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STATISTICAL QUALITY CONTROL

Majid Rafiee

Department of Industrial Engineering

Sharif University of Technology

How Statistical Process Control (SPC) works

Copyright Notice

• Parts (text & figures) of this lecture adopted from:

• Introduction to statistical quality control, 5th edition, by douglas C.

Montgomery , arizona state university

• Https://www.Wikipedia.Org

3

Outline

4

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Outline

5

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Introduction

• Statistical quality control

• Monitoring various stages of production

• Monitoring of process

• Means to determine major source of observed variation

• Chance variation

• Inevitable

• Assignable cause

• Detected & corrected by appropriate tools

6

Outline

7

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Chance & Assignable Cause of Quality Variation

8

Causes

Chance causes Assignable causes

Chance & Assignable Cause of Quality Variation

• Chance causes

• In random fashion

• inevitable

• Assignable causes

• Can be assigned to any particular cause

• Defective materials, defective labour, ..

9

Chance & Assignable Cause of Quality Variation

• A process is operating with only chance causes of variation present

is said to be in statistical control.

• A process that is operating in the presence of assignable causes is

said to be out of control.

10

Chance & Assignable Cause of Quality Variation

11

Outline

12

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Basic SPC Tools

13

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Basic SPC Tools

14

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Histogram or steam-and-leaf plot

• Displaying the relative density and shape of the data

• Giving the reader a quick overview of distribution

• Highlighting outliers

• Finding the mode

15

Basic SPC Tools

16

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Check sheet

17

• It is a form used to collect data (quantitative or qualitative)

• In real time

• At the location where the data is generated

Check sheet

18

Check sheet

19

Basic SPC Tools

20

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Pareto chart

• To assess the most frequently

occurring defects

• 80/20 rule

• ABC analysis

21

Pareto chart

• To assess the most frequently

occurring defects

• 80/20 rule

• ABC analysis

22

Various examples of Pareto charts

23

Basic SPC Tools

24

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Cause & effect diagram

• Visualization tool

• Categorizing the potential causes of a problem

• The design of the diagram looks like a skeleton of a fish

• Fishbone diagram

• Ishikawa diagram

25

Cause & effect diagram

26

How to Construct a Cause & effect diagram

• Define the problem to be analyzed

• Draw the effect box & the center line

• Specify major potential cause categories and join them to the center

line

• Identify possible causes & classify them into categories in previous

step

• Rank order the causes

• Take corrective actions

27

Basic SPC Tools

28

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Defect concentration diagram

• a graphical tool

• It is a drawing of the product

• all relevant views displayed

• Shows locations and

frequencies of various defects

29

Adopted from: http://www.syque.com/improvement/Location%20Plot.htm

Defect concentration diagram

30

Basic SPC Tools

31

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Scatter diagram

• To identify type of

relationship between two

quantitative variables

32

Basic SPC Tools

33

SPC tools

Histogram or steam-and-leaf

plot

Check sheet

Pareto chart

Cause & effect

diagram

Defect concentra

tion diagram

Scatter diagram

Control chart

Control chart & Specifications

• a statistical process control tool used to determine if a

manufacturing or business process is in a state of control.

• Specifications

• Lower specification limit

• Upper specification limit

• Target or nominal values

34

Control chart & Specifications

35

Outline

36

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Statistical Basis of Control chart & Specifications

• A point that plots within the control limits indicates the process is in

control

• No action is necessary

• A point that plots outside the control limits is evidence that the process

is out of control

• Investigation and corrective action are required to find and eliminate

assignable cause

• There is a close connection between control charts and hypothesis

testing

37

Shewhart Control Chart Model

• Let w be a sample statistic that measures some CTQ

• Let L be the “distance” of the control limits from the center line

• Expressed in standard deviation units

38

Photolithography Example

• Important quality characteristic in hard bake is resist flow width

• Process is monitored by average flow width sample of 5 wafers

• Process mean is 1.5 microns

• Process standard deviation is 0.15 microns

39

Photolithography Example

𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =𝝈𝒙

𝒏⇒ 𝝈𝒂𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝒙 =

𝟎. 𝟏𝟓

𝟓= 𝟎. 𝟎𝟔𝟕𝟏

To assume x-bar is approximately normally distributed, we would

assume 𝟏𝟎𝟎 𝟏 − 𝜶 % of the sample fall btw 𝟏. 𝟓 + 𝒁𝜶𝟐(𝟎. 𝟎𝟔𝟏𝟕) and

𝟏. 𝟓 − 𝒁𝜶𝟐(𝟎. 𝟎𝟔𝟏𝟕)

