Statistical Process Control

Managing for Quality

Dr. Ron Lembke

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

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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 Control Limits

n ni

i1

k

k

1400

7200

p Chart Control Limits16 + 7 +...+ 16

p X i

i1

k

ni

i1

k

121

14000.0864

n ni

i1

k

k

1400

7200

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

0.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

p Chart

UCL

LCL

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

R Chart Control Limits

R

k

ii

k

1 3 85 4 27 4 227

3 894. . .

.

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

. . ..

.

02468

1 2 3 4 5 6 7

R, Minutes

Day

R Chart Solution

UCL

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

From Table 6-13

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

X Chart Solution*

02468

1 2 3 4 5 6 7

`X, Minutes

Day

UCL

LCL

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.

N

Redesign the luggage delivery process Use TQM tools

Cause & effect diagrams Process flow charts Pareto charts

Solution

Method People

Material Equipment

Too Long