Control charts for variables

Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.
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Quality Control
Control Charts for
Variables
Dr. Mahmoud Abbas Mahmoud
Asst. Prof.
2016

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The variation concept is a law of nature in
that no two natural items in any category are
the same.
Variation

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 The variation may be quite large and easily
noticeable
 The variation may be very small. It may appear
that items are identical; however, precision
instruments will show difference
 The ability to measure variation is necessary
before it can be controlled
Variation

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There are three categories of variation in piece
part production:
1. Within-piece variation: Surface
2. Piece-to-piece variation: Among pieces
produced at the same time
3. Time-to-time variation: Difference in product
produced at different times of the day
Variation

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Materials
Tools
Operators Methods
Measurement
Instruments
Human
Inspection
Performance
EnvironmentMachines
INPUTS PROCESS OUTPUTS
Variation
Sources of Variation in production processes:

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Sources of variation are:
1. Equipment:
1. Toolwear
2. Machine vibration
3. Electrical fluctuations etc.
2. Material
1. Tensile strength
2. Ductility
3. Thickness
4. Porosity etc.
Variation

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3. Environment
1. Temperature
2. Light
3. Radiation
4. Humidity etc.
4. Operator
1. Personal problem
2. Physical problem etc.
Variation

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There is also a reported variation which is due
to the inspection activity.
Variation due to inspection should account for
one tenth of the four other sources of
variation.
Variation

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Variation may be due to chance causes
(random causes) or assignable causes.
When only chance causes are present, then the
process is said to be in a state of statistical
control. The process is stable and predictable.
Variation

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 Variable data
x-bar and R-charts
x-bar and s-charts
Charts for individuals (x-charts)
 Attribute data
For “defectives” (p-chart, np-chart)
For “defects” (c-chart, u-chart)
Control Charts

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Control
Charts
R
Chart
Variables
Charts
Attributes
Charts
X
Chart
P
Chart
C
Chart
Continuous
Numerical Data
Categorical or Discrete
Numerical Data
Control Charts

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The control chart for variables is a means of
visualizing the variations that occur in the
central tendency and the mean of a set of
observations. It shows whether or not a
process is in a stable state.
Control Charts for Variables

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Control Charts
Figure 5-1 Example of a control chart

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Control Charts
Figure 5-1 Example of a method of reporting inspection results

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The objectives of the variable control charts are:
1. For quality improvement
2. To determine the process capability
3. For decisions regarding product specifications
4. For current decisions on the production process
5. For current decisions on recently produced items
Variable Control Charts

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Procedure for establishing a pair of control charts for
the average Xbar and the range R:
1. Select the quality characteristic
2. Choose the rational subgroup
3. Collect the data
4. Determine the trial center line and control limits
5. Establish the revised central line and control limits
6. Achieve the objective
Control Chart Techniques

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The Quality characteristic must be measurable.
It can expressed in terms of the seven basic units:
1. Length
2. Mass
3. Time
4. Electrical current
5. Temperature
6. Substance
7. Luminosity
as appropriate.
Quality Characteristic

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A rational subgroup is one in which the variation
within a group is due only to chance causes.
Within-subgroup variation is used to determine the
control limits.
Variation between subgroups is used to evaluate
long-term stability.
Rational Subgroup

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There are two schemes for selecting the subgroup
samples:
1. Select subgroup samples from product or
service produced at one instant of time or as
close to that instant as possible (Instant-time
method)
2. Select from product or service produced over a
period of time that is representative of all the
products or services (Period-of-time method)
Rational Subgroup

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The first scheme will have a minimum variation
within a subgroup.
The second scheme will have a minimum variation
among subgroups.
The first scheme is the most commonly used since
it provides a particular time reference for
determining assignable causes.
The second scheme provides better overall results
and will provide a more accurate picture of the
quality.
Rational Subgroup

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 As the subgroup size increases, the control
limits become closer to the central value,
which make the control chart more sensitive
to small variations in the process average
 As the subgroup size increases, the inspection
cost per subgroup increases
 When destructive testing is used and the item
is expensive, a small subgroup size is required
Subgroup Size

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 From a statistical basis a distribution of
subgroup averages are nearly normal for
groups of 4 or more even when samples are
taken from a non-normal distribution
 When a subgroup size of 10 or more is used,
the s chart should be used instead of the R
chart.
 See Table 5-1 for sample sizes
Subgroup Size

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Data collection can be accomplished using the
type of figure shown in Figure 5-2.
It can also be collected using the method in
Table 5-2.
It is necessary to collect a minimum of 25
subgroups of data.
A run chart can be used to analyze the data in
the development stage of a product or prior to
a state of statistical control
Data Collection

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Run Chart
Figure 5-4 Run Chart for data of Table 5-2

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Trial Central Lines
Central Lines are obtained using:
1 1
g g
i i
i i
i
i
X R
X and R
g g
where
X average of subgroup averages
X average of the ith subgroup
g number of subgroups
R average of subgroup ranges
R range of the ith subgroup
 
 





 

