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Goal of Control Charts collect and present data visually allow us to see when trend appears see when “out of control” point occurs
0102030405060
1 2 3 4 5 6 7 8 9 10 11 12
Process Control Charts Graph of sample data plotted over time
UCL
LCL
Process Average ± 3
Time
X
0102030405060
1 2 3 4 5 6 7 8 9 10 11 12
Process Control Charts Graph of sample data plotted over time
Assignable Cause Variation
Natural Variation
UCL
LCL
Time
X
Definitions of Out of Control1. No points outside control limits
2. Same number above & below center line
3. Points seem to fall randomly above and below center line
4. Most are near the center line, only a few are close to control limits
1. 8 Consecutive pts on one side of centerline
2. 2 of 3 points in outer third
3. 4 of 5 in outer two-thirds region
Attributes vs. VariablesAttributes: Good / bad, works / doesn’t count % bad (P chart) count # defects / item (C chart)
Variables: measure length, weight, temperature (x-bar
chart) measure variability in length (R chart)
Attribute Control Charts Tell us whether points in tolerance or not
p chart: percentage with given characteristic (usually whether defective or not)
np chart: number of units with characteristic c chart: count # of occurrences in a fixed area of
opportunity (defects per car) u chart: # of events in a changeable area of
opportunity (sq. yards of paper drawn from a machine)
p Chart Control Limits
# Defective Items in Sample i
Sample iSize
UCLp p zp 1 p
n
p X i
i1
k
ni
i1
k
p Chart Control Limits
# Defective Items in Sample i
Sample iSize
z = 2 for 95.5% limits; z = 3 for 99.7% limits
# Samples
n
ppzpUCLp
1
p X i
i1
k
ni
i1
k
n ni
i1
k
k
p Chart Control Limits
# Defective Items in Sample i
# Samples
Sample iSize
z = 2 for 95.5% limits; z = 3 for 99.7% limits
n
ppzpUCLp
1
n
ppzpLCLp
1
n ni
i1
k
k
p X i
i1
k
ni
i1
k
p Chart ExampleYou’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)?
© 1995 Corel Corp.
p Chart Hotel DataNo. No. Not
Day Rooms Ready Proportion
1 200 16 16/200 = .0802 200 7 .0353 200 21 .1054 200 17 .0855 200 25 .1256 200 19 .0957 200 16 .080
p Chart Solution16 + 7 +...+ 16
p X i
i1
k
ni
i1
k
121
14000.0864
n ni
i1
k
k
1400
7200
p zp 1 p
n 0.0864 3
0.0864 1 0.0864 200
p Chart Solution16 + 7 +...+ 16
p zp 1 p
n 0.0864 3
0.0864 1 0.0864 200
0.0864 3* 0.01984 0.0864 0.01984
0.1460, and 0.0268
p X i
i1
k
ni
i1
k
121
14000.0864
n ni
i1
k
k
1400
7200
R Chart Type of variables control chart
Interval or ratio scaled numerical data
Shows sample ranges over time Difference between smallest & largest values
in inspection sample
Monitors variability in process Example: Weigh samples of coffee &
compute ranges of samples; Plot
You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?
Hotel Example
Hotel Data
Day Delivery Time
1 7.30 4.20 6.10 3.455.552 4.60 8.70 7.60 4.437.623 5.98 2.92 6.20 4.205.104 7.20 5.10 5.19 6.804.215 4.00 4.50 5.50 1.894.466 10.10 8.10 6.50 5.066.947 6.77 5.08 5.90 6.909.30
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
1 7.30 4.20 6.10 3.45 5.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55
5Sample Mean =
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
1 7.30 4.20 6.10 3.45 5.555.32 3.85
7.30 - 3.45Sample Range =
Largest Smallest
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
1 7.30 4.20 6.10 3.45 5.555.32 3.85
2 4.60 8.70 7.60 4.43 7.626.59 4.27
3 5.98 2.92 6.20 4.20 5.104.88 3.28
4 7.20 5.10 5.19 6.80 4.215.70 2.99
5 4.00 4.50 5.50 1.89 4.464.07 3.61
6 10.10 8.10 6.50 5.06 6.947.34 5.04
7 6.77 5.08 5.90 6.90 9.306.79 4.22
R Chart Control Limits
UCL D R
LCL D R
R
R
k
R
R
ii
k
4
3
1
Sample Range at Time i
# Samples
From Exhibit 6.13
Control Chart Limits
n A2 D3 D4
2 1.88 0 3.278
3 1.02 0 2.57
4 0.73 0 2.28
5 0.58 0 2.11
6 0.48 0 2.00
7 0.42 0.08 1.92
R Chart Solution
From 6.13 (n = 5)
R
R
k
UCL D R
LCL D R
ii
k
R
R
1
4
3
3 85 4 27 4 227
3 894
(2.11) (3.894) 8 232
(0)(3.894) 0
. . ..
.
X Chart Control Limits
k
RR
k
XX
RAXUCL
k
ii
k
ii
X
11
2
Sample Range at Time i
# Samples
Sample Mean at Time i
X Chart Control Limits
UCL X A R
LCL X A R
X
X
kR
R
k
X
X
ii
k
ii
k
2
2
1 1
Sample Range at Time i
# Samples
Sample Mean at Time i
From 6.13
Exhibit 6.13 Limits
n A2 D3 D4
2 1.88 0 3.278
3 1.02 0 2.57
4 0.73 0 2.28
5 0.58 0 2.11
6 0.48 0 2.00
7 0.42 0.08 1.92
R &X Chart Hotel Data
SampleDay Delivery TimeMean Range
1 7.30 4.20 6.10 3.45 5.555.32 3.85
2 4.60 8.70 7.60 4.43 7.626.59 4.27
3 5.98 2.92 6.20 4.20 5.104.88 3.28
4 7.20 5.10 5.19 6.80 4.215.70 2.99
5 4.00 4.50 5.50 1.89 4.464.07 3.61
6 10.10 8.10 6.50 5.06 6.947.34 5.04
7 6.77 5.08 5.90 6.90 9.306.79 4.22
X Chart Control Limits
X
X
k
R
R
k
ii
k
ii
k
1
1
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
. . ..
. . ..
X Chart Control Limits
From 6.13 (n = 5)
X
X
k
R
R
k
UCL X A R
ii
k
ii
k
X
1
1
2
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
5 813 0 58 * 3 894 8 060
. . ..
. . ..
. . . .
X Chart Solution
From 6.13 (n = 5)
X
X
k
R
R
k
UCL X A R
LCL X A R
ii
k
ii
k
X
X
1
1
2
2
5 32 6 59 6 797
5 813
3 85 4 27 4 227
3 894
5 813 (0 58)
5 813 (0 58)(3.894) = 3.566
. . ..
. . ..
. .
. .
(3.894) = 8.060
Thinking ChallengeYou’re manager of a 500-room hotel. The hotel owner tells you that it takes too long to deliver luggage to the room (even if the process may be in control). What do you do?
© 1995 Corel Corp.
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