Control Charts

1
Control Charts

• The concept of process variability forms the
  heart of statistical process control.
• Statistical process control is concerned with
  print production systems operating on target with
  minimum variance.

2
Two types of Process Variation

• Natural process variation, frequently called
  common cause or system variation, is the
  naturally occurring fluctuation or variation
  inherent in all processes.
• Special cause variation is typically caused by
  some problem or anomaly in the system.

3
Control Charts are a methodology

• for analyzing a process and quickly determining
  when it is out of control
• It is out of control when a special cause
  variation is present because something unusual is
  occurring in the process

4
A process is controlled when through the use
of past experiencewe can predict how the
process is expected to behave in the future.
5
Control Charts Defined

• Control charts are used to analyze and control
  the output of a process by measuring and
  monitoring key characteristics of the product.
• When the characteristics are measured or counted,
  the results are plotted on a chart that contains
  control limits.
• The location of the data points relative to the
  control limits indicates whether the process is,
  at that moment, in control.

6
Stability is a time-related

• A histogram gives a picture of overall
  variability, but tells nothing about time-related
  variation, or stability.
• The control chart is similar in concept to the
  histogram, with the important difference that the
  data is displayed as a function of time.
• The time-related nature of process behavior is
  discovered when the order of production is
  preserved.
• Decisions on whether to change a process or leave
  it alone are often best made on the basis of
  immediate past performance.

7
Types of Control Charts
There are many different types of control charts
depending on the type of data being analyzed. We
will cover several based on variables.

• X Chart data collected as individual
  measurements
• Xbar and R Charts data collected in small units
  called subgroups
• (sampling
  distributions)

8
To implement Control Charts -these concepts must
be understood

• Mean the average value for all the data
• Standard Deviation average variation around the
  mean
• Central Limit Theorem average of means from
  multiple samples are normally distributed

9
Normal DistributionLike most statistical process
control, time charts are based on the bell shaped
pattern common to the measurements from
industrial processes

• Measurements tend to cluster around a central
  point called the mean - xbar
• Data are symmetrical around the mean
• Standard deviations are located at equal
  distances from each other

10
Normal Distribution

• The proportion of measurements located between
  the mean and the standard deviation is a
  constant, where
• Mean - 1s 68
• Mean - 2s 95
• Mean - 3s 99.7

11
Constructing a X Control Chart based on
individual measurements

• Collect the data one at a time, based on a
  systematic random sample, and record in the order
  of occurrence
• Calculate the average of the data and draw it in
  as the center line on the graph
• Plot the data in a sequence obtained from left to
  right
• The limits are based on the average process
  variation, known as the standard deviation

12
In general the chart contains a center line that
represents the mean value for the in-control
process. Two other horizontal lines, called the
upper control limit (UCL) and the lower control
limit (LCL) are also shown on the chart.
13
Plus or minus "3 sigma" limits are typical

• In the U.S it is an acceptable practice to base
  the control limits upon a multiple of the
  standard deviation.
• Usually this multiple is 3 and thus the limits
  are called 3-sigma limits. ( / - 3 standard
  deviations)
• Approximately 99.7 of the measurements fall
  between /- 3 sigma from the mean
• There is only a .3 likelihood that a point
  falling outside of 3 sigma represents a natural
  occurrence, rather than a change in the process.

14

• The process above is in apparent statistical
  control. Notice that all points lie within the
  upper control limits (UCL) and the lower control
  limits (LCL). This process exhibits only common
  cause variation.

15

• The process above is out of statistical control
  at the fifth data point. This single point can be
  found outside the control limits (above them).
  This means that a source of special cause
  variation is present at that time. The likelihood
  of this happening by chance is only about 3 in
  1,000.

16
Typical cycle in SPC

• the process is highly variable and out of
  statistical control.
• as special causes of variation are found and
  fixed, the process comes into statistical control
• through process improvement, variation is
  reduced, control limits are narrowed

17

• Eliminating special cause variation keeps the
  process in control
• process improvement
• reduces the process variation and moves the
  control limits in toward the centerline of the
  process.

18
Strategies for detecting out-of-control
conditions

• If a data point falls outside the control limits,
  we assume that the process is probably out of
  control and that an investigation is warranted to
  find and eliminate the cause or causes.
• This does not mean that when all points fall
  within the limits, the process is in control.

