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Variables Control Charts for Subgroups (X-R X-s

Charts)

What is SPC? You Think You Know ...But Do

You Really?

Enough of Teasing .. Lets start to undo the

confusion.

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!

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

reproducibility)

Quality and Variability

What is Quality?

Quality is fitness for use

Product Control Model for Quality Control

Raw Material, Components Sub-Assemblies

Process

Product

Inspection

Pass

Fail

Ship

Rework

Scrap

Ship

Recycle

Disposal

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

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

control.

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

process

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

cause(s).

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

only. - 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

specs.

Usl-lsl

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

values

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

appropriate.

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?

0.135

Upper Control Limit

What does each area of 0.135 mean?

99.73

Center Line

Lower Control Limit

0.135

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

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

process?

Shifted Process

0.135

Upper Control Limit

Center Line

Lower Control Limit

0.135

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

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.

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

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

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

stable.

Why?

BREAK

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

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

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

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

side) - 2 out of 3 consecutive points beyond 2? (same

side)

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

subgroup

Can irrational subgrouping be a cause of

stratification?

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

Trend

Cycle

Shift / Run

Shift / Run

Stratification

Mixture

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

are - The Center Line and Control Limits of a S Chart

are

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

monitored - 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

flow - 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

limits - 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

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

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

Control, - Douglas C. Montgomery, John Wiley Sons,

- ISBN 0-471-30353-4