𝑼𝑪𝑳 = 𝟏. 𝟓 + 𝟑(𝟎. 𝟎𝟔𝟕𝟏)

𝑪𝑳 = 𝟏. 𝟓

𝑳𝑪𝑳 = 𝟏. 𝟓 − 𝟑(𝟎. 𝟎𝟔𝟕𝟏)

40

Photolithography Example

• Note that all plotted points

fall inside the control limits

• Process is considered to be

in statistical control

• Called three sigma

41

Distribution of X-bar vs X

42

Process improvement using control chart

• Using control charts consequently

• Identify assignable causes

• Eliminate causes

• Reducing variation

• Improving quality

43

Out of Control Action Plan (OCAP)

44

• a companion to the control chart

• reactions to out-of-control

situations

Out of Control Action Plan (OCAP)-1

45

Out of Control Action Plan (OCAP)-2

46

Types of Control Charts

47

Control Charts

Variables Control Charts Attributes Control Charts

Review over Classifying Data On Quality Characteristics

• Variables

• Often continuous measurements

• Length, voltage, viscosity

• Following continuous distribution

• Attributes

• Usually discrete data

• Often taking the form of counts

• Following discrete distribution

48

Types of Control Charts

• Variables Control Charts

• Applied to data following continuous distribution

• Attributes Control Charts

• Applied to data following discrete distribution

49

Types of Control Charts

50

Control Charts

Variables Control Charts Attributes Control Charts

Variables Control Charts

51

Variables Control Charts

X-bar chart R chart S chart S**2 chart

Types of Control Charts

52

Control Charts

Variables Control Charts Attributes Control Charts

Attributes Control Charts

53

Attributes Control Charts

C chart U chart Np chart P chart

Control chart design

54

Control chart design

selection of sample size & sampling frequency

control limits

Control chart design

55

Control chart design

selection of sample size & sampling frequency

control limits

Types of Process Variability

56

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

Types of Process Variability

57

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

Stationary & uncorrelated Process Variability

• Data vary around a fixed mean

in a stable or predictable

manner

58

Types of Process Variability

59

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

Stationary & auto correlated Process Variability

• Successive observations are

dependent with tendency to

move in long runs on either

side of mean

60

Types of Process Variability

61

Process Variability

Stationary & uncorrelated

NonstationaryStationary & auto correlated

Nonstationary Process Variability

• process drifts without any

sense of a stable or fixed

mean

62

Types of Errors

63

Error Types

Type I Type II

Types of Errors

• Type I

• denoted by αlpha

• is the rejection of a true null hypothesis

• Type II

• denoted by Beta

• is the failure to reject a false null hypothesis

64

Average Run Length (ARL)

• Average # of points that must be plotted before a point indicates an

out-of-control condition

• Let P be the probability that any point exceeds control limits

• If observations are uncorrelated, then

ARL = 1/P

65

Average Run Length (ARL) in Control

• In 3-sigma

• P=0.0027 (alpha)

• The probability that a single point falls out of limits, when process is in

control

• Then the average run length of x-bar chart when process is in control

is

𝐴𝑅𝐿0 =1

𝑃=

1

0.0027= 370

66

Average Run Length (ARL) Out of Control

• In our Photolithography Example

• Mean has been shifted to 1.75 instead of 1.5

• Process is out of control

• Probability of single point of x-bar being btw control limits via changed mean

is 0.5 (1-Beta)

• Then the average run length of x-bar chart when process is out of

control is

𝐴𝑅𝐿𝟏 =1

𝑃=

1

0. 𝟓= 𝟐

67

Average Run Length (ARL) Distribution

• Distribution of run length for a Shewhart control chart is geometric

distribution

• Standard deviation is very large

• Geometric distribution is very skewed

• Mean of distribution (ARL) is not necessarily a very typical value of run length

68

Average time to signal

• if samples are taken at fixed intervals of time that are h hours apart

ATS = ARL * h

• If the time interval btw samples is h=1 hour, the average time

required to detect the shift is

𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 2 1 = 2 ℎ𝑜𝑢𝑟𝑠

69

Sample Size & Sampling Frequency

• If sampling frequency is fixed, then as sample size increases, out of

control ARL & ATS decreases

• E.g. if n = 10, instead of n=5 :

