Variables Control Charts

1
Variables Control Charts for Subgroups (X-R X-s
Charts)
2
What is SPC? You Think You Know ...But Do
You Really?
3
Enough of Teasing .. Lets start to undo the
confusion.
4
Variability

• The Devil is in the Deviations. No two things can
  ever be made exactly alike, just like no two
  things are alike in nature.
• Variation cannot be avoided in life! Every
  process has variation. Every measurement. Every
  sample!

5
Sources of Variation

• Variability can come about due to changes in
• Material quality
• Machine settings or conditions
• Manpower standards
• Methods of processing
• Measurement
• Environment

6
Types of Variation

• One way of classifying variation is
• within unit (positional variation)
• between units (unit-unit variation)
• between lots (lot-lot variation)
• between lines (line-line variation)
• across time (time-time variation)
• measurement (gage repeatability
  reproducibility)

7
Quality and Variability
What is Quality?
Quality is fitness for use
8
Product Control Model for Quality Control
Raw Material, Components Sub-Assemblies
Process
Product
Inspection
Pass
Fail
Ship
Rework
Scrap
Ship
Recycle
Disposal
9
Process Control Model for Quality Control
Raw Material, Components Sub-Assemblies
Uncontrollable Inputs
Controllable Inputs
Process
Product
Observation Data Collection Evaluation Data
Analysis Diagnosis Fault Discovery Decision Form
ulate Action Implementation Take Action
10
Statistical Process Control

• The process control model shifts focus to the
  home front, i.e. the manufacturing process,
  taking a preventive instead of reactive mode.
• It also has something which the old concept of
  product control lacked - statistics. This allows
  use of samples to understand the entire process.
• The new emphasis had to have a name - Statistical
  Process Control (SPC).
• We owe the application of statistics as a tool
  for manufacturing to Dr Walter A. Shewhart.

11
Dr Walter A. ShewhartFather of Control Charts

• Physicist at Bell Telephone Labs., specializing
  in the Brownian movement.
• Asked to help in the war effort to design
  standard radio headset for army troops.

• Developed important descriptive statistics
• to aid in manufacturing, the most important of
  which was the X-R chart (invented in 1924).
• Presented to the outside world in a series of
  lectures at Stevens Institute of Technology. The
  lecture material became his well-known book,
  Economic Control of Quality of Manufactured
  Product (1931).

12
Success in Manufacturing

• The key to success in manufacturing is an
  effective SPC program that continuously finds and
  eliminates problems.
• Central to an SPC program are the following
• Understand the causes of variability
• Shewhart found two basic causes of variability
• Chance causes of variability
• Assignable causes of variability
• Develop methods of recognizing these causes
• SPC charts

13
Introduction to SPC Charts
Concepts and Principles of Control Charts Lets
dive into them now ...
14
Two Basic Causes of Variability

• Chance Causes of Variation
• Due to the cumulative effect of many small
  unavoidable sources of variation.
• Also known as
• common variation
• random variation
• inherent variation
• natural variation
• A process operating with only chance causes of
  variation present is said to be in statistical
  control.

15
Two Basic Causes of Variability

• Assignable (or Special) Causes of Variation
• Variation in a process that is different from
  from chance variation disturbs a process so that
  what it produces seems unnatural.
• Examples of such causes of variation are
• improperly adjusted machine
• excessive tool wear
• defective raw material
• A process operating in the presence
• of assignable causes of variation is
• said to be out-of-control.

16
Objectives of SPC Charts

• All control charts have one primary purpose!
• To detect assignable causes of variation
• that cause significant process shift, so that
• investigation and corrective action may be
  undertaken to rid the process of the assignable
  causes of variation before too many
  non-conforming units are produced.
• in other words, to keep the process in
  statistical control.

17
Objectives of SPC Charts

• The following are secondary objectives or direct
  benefits of the primary objective
• To reduce variability in a process.
• To help estimate the parameters of a process and
  establish its process capability.

