Statistical Process Control

Dr. Ron Lembke

Statistics

Measures of Variability Range: difference between largest and smallest

values in a sample Very simple measure of dispersion R = max - min

Variance: Average squared distance from the mean Population (the entire universe of values) variance:

divide by N Sample (a sample of the universe) var.: divide by N-1

Standard deviation: square root of variance

Skewness Lack of symmetry Pearson’s coefficient

of skewness:0246810121416

0246810121416

0246810121416

Skewness = 0 Negative Skew < 0

Positive Skew > 0

s

Medianx )(3

Kurtosis Amount of peakedness

or flatness

Kurtosis < 0 Kurtosis > 0

Kurtosis = 04

4)(

ns

xx

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-6 -4 -2 0 2 4 6

Table 14.2 Is everything

OK? How can we

decide? Does it look

normal?

Histogram of 150 data points Histogram looks pretty normal, but definitely not perfect. Looks like 2 peaks, actually, but pretty normal.

0

5

10

15

20

25

30

35

40

45

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Is it Normal? Sorted all 150 data points, plotted Mean = 0.7616, stdv = 0.0738

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145

Compute 1,2,3 sigma limits

Is it Normal?

F(m-3s) = 0 F(m+1s) = 129/150 = 0.86 F(m-2s) = 7/150 = 0.0467 F(m+2s) = 149/150 = 0.993 F(m-1s) = 24/150 = 0.16 F(m+3s) = 150/150 = 1.00 F(m) = 71/150 = 0.473

Is it Normal? Data Theoretical F(m-3s) = 0 0.001 F(m-2s) = 0.0467 0.023 F(m-1s) = 0.160 0.159 F(m) = 0.473 0.500 F(m+1s) = 0.860 0.841 F(m+2s) = 0.993 0.977 F(m+3s) = 1.000 0.999

Values we get are pretty close to a normal distribution.

Real Test of Normality Kolmologorov-Smirnov Anderson-Darling

Sadly, we don’t have time for either today You need SAS or something like it. Excel can’t do everything.

Process Capability UTL = 1.0 LTL = 0.5 Cpk > 1.0 Process

capable, but barely

Is everything OK?

0

5

10

15

20

25

30

35

40

45

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05

Plot Data over time

No significant trend to data?

y = 0.0005x + 0.7237

R2 = 0.0804

0

0.2

0.4

0.6

0.8

1

1.2

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141

Series1

Linear (Series1)

Plot Data over time

Data is in sets of 5, all taken at same time. Plotting individual points makes us see

trends that aren’t really there. Solution – plot averages of each sample

Sample Means

0.650

0.670

0.690

0.710

0.730

0.750

0.770

0.790

0.810

0.830

0.850

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Series1

Control Chart Control Limits are mean +/- 3 std. dev.

0.650

0.670

0.690

0.710

0.730

0.750

0.770

0.790

0.810

0.830

0.850

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Data

LCL

UCL

We have Out-of-Control Points Looks like the mean has shifted. Something is definitely wrong.

Control Limits catch early In fact, we

should compute control limits using first 17 data points

0.650

0.670

0.690

0.710

0.730

0.750

0.770

0.790

0.810

0.830

0.850

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Series1

Series2

Series3

Revise Control Limits New control

limits using first 16 data points.

0.600

0.650

0.700

0.750

0.800

0.850

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Mean

LCL2

UCL2

Control Chart Usage Only data from one process on each chart Putting multiple processes on one chart

only causes confusion 10 identical machines: all on same chart

or not?

In Control A process is “in control” if it is not affected

by any unusual forces Compute Control Limits, Plot points

X-bar Chart for Averages

RAXLCL

RAXUCL

X

X

2

2

k

RR

k

XX

k

ii

k

ii

11

Definitions of Out of Control1. No points outside control limits2. Same number above & below center line3. Points seem to fall randomly above and

below center line4. Most are near the center line, only a few

are close to control limits1. 8 Consecutive pts on one side of centerline2. 2 of 3 points in outer third3. 4 of 5 in outer two-thirds region

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 10.108.10 6.50 5.066.947 6.77 5.08 5.90 6.909.30

R &X Chart Hotel Data

Sample

Day 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 5

Sample Mean =

R &X Chart Hotel Data

Sample

Day 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

Sample

Day 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 10.108.10 6.50 5.06 6.947.34 5.04

7 6.77 5.08 5.90 6.90 9.306.79 4.22

R

R Chart Control Limits

R

k

ii

k

1 3 85 4 27 4 227

3 894. . .

.