19
If the plot looks non-random it also is not
in-control

• if the data points exhibit some form of
  systematic behavior, there is still something
  wrong
• "in control" implies that all points are between
  the control limits and they form a random pattern

20
General rules for detecting out of control
situations
21
If the plot looks non-random it also is not
in-control

• Any Point Above or below 3 Sigma
• 2 Out of the Last 3 Points Above or below 2 Sigma
• 4 Out of the Last 5 Points Above or below 1
  Sigma
• 8 Consecutive Points Above or below Control Line
• 6 in a row trending up or down
• 14 in a row alternating up and down

22

Trend Rules for detecting out of control or
non-random situations
6 in a row trending up or down
14 in a row alternating up and down
23
These Rules are based on Probability

• For a normal distribution, the probability of
  encountering a point outside 3 sigma is 0.3.
• This is a rare event (about 3 out of 1000)
• Therefore, if we observe a point outside the
  control limits, we conclude the process has
  shifted and is unstable.

24
Rules are based on Probability

• Similarly, we can identify other events that are
  equally rare and use them as flags for
  instability.
• The probability of observing
• two points out of three in a row
• between 2
  sigma and 3 sigma
• or
• four points out of five in a
  row
• between 1
  sigma and 2 sigma
• is also
  about 0.3.

25
Types of Errors 3-sigma Control Limits are a
good balance point between two types of errors

• Type I or alpha errors occur when a point falls
  outside the control limits even though no special
  cause exists. The result is a witch-hunt for
  special causes and unnecessary process
  adjustment. The tampering usually distorts a
  stable process as well as wasting time and
  energy.
• Type II or beta errors occur when you miss a
  special cause because the chart isn't sensitive
  enough to detect it. In this case, you will go
  along unaware that the problem exists and thus
  unable to fix it.

26
Types of errors

• All process control is vulnerable to these Type I
  and Type II errors.
• The reason that 3-sigma control limits balance
  the risk of error is that, for normally
  distributed data, data points will fall inside
  3-sigma limits 99.7 of the time when a process
  is in control.
• This makes the witch hunts infrequent but still
  makes it likely that unusual causes of variation
  will be detected.

27
Control Limits vs. Specifications

• Control Limits are used to determine if the
  process is in a state of statistical control.
• (is producing consistent output)
• Specification Limits are used to determine if the
  product will function in the intended fashion.

28
Control Limits vs. Specifications

• Control limits are derived from the actual output
  of the process, while specification limits are
  imposed on the process
• Control limits should be well within the
  specification limits
• Control limits should be centered on the midpoint
  of the specification

29
Control Limits vs. Specifications

• A capable process is one where almost all the
  measurements fall inside the specification
  limits.

30
Process Capability Index

• We are often required to compare the output of a
  stable process with the process specifications to
  make a statement about how well the process meets
  specification.
• To do this we compare the process specification
  limits with the natural variability of a stable
  process. 

Process Capability Index Cp USL LSL
6 s
31
Value of Cp for Different Processes

• Cp lt 1.0 the process is not capable
• When Cp 1.00 the process is just capable
• As Cp gt 1.00 the process becomes more capable

32
All control charts have three basic components

• a centerline, usually the mathematical average of
  all the samples plotted.
• upper and lower statistical control limits that
  define the constraints of common cause
  variations.
• performance data plotted over time.

33
Control Charts based on Sampling
Distributions

• X-bar chart - the sample means are plotted in
  order to control the average value
• R chart - the sample ranges are plotted in order
  to control the variability

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Importance of Subgroups and Averages

• Individual readings taken from a process may not
  always be normally distributed
• If data is collected from a process in subgroups
  and the average of these is used instead of
  individual readings, the resulting distribution
  will always approach a normal distribution
• This is The Central Limit Theorem

36
X-Bar and R Charts
The two types of charts go together when
monitoring variables, because they measure the
two critical parameters central tendency and
dispersion.
37
It is possible that an out-of-control signal will
appear on one kind of chart and not the other.
38
All control charts have three basic components

• a centerline, usually the mathematical average of
  all the samples plotted.
• upper and lower statistical control limits that
  define the constraints of common cause
  variations.
• performance data plotted over time.