18
General Form of SPC Charts

• Graphical comparison of a quality characteristic
  against computed control limits.
• Usually, its sample statistic is plotted over
  time. Sometimes, the actual value of the quality
  characteristic is plotted.

Each point is usually a sample statistic (such as
subgroup average) of the quality characteristic
19
General Form of SPC Charts

• Control charts plot variation over time.
• Control limits, Upper Control Limit (UCL) and
  Lower Control Limit (LCL), help us distinguish
  between the two basic causes of variability.

Center Line represents mean operating level of
process
UCL LCL are vital guidelines for deciding when
action should be taken in a process
20
General Form of SPC Charts

• A point outside of UCL or LCL is evidence that
  process is out of control
• Investigation and corrective action are required
  to eliminate the assignable cause(s).
• Assignable cause(s) may be measuring error,
  plotting error, special variation from some
  process input, etc.

Out-of-control signal Investigate assignable
cause(s).
21
Process Control vs Process Capability
At this juncture, lets distinguish between
process control and process capability ...
22
Process Control

• Means that chance causes are the only source of
  variation present.
• Refers to voice of the process, i.e. we only
  need data from the process to determine if a
  process is in control.
• Quality characteristic is monitored to verify if
  it forms a stable distribution over time, with
  control limits computed from the process data
  only.
• Just because a process is in control does not
  necessarily mean it is a capable process.

23
Process Capability

• The goodness of a process is measured by its
  process capability.
• Compares voice of the process with voice of
  the customer, which is given in terms of
  customer specs. or requirements.
• Measures how well a stable distribution (process
  in control) meets customer requirements by the
  proportion of products within or out of customer
  specs.

Usl-lsl
6 s
24
Control Limits vs Spec. Limits

• Specification Limits (USL , LSL)
• determined by design considerations
• represent the tolerable limits of individual
  values of a product
• usually external to variability of the process
• Control Limits (UCL , LCL) base on data
• derived based on variability of the process
• usually apply to sample statistics such as
  subgroup average or range, rather than individual
  values

25
Shewhart Control Charts - Overview
26
Shewhart Control Charts - Overview

• Shewhart control charts are characterized by
  having control limits set at ks distance from
  process mean. A usual value of k is 3, giving
• Upper Control Limit ?w 3?w
• Center Line ?w
• Lower Control Limit ?w 3?w
• Whether the data is variable or attribute,
  Shewhart control charts plot the sample statistic
  of the quality characteristic of interest.

27
Shewhart Variables Control Charts for Subgroups
28
Introduction to X-R Charts
29
Central Limit Theorem and Normal Distribution

• Shewhart variables control charts for subgroups
  work because of two important principles
• Central Limit Theorem
• Normal Distribution
• Shewhart found that when the averages of
  subgroups from a constant-cause system are
  plotted in the form of a histogram, the normal
  distribution appears.

30
Central Limit Theorem and Normal Distribution

• The constant-cause system does not itself have to
  be normally distributed. It can be skewed,
  rectangular or even inverted pyramid.
• As long as the sample size is adequately large,
  the averages of the subgroups will show a central
  tendency and variation that tend to follow the
  normal curve.
• This is called the Central Limit Theorem.

31
Central Limit Theorem and Normal Distribution

• This discovery means that a process can be
  monitored over time by measuring the averages of
  a subgroup of parts (basis for X-chart).
• If the process is a constant-cause system, these
  averages would fall within a normal curve. The
  variability is entirely due to common causes.
• When assignable causes appear, they will affect
  the averages to the point where these averages
  will probably not fit within the normal curve.

32
Central Limit Theorem and Normal Distribution

• Important Information from Central Limit Theorem
• If k observations of sample size n are taken,
  the distribution of x1, x2, , xk will
  approximate a normal distribution N(?x,?x)
  distribution, with

33
Construction of X-R Charts

• The X-R chart is the most versatile of control
  charts, and is used in most applications.
• Charting of averages and charting of ranges are
  used to check if a constant-cause system exists.