𝐴𝑅𝐿1=

1

𝑝

=1

0.9

= 1.11

𝐴𝑇𝑆1 = 𝐴𝑅𝐿1ℎ = 1.11 1 = 1.11 ℎ𝑜𝑢𝑟𝑠

70

Sample Size & Sampling Frequency

• If it became important to detect shift in approximately first hour

after it occurred, two control chart designs would work:

• Design1

• Sampling size: n=5

• Sampling Frequency: every half hour

• Design2

• Sampling size: n=10

• Sampling Frequency: every hour

71

Control chart design

72

Control chart design

selection of sample size & sampling frequency

control limits

Choice of Control Limits

• 3-Sigma Control Limits

• Probability of type I error is 0.0027

• Probability Limits

• Type I error probability is chosen directly

• For example, 0.001 gives 3.09-sigma control limits

• Warning Limits

• Typically selected as 2-sigma limits

73

Choice of Control Limits

74

rational subgroup

• Subgroups or samples should be selected so that

• If assignable causes are present, chance for differences between

subgroups will be maximized

• Chance for difference due to assignable causes within a subgroup will be

minimized.

• Two general approaches for constructing rational subgroups:

• Consecutive units

• Random sample of all process output over sampling interval

75

consecutive units

• Sample consists of units produced at the same time

• Primary purpose is to detect process shifts

76

random sample

• random sample of all process output over sampling interval

• Sample consists of units that are representative of all units

produced since last sample

• Often used to make decisions about acceptance of product

• Effective at detecting shifts to out-of-control state and back into

in-control state between samples

• we can often make any process appear to be in statistical control

just by stretching out the interval between observations in the

sample.

77

Different Approaches for Sampling

78

Analysis of Patterns on Control Chart

79

• Pattern is very nonrandom in

appearance

• 19 of 25 points plot below the center

line, while only 6 plot above

• Following 4th point, 5 points in a row

increase in magnitude, a run up

• There is also an unusually long run

down beginning with 18th point

Cyclic Pattern

80

how to detect nonrandom patterns?

• Western Electronic Handbook, Set of decision rules

• 1. Any single data point falls outside the 3σ-limit from the centerline

• 2. Two out of three consecutive points fall beyond the 2σ-limit, on the

same side of the centerline

• 3. Four out of five consecutive points fall beyond the 1σ-limit, on the

same side of the centerline

• 4. Nine consecutive points fall on the same side of the centerline

81

Discussion of Sensitizing Rules for Control Chart

• Suppose that analyst uses k decision rules & that criterion I has type

I error probability 𝛼𝑖 , then the overall type I error based on k test is:

𝛼 = 1 −ෑ

𝑖=1

𝑘

(1 − 𝛼𝑖)

• Q) What are the consequences of increasing overall error type I?

82

Control Chart Application

83

Control Chart Application

Phase I Phase II

Control Chart Application

84

Control Chart Application

Phase I Phase II

Phase I

• Phase I is a retrospective analysis of process data to construct trial

control limits

• Charts are effective at detecting large, sustained shifts in process

parameters, outliers, measurement errors, data entry errors, etc.

• Facilitates identification and removal of assignable causes

85

Control Chart Application

86

Control Chart Application

Phase I Phase II

Phase II

• In this phase, the control chart is used to monitor the process

• Process is assumed to be reasonably stable

• Emphasis is on process monitoring, not on bringing an unruly

process into control

87

Outline

88

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Implementing SPC

• Elements of a successful SPC program

• Management leadership

• A team approach

• Education of employees at all levels

• Emphasis on reducing variability

• Measuring success in quantitative terms

• A mechanism for communicating successful results

throughout the organization

89

Outline

90

Introduction

Chance & Assignable Cause of Quality Variation

Magnificent seven

Implementing SPC in a quality Improvement program

Application of SPC

Statistical basis of control chart

Application of SPC

• Industrial applications

• Nonmanufacturing applications

• Sometimes require ingenuity

• Mostly do not have a natural measurement system

• The observability of the process may be fairly low

91

process mapping

• Flow charts & operation process charts are particularly useful in

developing process definition & process understanding.

• This is sometimes called process mapping.

• Used to identify value-added versus nonvalue-added activity

• To eliminate non-value added activities

92

Flow chart

93

Back up

94