39
Common Cause
UCL
X
SCALE
LCL
40
Special Cause
UCL
X
SCALE
LCL
41
Variables Control Chart
Out ofcontrol
Abnormal variationdue to assignable sources
1020
UCL
1010
1000
Mean
Normal variationdue to chance
990
LCL
980
Abnormal variationdue to assignable sources
970
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Sample number
42
If the plot looks non-random it also is not
in-control

• Any Point Above or below 3 Sigma
• 2 Out of the Last 3 Points Above or below 2 Sigma
• 4 Out of the Last 5 Points Above or below 1
  Sigma
• 8 Consecutive Points Above or below Control Line
• 6 in a row trending up or down
• 14 in a row alternating up and down

43
Control Charts based on Sampling Distributions

• X-bar chart - the sample means are plotted in
  order to control the average value
• R chart - the sample ranges are plotted in order
  to control the variability

44
Importance of Subgroups and Averages

• Individual readings taken from a process may not
  always be normally distributed
• If data is collected from a process in subgroups
  and the average of these is used instead of
  individual readings, the resulting distribution
  will always approach a normal distribution This
  is
• The Central Limit Theorem

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46
Observations from Sample Distribution
Sample number
47
X-Bar and R Charts
The two types of charts go together when
monitoring variables, because they measure the
two critical parameters central tendency and
dispersion.
48
It is possible that an out-of-control signal will
appear on one kind of chart and not the other.
49
Mean and Range Charts
Detects shift
Does notdetect shift
R-chart
50
Mean and Range Charts
UCL
LCL
Does notdetect shift
51
X-Bar and R Control Charts
The X-Bar Chart and the R Chart are not only used
together, they are calculated from the same raw
data.
Consider that we have a precision made piece
coming off of an assembly line. We wish to see if
the process resulting in the object diameter is
in control.
52
X-Bar and R Control Charts Procedure
Take a sample of FIVE objects and measure each.
Calculate the average of the five. This is one
data point for the X-Bar Chart. Calculate the
range (largest minus smallest) of the five. This
is one data point for the R Chart.
53
X-Bar and R Control Charts Procedure
Repeat these two steps twenty (20) times. You
will have 20 "X-Bar" points and 20 "R" points.
54
X-Bar and R Control Charts Procedure
Calculate the average of the 20 X bar points-
yes, the average of the averages. This value is
X Double Bar, and is the centerline of the
X-Bar Chart.
55
X-Bar and R Control Charts Procedure
Calculate the average of the twenty R points.
This is called R Bar This is the centerline of
the R chart, and also is used in calculating the
control limits for both the X-Bar chart and the R
chart.
56
The Only Tricky Part

• Calculating the upper and lower control limits
  for the X-Bar and R control charts, and the
  process standard deviation.
• Use the following equations for our example
• diameter of precision made piece coming off of an
  assembly line
• (sample size of
  FIVE)

57
X-Bar Control Limits
Upper Control Limit for X-Bar chart X double
bar .577 R bar Lower Control Limit for
X-Bar chart X double bar - .577 R bar
58
R Control Limits
Upper Control Limit for R Chart 2.11
Rbar Lower Control Limit for R Chart 0
Rbar
59
X-Bar and R Control Charts Procedure
Plot the original X Bar and R points, twenty of
each, on the respective X and R control
charts. Identify points which fall outside of
the control limits. These points are due to
unanticipated or unacceptable causes.
60
X-Bar and R Control Charts Procedure
The sample size is traditionally 5, as in our
example. Control charts can and do use sample
sizes other than 5, but the control limit
factors, derived by statisticians, change (for
our example .577 for the X-Bar chart)
61
X-Bar Control Charts
Find the UCL and LCL using the following
equations UCL X double bar (A2)Rbar CL X
double bar LCL X double bar - (A2)Rbar
62
R Control Charts
Find the UCL and LCL with the following formulas
UCL (D4)Rbar CL Rbar LCL(D3)Rbar with
D3 and D4 can be found in the following table
63
Samples of 5
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X-Bar Control Limits

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R Control Limits
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X Bar-r-chart

• The process standard deviation for each graph is
  calculated by a formula based on the average of
  the range values, (Rbar), and another table.