X-chart measures variability between samples
R-chart measures variability within samples
R Always screw
34
Construction of X-R Charts

• The control limits are the estimated /-3 sigma
  limits for the process.
• Tables of constants were developed to make the
  sigma calculations simple and to reduce error.

35
Construction of X-R Charts

• The Center Line and Control Limits of a X-chart
• The Center Line and Control Limits of a R-chart

36
Construction of X-R Charts
Shewhart Constants
For sample size n gt 10, R loses its efficiency in
estimating process sigma and R-chart may not be
appropriate.
37
Control Charts Sampling Risks

• Since the control limits are the /-3 sigma
  limits for the process, the interval between the
  limits cover 99.73 of the normal distribution.

38
Control Charts Sampling Risks

• If there is no change in the process, there is
  still a chance of getting a point out of the 3s
  control limits. What is the implication?

0.135
Upper Control Limit
What does each area of 0.135 mean?
99.73
Center Line
Lower Control Limit
0.135
39
Control Charts Sampling Risks

• Type I Error reject good lot over reject
• Concluding that the process is out of control
  when it is really in control
• ? probability of making Type I error
• commonly known as the producers risk
• total of 0.27 for control limits of
  /- 3s

Is process really out of control? Or is the point
outside due to random variation?
0.135
0.135
40
Control Charts Sampling Risks

• Type I Error and Tampering
• If the process is really in control, and process
  adjustment is made because of Type I error, it is
  called tampering with the process.
• Tampering has been shown to actually increase the
  variability of the process!

41
Control Charts Sampling Risks

• Type II Error accept fail lot
• Concluding that the process is in control when it
  is really out of control
• ? probability of making Type II error
• commonly known as the consumers risk

Is process really in control? Or is the point
inside due to random variation of the shifted
process?
Shifted Process
0.135
Upper Control Limit
Center Line
Lower Control Limit
0.135
Sample Number or Time
42
Control Charts Sampling Risks

• The control chart is a test of the hypothesis
  that the process is in statistical control.

In-control signal Accept H0 - Process remains
unchanged - No assignable causes present
Out-of-control signal Reject H0 - Process has
shifted - Assignable causes present
43
Control Limits Sampling Risks

• By moving the control limits further from the
  center line, the risk of a Type I error is
  reduced.
• However, widening the control limits will
  increase the risk of a Type II error.
• For a given Type I error (control limits
  interval), the risk of a Type II error can
• be reduced by increasing the
• sample size.

44
Lets try an example of X-R chart
45
Example 1 X-R Chart
Piston rings for an automotive engine are forged.
20 preliminary samples, each of size 5, were
obtained. The inside diameter of these rings are
shown here. Verify if the forging process is in
statistical control. The data are found in SPC
Charts.MTW.

• S/N X1 X2 X3 X4 X5
• 1 74.030 74.002 74.019 73.992 74.008
• 2 73.995 73.992 74.001 74.011 74.004
• 3 73.988 74.024 74.021 74.005 74.002
• 4 74.002 73.996 73.993 74.015 74.009
• 5 73.992 74.007 74.015 73.989 74.014
• 6 74.009 73.994 73.997 73.985 73.993
• 7 73.995 74.006 73.994 74.000 74.005
• 8 73.985 74.003 73.993 74.015 73.998
• 9 74.008 73.995 74.009 74.005 74.004
• 10 73.998 74.000 73.990 74.007 73.995
• 11 73.994 73.998 73.994 73.995 73.990
• 12 74.004 74.000 74.007 74.000 73.996
• 13 73.983 74.002 73.998 73.997 74.012
• 14 74.006 73.967 73.994 74.000 73.984
• 15 74.012 74.014 73.998 73.999 74.007
• 16 74.000 73.984 74.005 73.998 73.996
• 17 73.994 74.012 73.986 74.005 74.007
• 18 74.006 74.010 74.018 74.003 74.000

