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Variables Control Charts


Variables Control Charts for Subgroups (X-R & X-s Charts) Variability Sources of Variation Variability can come about due to changes in: Material quality Machine ... – PowerPoint PPT presentation

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Title: Variables Control Charts

Variables Control Charts for Subgroups (X-R X-s
What is SPC? You Think You Know ...But Do
You Really?
Enough of Teasing .. Lets start to undo the
  • 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

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

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

Quality and Variability
What is Quality?
Quality is fitness for use
Product Control Model for Quality Control
Raw Material, Components Sub-Assemblies
Process Control Model for Quality Control
Raw Material, Components Sub-Assemblies
Uncontrollable Inputs
Controllable Inputs
Observation Data Collection Evaluation Data
Analysis Diagnosis Fault Discovery Decision Form
ulate Action Implementation Take Action
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.

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).

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

Introduction to SPC Charts
Concepts and Principles of Control Charts Lets
dive into them now ...
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

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.

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.

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.

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
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
UCL LCL are vital guidelines for deciding when
action should be taken in a process
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
Process Control vs Process Capability
At this juncture, lets distinguish between
process control and process capability ...
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
  • Just because a process is in control does not
    necessarily mean it is a capable process.

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

6 s
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

Shewhart Control Charts - Overview
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.

Shewhart Variables Control Charts for Subgroups
Introduction to X-R Charts
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.

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.

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.

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

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
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.

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

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
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.

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?

Upper Control Limit
What does each area of 0.135 mean?
Center Line
Lower Control Limit
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?
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!

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
Shifted Process
Upper Control Limit
Center Line
Lower Control Limit
Sample Number or Time
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
Control Limits Sampling Risks
  • By moving the control limits further from the
    center line, the risk of a Type I error is
  • 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.

Lets try an example of X-R chart
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
  • 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

Example 1 X-R Chart
  • MiniTab
  • Stat ? Control Charts ?Xbar-R

Example 1 X-R Chart
Is process in control?
Why are the 2 distances different in value?
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.

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

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.

Revising Control Limits and Center Lines
  • Some users will replace the center line of the
    X-chart with a target value, such as nominal
  • 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
  • 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
Indicators of Instability
  • Primary Indicators
  • any point outside of a control limit
  • Secondary Indicators
  • any non-random pattern of points on a control
  • shift or run
  • trend
  • stratification
  • mixture
  • periodicity

Primary Indicators of Instability
  • Any point outside a control limit
  • 1 point beyond 3? limits

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

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
  • 2 out of 3 consecutive points beyond 2? (same

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

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

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

Secondary Indicators of Instability
  • Stratification
  • points hugging the center line, usually within
    1? limits

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

Can irrational subgrouping be a cause of
Secondary Indicators of Instability
  • Mixture
  • points hugging the control limits

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

Secondary Indicators of Instability
  • Cycle or Periodicity
  • any ongoing, repeating pattern

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

MiniTabs Tests for Instability
Primary Indicator
Secondary Indicators
MiniTabs Tests for Instability
Shift / Run
Shift / Run
Shift / Run
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

Relevance of Shut-Down Rules
Suitable for all charts
Suitable only for X-Chart
X-S Charts
  • The Center Line and Control Limits of a X Chart
  • The Center Line and Control Limits of a S Chart

Shewhart Constants
For n gt 25
Example 2
  • MiniTabs Stat ? Control Charts ?Xbar-S

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.

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
  • Goal Monitor and control process inputs and,
    over time, eliminate the need for SPC charts

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

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
  • Introduction of new operators
  • Change in sample size

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

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

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

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
  • Plot the data collected on the chart

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

Implementing the Control Chart
  • Use the Control Chart as a Process Monitoring
  • 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

Implementing the Control Chart
Implementing the Control Chart
Measurement Variation Affects the Control Chart!
Inadequate Discrimination
Adequate Discrimination
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

  • End of Topic
  • What question do you have?

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