68
Pareto Principle

• The Vital Few and Trivial Many Rule
• Predictable Imbalance
• 8020 Rule

69
Named after Vilfredo Pareto -an Italian economist

• He observed in 1906 that 20 of the Italian
  population owned 80 of Italy's wealth
• He then noticed that 20 of the pea pods in his
  garden accounted for 80 of his pea crop each
  year

70
The Pareto Principle

• A small number of causes is responsible for a
  large percentage of the effect-
• -usually a 20-percent to 80-percent ratio.
• This basic principle translates well into quality
  problems - most quality problems result from a
  small number of causes.
• You can apply this ratio to almost anything, from
  the science of management to the physical world

71
Addressing the most troublesome 20 of the
problem will solve 80 of it.   Within your
process, 20 of the individuals will cause 80 of
your headaches.   Of all the solutions you
identify, about 20 are likely to remain viable
after adequate analysis. 80 of the work is
usually done by 20 of the people.
72
80 of the quality can be gotten in 20 of the
time -- perfection takes 5 times longer 20 of
the defects cause 80 of the problems. Project
Managers know that 20 of the work (the first 10
and the last 10) consume 80 of the time and
resources.
73
A Pareto chart is a useful tool for graphically
depicting these and other relationships It is a
simple Histogram style graph that ranks problems
in order of magnitude to determine the priorities
for improvement activities The goal is to target
the largest potential improvement area then move
on to the next, then next, and in so doing
address the area of most benefit first The chart
can help show you where allocating time, human,
and financial resources will yield the best
results.
74
While the rule is not an absolute, one should use
it as a guide and reference point to ask whether
or not you are truly focusing on 20 - The
Vital Few or 80 - The Trivial Many True
progress results from a consistent focus on the
20 most critical objectives.
the
75
The simplicity of the Pareto concept makes it
prone to be underestimated and overlooked as a
key tool for quality improvement. Generally,
individuals tend to think they know the important
problem areas requiring attention if they
really know, why do problem areas still exist?
76
Although the idea is quite simple, to gain a
working knowledge of the Pareto Principle and its
application, it is necessary to understand the
following basic elements
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Pareto Analysis

• Creating an tabular array of representative
  sample data that ranks the parts to the whole
• with the objective to use the facts to find the
  highest concentration of quality improvement
  potential in the fewest number of projects or
  remedies
• Thus achieving the highest return for the
  investment.

78
Pareto Analysis of Printing Defects
79
Pareto Diagram

• The Category Contribution, the causes of whatever
  is being investigated, are listed across the
  bottom, and a percentage is assigned for each
  (Relative Frequency) to total 100. A vertical
  bar chart is constructed, from left to right, in
  order of magnitude, using the percentages for
  each category.

80
Pareto Diagram is a combined bar chart and line
diagram based on cumulative percentages.
80 improvement in quality or performance can
reasonably be expected by eliminating 20 of the
causes of unacceptable quality or performance
81
Pareto Diagram of Total Printing Defects
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Relative Frequency

• (Category Contribution) / (Total of all
  Categories) x 100 expressed in bar chart form.

84
Cumulative Frequency

• (Relative Frequency of Category Contribution)
  (Previous Cumulative Frequency) expressed as a
  line graph

85
Break Point

• The percentage point on the line graph for
  Cumulative Frequency at which there is a
  significant decrease in the slope of the plotted
  line

86
Vital Few

• Category Contributions that appear to the left of
  the Break Point account for the bulk of the
  effect

87
Trivial Many

• Category Contributions that appear to the right
  of the Break Point, which account for the least
  of the effect.

88
Pareto Diagram Analysis

• Pareto analysis provides the mechanism to control
  and direct effort by fact, not by emotion.
• It helps to clearly establish top priorities and
  to identify both profitable and unprofitable
  targets.
• In addition to selecting and defining key
  quality improvement programs

89

• Prioritize problems, goals, and objectives
• Identify root causes
• Select key customer relations and service
  programs
• Select key employee relations improvement
  programs
• Select and define key performance improvement
  programs
• Address the Vital Few and the Trivial Many causes
  of nonconformance
• Maximize research and product development time
• Verify operating procedures and manufacturing
  processes
• Product or services sales and distribution
• Allocate physical, financial and human resources

90
For a General Manager
The value of the Pareto Principle is that it
focuses efforts on the 20 percent that matters.
Of the things you do during your day, only 20
percent really matter. Those 20 percent produce
80 percent of your results. Identify and focus
on those things.
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To Create a Pareto Chart   Select the items
(problems, issues, actions, defects, etc.) to be
compared.   Select a standard for measurement.
  Gather necessary data Arrange the items on
the horizontal axis in a descending order
according to the measurements you selected.
  Draw a bar graph where the height is the
measurement you selected.