46
Example 1 X-R Chart

• MiniTab
• Stat ? Control Charts ?Xbar-R

47
Example 1 X-R Chart
Is process in control?
Why are the 2 distances different in value?
48
Interpreting X-R Chart Together

• The X-R chart must be interpreted together as
• well as separately.
• Read the R-chart first to determine if it is in
  control, i.e. no points out of the control limits
  or non-random pattern (to be discussed later).
• The R-chart is more sensitive to changes in
  uniformity or consistency. Anything that
  introduces changes to the process variability,
  such as poor material or lack of maintenance,
  will affect the R-chart.

49
Interpreting X-R Chart Together

• Some assignable causes show up on both the X and
  R charts. Work on the R-chart first.
• Never attempt to interpret the X-chart when the
  R-chart indicates an out-of-control condition,
  i.e. when the within-subgroup variability is not
  stable.

Why?
50
BREAK
51
Revising Control Limits and Center Lines

• The initial trial control limits should be
  treated as subject to possible subsequent
  revision. The control chart should always reflect
  accurately the present conditions of the process.
• A sustained change in the level of either chart,
  usually for at least 20 points, may call for
  revision of the control limits to recognize the
  permanent change.
• Some practitioners establish regular periods for
  review of the control limits, such as every week,
  month, or every 50 samples, etc.

52
Revising Control Limits and Center Lines

• Some users will replace the center line of the
  X-chart with a target value, such as nominal
  spec.
• If the process mean can be easily adjusted by
  manipulating some process inputs, it may be
  helpful to shift the process mean to the desired
  value.
• If the mean is not easily influenced by a simple
  process adjustment, such as flatness of a
  machined part, forcing a target value can result
  in many points out of the control limits.

What about changing the sample size? revise
control limit
53
Indicators of Instability

• Primary Indicators
• any point outside of a control limit
• Secondary Indicators
• any non-random pattern of points on a control
  chart
• shift or run
• trend
• stratification
• mixture
• periodicity

54
Primary Indicators of Instability

• Any point outside a control limit
• 1 point beyond 3? limits

55
Primary Indicators of Instability

• Common Causes
• new workers, methods, raw materials or machines
• change in inspection methods or standards
• change in skill and/or motivation of operators

56
Secondary Indicators of Instability

• Shift or Run
• k consecutive points (usually 7, 8 or 9) on the
  same side of the center line
• 4 out of 5 consecutive points beyond 1? (same
  side)
• 2 out of 3 consecutive points beyond 2? (same
  side)

57
Secondary Indicators of Instability

• Common Causes of Shift or Run
• new workers, methods, raw materials or machines
• change in inspection methods or standards
• change in skill and/or motivation of operators

58
Secondary Indicators of Instability

• Trend
• k consecutive points (usually 5, 6 or 7) moving
  in the same direction

59
Secondary Indicators of Instability

• Common Causes of Trend
• new workers, methods, raw materials or machines
• change in inspection methods or standards
• change in skill and/or motivation of operators

60
Secondary Indicators of Instability

• Stratification
• points hugging the center line, usually within
  1? limits

61
Secondary Indicators of Instability

• Common Causes of Stratification
• incorrect calculation of control limits
• sampling process collects one or more units from
  different underlying distributions within each
  subgroup

Can irrational subgrouping be a cause of
stratification?
62
Secondary Indicators of Instability

• Mixture
• points hugging the control limits

63
Secondary Indicators of Instability

• Common Causes of Mixture
• two (or more) overlapping distributions
• over-control by operators

64
Secondary Indicators of Instability

• Cycle or Periodicity
• any ongoing, repeating pattern

65
Secondary Indicators of Instability

• Common Causes of Cycle or Periodicity
• systematic environmental changes
• temperature
• operator fatigue
• rotation of operators
• fluctuation in machine settings
• maintenance schedules
• tool wear