R Chart Control Limits Solution

From B-1 (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

(2.114) (3.894) 8 232

(0)(3.894) 0

. . ..

.

02468

1 2 3 4 5 6 7

R, Minutes

Day

R Chart Control 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 B-1

R &X Chart Hotel Data

Sample

Day 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 10.108.10 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 B-1 (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 Control Limits Solution

From Table B-1 (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 ChartControl Chart Solution*

02468

1 2 3 4 5 6 7

` X, Minutes

Day

UCL

LCL

Subgroup Size All data plotted on a control chart represents the

information about a small number of data points, called a subgroup.

Variability occurs within each group Only plot average, range, etc. of subgroup Usually do not plot individual data points Larger group: more variability Smaller group: less variability Control limits adjusted to compensate Larger groups mean more data collection costs

General Considerations, X-bar, R Operational definitions of measuring

techniques & equipment important, as is calibration of equipment

X-bar and R used with subgroups of 4-9 most frequently 2-3 if sampling is very expensive 6-14 ideal

Sample sizes >= 10 use s chart instead of R chart.

Single Data Points? What if you only have one data point on a

process? Inspect every single item There is no range. R=0?

Charts for Individuals (x-Chart) CL: x-bar +/- 3R-bar/d2 R = difference between consecutive units Draw control limits on the chart We can also put User specification limits on, for

reference purposes Doesn’t catch trends as quickly Normality assumption must hold

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

UCLp p zp 1 p

n

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

UCLp p zp 1 p

n

LCLp p zp 1 p

n

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 Data

No. No. NotDay Rooms Ready Proportion1 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 Limits

16 + 7 +...+ 16

p X i

i1

k

ni

i1

k

121

14000.0864

n ni

i1

k

k

1400

7200

p Chart Control Limits Solution

16 + 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 Control Limits Solution

16 + 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 Control Chart Solution

UCL

LCL

P-Chart Example Enter the data, compute the average, calculate

standard deviation, plot lines

P Chart of Number Cracked

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Sample

Pro

port

ion

Dealing with out of control Two points were out of control. Were there

any “assignable causes?” Can we blame these two on anything

special? Different guy drove the truck just those 2 days. Remove 1 and 14 from calculations. p-bar down to 5.5% from 6.1%, st dev, UCL,

LCL, new graph

Revised p-ChartP Chart of Number Cracked

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Sample

Pro

port

ion

Different Sample sizes Standard error varies inversely with sample size Only difference is re-compute for each data

point, using its sample size, n. Why do this? The bigger the sample is, the more

variability we expect to see in the sample. So, larger samples should have wider control limits.

If we use the same limits for all points, there could be small-sample-size points that are really out of control, but don’t look that way, or huge sample-size point that are not out of control, but look like they are.

Judging high school players by Olympic/NBA/NFL standards.

n

pp )1(

Different Sample SizesP Chart of Exact Change, p.202

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.550

0.600

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Sample

Pro

portio

n

How not to do it If we calculate n-bar, average sample size,

and use that to calculate a standard deviation value which we use for every period, we get: One point that really is out of control, does not

appear to be OOC 4 points appear to be OOC, and really are not.

Only potentially do this if all values fall within 25% of the average But with computers, why not do it right?

5 false readingsFig. 7.6 DONE WRONG!!

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Sample

Pro

port

ion

np Charts – Number Nonconforming Counts number of defectives per sample Sample size must be constant

C-Chart Control Limits # defects per item needs a new chart How many possible paint defects could

you have on a car? C = average number defects / unit Each unit has to have same number of

“chances” or “opportunities” for failure

UCL c zC c

LCL zC cc

Paint BlemishesC Chart Blemishes, p.211

0

2

4

6

8

10

12

14

16

1 3 5 7 9 11 13 15 17 19 21 23 25

Sample

Sam

ple

Number of data points Ideally have at least 2 defective points per

sample for p, c charts Need to have enough from each shift, etc.,

to get a clear picture of that environment At least 25 separate subgroups for p or np

charts

Small Average Counts For small averages, data likely not

symmetrical. Use Table 7.8 to avoid calculating UCL, LCL

for averages < 20 defects per sample Aside:

Everyone has to have same definitions of “defect” and “defective”

Operational Definitions: we all have to agree on what terms mean, exactly.

U charts: areas of opportunity vary Like C chart, counts number of defects per

sample No. opportunities per sample may differ Calculate defects / opportunity, plot this. Number of opportunities is different for every

data point ni = # square feet in sample i

in

uu 3

U Chart of Defects, p. 222

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Sample