 78

0

. . .

(c) The ranges are ok; the means are not in control.

S6.10 X= 10, R = 3 3.

(a) standard deviation = 1.36, x = =136 5 0 61. .

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(b) Using x

UCL

LCL

= + =

= =

( )

( )

10 3 0 61 1183

10 3 061 817

. .

. .

Using A2 0 577= .

UCL

LCL

= + =

= =

( )

( )

10 33 0 577 1190

10 33 0 577 810

. . .

. . .

(c) UCL

LCL

= =

= =

( )

( )

2115 33 6 98

0 33 0

. . .

.

(d) Yes, both mean and range charts indicate process is in control.

S6.11 RDesired = 3 5. , XDesired = 50 , n = 6

UCL X A R

LCL X A R

UCL D R

LCL D R

X

X

R

R

= + = + =

= = =

= = =

= = =

2

2

3

4

50 0 483 35 5169

50 0 483 3 5 48 31

2 004 3 5 7 014

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 apparentlyincreasing. Immediate action is needed to correct the problem and get the average within thecontrol limits again.

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