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Session IX
“Control Charts for Variables”
Variation
The variation concept is a law of nature in that no
two natural items are the same.
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
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”
Materials
Tools
Operators MethodsMeasurement
Instruments
HumanInspectionPerformance
EnvironmentMachines
INPUTS PROCESS OUTPUTS
Sources of Variation in production processes:
Variation
Sources of Variation1. Equipment:Tool wearMachine vibrationElectrical fluctuations etc.
2. MaterialTensile strengthDuctilityThicknessPorosity etc.
Sources of Variation
3. Environment Temperature Light Radiation Humidity etc.
3. Operator Personal problem Physical problem etc.
Sources of Variation
There is also a reported variation which is due to the
inspection activity which should be one tenth of the
four other sources of variation.
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.
Control Charts
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)
ControlCharts
RChart
VariablesCharts
AttributesCharts
XChart
PChart
CChart
Continuous
Numerical DataCategorical or Discrete
Numerical Data
Control Charts
Control Charts for Variables
The control chart for variables is a mean 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
Example of a control chart
Control Charts
Example of a method of reporting inspection
results
Control Charts for Variables
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
Control Charts for Variables
Procedure for establishing a pair of control charts for
the average X-bar 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
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. Luminosity
Quality Characteristic
Rational Subgroup
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
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
2. Select from product or service produced over a
period of time that is representative of all the
products or services
Rational Subgroup
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.
Subgroup Size
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
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 6-1 for sample sizes
Data Collection
• Data collection can be accomplished using the
type of figure shown in Figure 6-2. It can also be
collected using the method in Table 6-2.
• 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
Run Chart
Run Chart for data of Table 6-2
1 1
g g
i i
i i
i
i
X R
X and Rg 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
Trial Central Lines
Central Lines are obtained using:
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
Trial Control Limits
Trial control limits are established at ±3 standard
deviations from the central value
2 4
2 3
RX
RX
UCL X A R UCL D R
LCL X A R LCL D R
Trial Control Limits
In practice calculations are simplified by using the
following equations where A2,D3 and D4 are factors
that vary with the subgroup size and are found in
Table B of the Appendix.
Trial Control Limits
X-bar and R chart for preliminary data with trial control limits
d dnew new
d d
d
d
d
X X R RX and R
g g g g
where
X discarded subgroup averages
g number of discarded subgroups
R discarded subgroup ranges
Revised Central Lines
Standard Values
00 0 0
2
new new
RX 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
28
Trial control limits and revised control limits for X-bar and R charts
Achieve the Objective
Continuing use of control charts, showing improved quality
d dnew new
d d
d
d
d
X X R RX and R
g g g g
where
X discarded subgroup averages
g number of discarded subgroups
R discarded subgroup ranges
Revised Central Lines
1 1
3 4
3 3
g g
i ii i
sX
sX
s Xs X
g g
UCL X A s UCL B s
LCL X A s LCL B s
Sample Standard Deviation Control Chart
For subgroup sizes >=10, a s chart is more accurate
than an R Chart. Trial control limits are given by:
0
00 0
4
0 0 6 0
0 0 5 0
4 5 6, , ,
dnew
d
d
new
d
sX
sX
d
X XX X
g g
s s ss 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
Revised Limits for s chart