Process Improvement Dr. Ron Tibben-Lembke. Statistics.

Process Improvement

Dr. Ron Tibben-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

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

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

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?

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

UCL p zp

n

p

X

n

p

ii

k

ii

k

(1 - p)

1

1



Sample iSize

UCL p zp p)

n

p

X

n

p

ii

k

ii

k

(1

1

1

z = 2 for 95.5% limits; z = 3 for 99.7% limits

# Samples

n

n

k

ii

k

1



# Samples

Sample iSize

z = 2 for 95.5% limits; z = 3 for 99.7% limits

UCL p z

LCL p z

n

n

kp

X

n

p

p

ii

k

ii

k

ii

k

1 1

1

and

n

p p) (1

p p)

n

(1

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 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


n

n

k

ii

k

1 14007

200


16 + 7 +...+ 16

p

X

n

ii

k

ii

k

1

1

1211400

0864.n

n

k

ii

k

1 14007

200

p Chart Control Limits Solution

pp 3 0864 3.n

p p) (1

200

.0864 * (1-.0864)

p

X

n

ii

k

ii

k

1

1

1211400

0864.n

n

k

ii

k

1 14007

200

16 + 7 +...+ 16

p Chart Control Limits Solution

0864 0596 1460. . . or & .0268

pp 3 0864 3.n

p p) (1

200

.0864 * (1-.0864)

p

X

n

ii

k

ii

k

1

1

1211400

0864.n

n

k

ii

k

1 14007

200

16 + 7 +...+ 16

0.00

0.05

0.10

0.15

1 2 3 4 5 6 7

P

Day

p Chart Control Chart Solution

UCL

LCL

Table 7.1 p.193 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

Figure 7.4, p. 196P 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(

Fig. 7.6P 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.

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

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

Figure 7.9C 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

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

Table 7.13

ia

uu

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

Variable Control Charts Focus on unit-to-unit variability

x chart: subgroup average R chart: subgroup range I chart: average, subgroup size of one MR moving range chart: one data point per

subgroup s chart: standard deviation with more than 10

samples per subgroup

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 Calculate the range of each data sample:

Maximum – Minimum Calculate average range:

k

RR

k

ii

1

R Chart – Control Limits How much variability

should there be in the R values?

Depends on process variability,

We don’t know that, only the R values.

We could get it from here:

233 / dRddR

RDLCL

RDUCL

R

R

3

4

But this seems a lot easier:

Look up values in Table B-1, p. 786

Control Chart Limits

n A2 D3 D4

2 1.880 0 3.267

3 1.023 0 2.574

4 0.729 0 2.282

5 0.577 0 2.114

6 0.483 0 2.004

7 0.419 0.076 1.924

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 3.45 5.552 4.60 8.70 7.60 4.43 7.623 5.98 2.92 6.20 4.20 5.104 7.20 5.10 5.19 6.80 4.215 4.00 4.50 5.50 1.89 4.466 10.10 8.10 6.50 5.06 6.947 6.77 5.08 5.90 6.90 9.30

R &X Chart Hotel Data

SampleDay Delivery Time Mean Range

1 7.30 4.20 6.10 3.45 5.55 5.32

7.30 + 4.20 + 6.10 + 3.45 + 5.55 5

Sample Mean =



1 7.30 4.20 6.10 3.45 5.55 5.32 3.85

7.30 - 3.45Sample Range =

Largest Smallest



1 7.30 4.20 6.10 3.45 5.55 5.32 3.852 4.60 8.70 7.60 4.43 7.62 6.59 4.273 5.98 2.92 6.20 4.20 5.10 4.88 3.284 7.20 5.10 5.19 6.80 4.21 5.70 2.995 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.047 6.77 5.08 5.90 6.90 9.30 6.79 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


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



1 7.30 4.20 6.10 3.45 5.55 5.32 3.852 4.60 8.70 7.60 4.43 7.62 6.59 4.273 5.98 2.92 6.20 4.20 5.10 4.88 3.284 7.20 5.10 5.19 6.80 4.21 5.70 2.995 4.00 4.50 5.50 1.89 4.46 4.07 3.616 10.10 8.10 6.50 5.06 6.94 7.34 5.047 6.77 5.08 5.90 6.90 9.30 6.79 4.22


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

. . ..

. . ..


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

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 is sampling is very expensive 6-14 ideal

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

Process Improvement Dr. Ron Tibben-Lembke. Statistics.

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