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Trial Control Limits
Trial control limits are established at ±3 standard
deviations from the central value
3 3
3 3
R RX X
R RX X
X
R
UCL X UCL R
LCL X LCL R
where
UCL=upper control limit
LCL=lower control limit
population standard deviation of the subgroup averages
population standard deviation of the range
 
 


   
   



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In practice calculations are simplified by using
the following equations where A2,D3 and D4 are
factors that vary with the subgroupsize and are
found in Table B of the Appendix.
2 4
2 3
RX
RX
UCL X A R UCL D R
LCL X A R LCL D R
  
  

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Figure 5-5 Xbar and R chart for preliminary data with trial control limits

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Revised Central Lines
d d
new new
d d
d
d
d
X X R R
X and R
g g g g
where
X discarded subgroup averages
g number of discarded subgroups
R discarded subgroup ranges
 
 
 



 

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Discarded Points

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Standard Values
0
0 0 0
2
new new
R
X X R R and
d
  
0 0 2 0
0 0 1 0
RX
RX
UCL X A UCL D
LCL X A LCL D
 
 
  
  

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Figure 5-6 Trial control limits and revised control limits for Xbar and R charts

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Achieve the Objective
Figure 5-7 Continuing use of control charts, showing improved quality

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Sample Standard Deviation
Control Chart
For subgroup sizes >=10, an s chart is more
accurate than an R Chart. Trial control limits are
given by:
1 1
3 4
3 3
g g
i ii i
sX
sX
s X
s X
g g
UCL X A s UCL B s
LCL X A s LCL B s
 
 
  
  
 

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Revised Limits for s chart
0
0
0 0
4
0 0 6 0
0 0 5 0
4 5 6, , ,
d
new
d
d
new
d
sX
sX
d
X X
X X
g g
s s s
s s
g g c
UCL X A UCL B
LCL X A LCL B
where
s discarded subgroup averages
c A B B factors found in Table B

 
 

 


  

  
  





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Process in Control
 When special causes have been eliminated
from the process to the extent that the points
plotted on the control chart remain within the
control limits, the process is in a state of
control
 When a process is in control, there occurs a
natural pattern of variation
State of Control

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State of Control
Figure 5-9 Natural pattern of variation of a control chart

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Types of errors:
 Type I, occurs when looking for a special
cause of variation when in reality a common
cause is present
 Type II, occurs when assuming that a common
cause of variation is present when in reality
there is a special cause
State of Control

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When the process is in control:
1. Individual units of the product or service will
be more uniform
2. Since the product is more uniform, fewer
samples are needed to judge the quality
3. The process capability or spread of the
process is easily attained from 6ơ
4. Trouble can be anticipated before it occurs
State of Control

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When the process is in control:
5. The % of product that falls within any pair of
values is more predictable
6. It allows the consumer to use the producer’s
data
7. It is an indication that the operator is
performing satisfactorily
State of Control

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Common
Causes
Special
Causes
45

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State of Control
Figure 5-11 Frequency Distribution of subgroup averages with control limits

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When a point (subgroup value) falls outside its
control limits, the process is out of control.
Out of control means a change in the process due
to a special cause. A process can also be
considered out of control even when the points
fall inside the 3ơ limits
State of out-of-Control

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 It is not natural for seven or more consecutive
points to be above or below the central line.
 Also when 10 out of 11 points or 12 out of 14
points are located on one side of the central
line, it is unnatural.
 Six points in a row are steadily increasing or
decreasing indicate an out of control situation
State of out-of-Control

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1. Change or jump in level.
2. Trend or steady change in level
3. Recurring cycles
4. Two populations (also called mixture)
5. Mistakes
Out-of-Control Condition

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Patterns in out-of-Control Charts
Figure 5-12 Some unnatural runs-process out of control

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Patterns in out-of-Control Charts
Figure 5-13 Simplified rule for out-of-control pattern

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Out-of-Control Patterns
Change or jump in level Trend or steady change in level
Recurring cycles Two populations

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Specifications
Figure 5-18 Comparison of individual values compared to averages

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Calculations of the average for both the individual
values and for the subgroup averages are the
same. However the sample standard deviation is
different.
Specifications
X
X
n
where
population standard deviation of subgroup averages
population standard deviation of individual values
n=subgroup size
If we assume normality, then the population standard deviation
can be







4
s
estimated from
c
 

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If the population from which samples are taken
is not normal, the distribution of sample
averages will tend toward normality provided
that the sample size, n, is at least 4. This
tendency gets better and better as the sample
size gets larger. The standardized normal can
be used for the distribution averages with the
modification.
Central Limit Theorem
X
X X
Z
n
 
 
 
 

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Figure 5-19 Illustration of central limit theorem

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Figure 5-20 Dice illustration of central limit theorem

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Figure 5-21 Relationship of limits, specifications, and distributions
Control Limits & Specifications

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 The control limits are established as a function
of the average
 Specifications are the permissible variation in
the size of the part and are, therefore, for
individual values
 The specifications or tolerance limits are
established by design engineers to meet a
particular function
Control Limits & Specifications