66
MiniTabs Tests for Instability
Primary Indicator
Secondary Indicators
67
MiniTabs Tests for Instability
Shift / Run
Trend
Cycle
Shift / Run
Shift / Run
Stratification
Mixture
68
Tests for Instability

• CAUTION Do not apply tests blindly
• Not every test is relevant for all charts
• Excessive number of tests ? Increased ?-error
• Nature of application

69
Relevance of Shut-Down Rules
Suitable for all charts
Suitable only for X-Chart
_
70
X-S Charts
_
_

• The Center Line and Control Limits of a X Chart
  are
• The Center Line and Control Limits of a S Chart
  are

71
Shewhart Constants
For n gt 25
72
Example 2

• MiniTabs Stat ? Control Charts ?Xbar-S

73
R Chart vs S Chart

• For ease of computation, the R Chart is preferred
• The S Chart may be used when n is not constant
• For large sample size (n ? 10), the range loses
  its efficiency as an estimator of ?
• Larger sample size is required when
• lower sampling risks are required
• greater drift sensitivity is required
• quality characteristic is non-normal
• Historical Note When Shewhart developed thest
  charts in the 1920s, there was no easy
• way to calculate the standard deviation.
  Thus, the range approach became
• ingrained in SPC application.

74
Using SPC

• Place charts only where necessary based on
  project scope
• Remove charts that are not value-added
• Initially, the process outputs may need to be
  monitored
• Goal Monitor and control process inputs and,
  over time, eliminate the need for SPC charts

75
Where to Use SPC Charts

• When a mistake-proofing device is not feasible
• Identify processes with high RPNs from FMEA
• Evaluate the Current Controls column to
  determine gaps in the control plan. Does SPC
  make sense?
• Identify critical variables based on DOE
• Customer requirements
• Management commitments

76
Updating Control Limits

• Control Limits should be updated when
• Change in supplier for a critical material
• Change in process machinery
• Engineering change orders that affect process
  flow
• Introduction of new operators
• Change in sample size

77
Implementing the Control Chart

• 1) Preparation of Sampling
• 2) Data Collection
• 3) Construct the Control Chart
• 4) Analysis Interpretation
• 5) Use the Control Chart as a Process
  Monitoring Tool

78
Implementing the Control Chart

• Preparation of Sampling
• Choose the quality characteristic to be measured
• measurements taken on the final product
• measurements taken on the in-process product
• measurements taken on the process variables
• Determine the basis, size and frequency

79
Implementing the Control Chart

• Data Collection
• Record the data
• Calculate the relevant statistics mean, range,
  proportion, etc

80
Implementing the Control Chart

• Construct the Control Chart
• Calculate the trial center line and the trial
  control limits
• Plot the trial center line and the trial control
  limits
• Plot the data collected on the chart

81
Implementing the Control Chart

• Analysis Interpretation
• Investigate the chart for lack of control
• Eliminate out-of-control points if required
• Recompute control limits if necessary
• Determine process capability

82
Implementing the Control Chart

• Use the Control Chart as a Process Monitoring
  Tool
• Continue data collection and plotting
• Identify out-of-control situations and take
  correction action
• If a permanent process shift has occurred,
  recalculate the new center line and control limits

83
Implementing the Control Chart
84
Implementing the Control Chart
Measurement Variation Affects the Control Chart!
Inadequate Discrimination
Adequate Discrimination
85
Statistical Process Control

• A state of statistical control is not a natural
  state for a manufacturing process. It is an
  achievement, arrived at by elimination one by
  one, by determined effort, of special causes of
  excessive variation.
• There is no process capability and no meaningful
• specifications, except in statistical control.
• - William Edwards Deming

86

• End of Topic
• What question do you have?

87
Reading Reference

• Introduction to Statistical Quality
  Control,
• Douglas C. Montgomery, John Wiley Sons,
  
• ISBN 0-471-30353-4