1

Chapter 6Chapter 6

Statistical Statistical Process Control Process Control

(SPC)(SPC)

2

Descriptive Statistics

1. Measures of Central Tendencies (Location)

• Mean

• Median = The middle value

• Mode - The most frequent number

2. Measures of Dispersion (Spread)

• Range R=Maximum-Minimum

• Standard Deviation

• Variance

N

xi

N

)x( 2i

2

1 2 3 4 5 6 7 8

xxxx x

µ

(x-µ)

N

xi2)(

The Standard Deviation

River Crossing Problem

River A B C

1 1 1

2 1 1

3 3 6

3 3 1

3 3 6

3 3 1

3 6 6

3 2 1

2 2 1

2 1 1

Average 2.5 2.5 2.5

Range 2 5 5

St Dev 0.7071 1.5092 2.4152

5

Inferential Statistics

Population (N) Parameters

Samples (n)

Statistics

1. Central Tendency: 1. Central Tendency:

2. Dispersion: 2. Dispersion:

x

iN n

xX i

n

ix

N

( )2

s

x X

n

n

i

1

2

1

( )

6

The Normal (Gaussian) Curve

-3 -2 -1 +1 +2 +3

68.26%

95.46%

99.73%

7

0

2

4

6

8

10

12

14

16

18

20

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

Series1

Series2

Series3

Series4

Series5

Red Bead Experiment

8

Types of Control Charts

Quality Characteristic

n>6

Variable Attribute

Type of Attribute

Constantsamplesize?

Constantsampling

unit?

p-chart

np-chart

u-chart

c-chart

X and MR

chart

X-bar and R chart

X-bar and s chart

Defective DefectYes

Yes

YesYes

No

No

No

No

n>1

9

Data

Information

1. Central Tendency

2. Dispersion

3. Shape

Action

Stats

Decision No Action

10

The Shape of the Data Distribution

mean = median = mode

modemean

median

Skewed to the right (positively skewed)

medianmode

mean

Skewed to the left (negatively skewed)

)(3 medianmean

SK

• “Box-and-Whisker” Plot

• Pearsonian Coefficient of Skewness

11

Control Charts

+3+3σσ

AveragAveragee

-3-3σσ

Common Cause(Chance or Random)

Special Cause(Assignable)

Special Cause(Assignable)

12

Central Limit Theorem

Standard Error of the Mean

xpop

n .

Population (individual) Distribution

Sample (x-bar) Distribution

X

μ

13

X-Bar and R Example

1 .164 .162 .161 .163 .163 .166

2 .168 .164 .167 .166 .164 .165

3 .165 .164 .164 .166 .161 .165

4 .169 .164 .164 .163 .167 .167

5 .167 .168 .165 .162 .164 .168 X-Double Bar

X-Bar .1666 .1664 .1642 .1640 .1638 .1662 .16487 R-Bar

R .005 .006 .006 .003 .006 .003 .00483

Rational Subgroup

Subgroup Interval

14

X-Bar and R Control Chart Limits

RAX 2

RD4

n A2 D4 d2

2 1.880 3.268 1.128

3 1.023 2.574 1.693

4 .729 2.282 2.059

5 .577 2.114 2.326

6 .483 2.004 2.534

RAX 2

UCLx-Bar .16487 + (.577 x .00483) = .1676

LCLx-Bar.16487 - (.577 x .00483) = .1621

UCLR2.114 x .00483 = .0102

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16

Attribute Control Chart Limits

Defectives Defects

ChangingSample Size

FixedSample Size

n

ppp

)1(3

)1(3 pnpnp

n

uu 3

cc 3

17

n 235 250 200 250 260 225 270 269 237 240 *n-bar =

243.6

p .0766 .0600 .1100 .0200 .0462 .0667 .0815 .0409 .0820 .0417 p-bar=

.06238

n

ppp

)1(3

p-Chart Example

UCLp

LCLp

n

ppp

)1(3

1089.6.243

)06238.1(06238.306238.

0159.6.243

)06238.1(06238.306238.

*Note: Use n-bar if all n’s are within 20% of n-bar

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19

The α and β on Control Charts

+3+3σσ

AveragAveragee

-3-3σσ

α = .00135

α = .00135

β

β

β

20

Out of Control Patterns

2 of 3 successive points outside 2

4 of 5 successive points outside 1

8 successive points same side of centerline

-3

3

2

1

-1

-2

Average

21

Control Chart Patterns

Gradual Trend

“Freaks”

Sudden Shifts

Cycles

Instability

“Hugging” Centerline “Hugging Control Limits”

22

Six Sigma Process Capability

Cpk = 1.5 3.4 ppm

USLLSL1.5

Cp = 2.0 .54 ppm

23

Cause and Effect Diagrama.k.a. Ishikawa Diagram, Fishbone Diagram

Process

Person Procedures

Material Equipment

B CA

24

Pareto Charta.k.a. 80/20 Rule

Vital Few

Trivial (Useful) Many

25

26

27

28

29

Taguchi Loss Function

.500 .520.480

The Taguchi Loss Function: L (x) = k (x-T)2

Los

s ($

)

.500 .520.480

Traditional Loss Function:

Los

s ($

)

30

Response Curves

Most “Robust” Setting