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 The process spread will be referred to as the
process capability and is equal to 6σ
 The difference between specifications is called
the tolerance
 When the tolerance is established by the design
engineer without regard to the spread of the
process, undesirable situations can result
Process Capability & Tolerance

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Three situations are possible:
 Case I: When the process capability is less
than the tolerance 6σ<USL-LSL
 Case II: When the process capability is equal
to the tolerance 6σ=USL-LSL
 Case III: When the process capability is
greater than the tolerance 6σ >USL-LSL

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Case I: When the process capability is less than
the tolerance 6σ<USL-LSL
Figure 5-24 Case I 6σ<USL-LSL

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 The range over which the natural variation of
a process occurs as determined by the
system of common causes
 Measured by the proportion of output that
can be produced within design specifications
Process Capability

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This following method of calculating the process
capability assumes that the process is stable or in
statistical control:
 Take 25 (g) subgroups of size 4 for a total
of 100 measurements
 Calculate the range, R, for each subgroup
 Calculate the average range, RBar= ΣR/g
 Calculate the estimate of the population
standard deviation
 Process capability will equal 6σ0
Process Capability
0
2
R
d
 

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The process capability can also be obtained by using
the standard deviation:
 Take 25 (g) subgroups of size 4 for a total of
100 measurements
 Calculate the sample standard deviation, s, for
each subgroup
 Calculate the average sample standard
deviation, sbar = Σs/g
 Calculate the estimate of the population
standard deviation
Process capability will equal 6σo
Process Capability
0
4
s
c
 

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Process capability and tolerance are combined to
form the capability index.
Capability Index
0
0
6
6
p
p
USL LSL
C
where C capabilityindex
USL LSL tolerance
process capability





 


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The capability index does not measure process
performance in terms of the nominal or target
value. This measure is accomplished by Cpk.
Capability Index
0
{( ) ( )
3
6
pk
p
Min USL X or X LSL
C
where C capabilityindex
USL LSL tolerance
process capability


 


 


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1. The Cp value does not change as the process
center changes
2. Cp=Cpk when the process is centered
3. Cpk is always equal to or less than Cp
4. A Cpk = 1 indicates that the process is
producing product that conforms to
specifications
5. A Cpk < 1 indicates that the process is
producing product that does not conform to
specifications
Capability Index

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6. A Cp < 1 indicates that the process is not
capable
7. A Cp=0 indicates the average is equal to
one of the specification limits
8. A negative Cpk value indicates that the
average is outside the specifications
Capability Index

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Six Sigma is both a quality management
philosophy and a methodology that focuses
on reducing variation, measuring defects, and
improving quality of products, processes and
services.
Six Sigma

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Charts for Better Operator Understanding:
1. Placing individual values on the chart: This
technique plots both the individual values and
the subgroup average. Not recommended since
it does not provide much information.
2. Chart for subgroup sums: This technique plots
the subgroup sum, ΣX, rather than the group
average, Xbar.
Different Control Charts
( )
( )
X X
X X
UCL n UCL
UCL n LCL



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Charts for Variable Subgroup Size:
Used when the sample size is not the same
 Different control limits for each subgroup
 As n increases, limits become narrower
 As n decreases, limits become wider apart
 Difficult to interpret and explain
 To be avoided

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Chart for Trends:
Used when the plotted points have an upward
or downward trend that can be attributed to
an unnatural pattern of variation or a natural
pattern such as tool wear.
The central line is on a slope, therefore its
equation must be determined.

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Use Least Square Calculations
2
2 2
2 2
( )( ) ( )( )
( )
( )( )
( )
X a bG
X G G G X
a
g G G
g G X G X
b
g G G
where
X subgroup average and represents the vertical axis
a= point of intercept on the vertical axis
b=slope of the line
G=subgroup number a
 







   
 
  
 
nd represents the horizontal axis
g=number of subgroups
Chart for Trends

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Used when we cannot have
multiple observations per
time period
Value Xbar R
44
46
54 48.00 10
38 46.00 16
49 47.00 16
46 44.33 11
45 46.67 4
31 40.67 15
55 43.67 24
37 41.00 24
42 44.67 18
43 40.67 6
47 44.00 5
51 47.00 8
X
X
n
R
R
n




NOTE: n here is equal to 12, NOT 14
Chart for Moving Average and
Moving Range
An example

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Extreme readings have a greater effect than in
conventional charts. An extreme value is used
several times in the calculations, the number of
times depends on the averaging period.
Chart for Moving Average and
Moving Range

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This is a simplified variable control chart.
 Minimizes calculations
 Easier to understand
 Can be easily maintained by operators
 Recommended to use a subgroup of 3, then
all data is used.
Chart for Median and Range

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2.660
2.660
3.267
(0)
x
x
R
R
X R
X R
g g
UCL X R
LCL X R
UCL R
LCL R
 
 
 


 
To use those equations, you have to use a moving range with n=2
Chart for Individual Values

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Charts with Non-Acceptance
Limits
Non-Acceptance limits have the same
Relationship to averages as specifications
have to individual values. Control Limits tell
what the process is capable of doing, and reject
limits tell when the product is conforming to
specifications.

Control charts for variables

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