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```

Variable - a single quality characteristic
that can be measured on a numerical scale.
When working with variables, we should
monitor both the mean value of the
characteristic and the variability associated
with the characteristic.
“One of the axioms or truisms of
manufacturing is that no two objects are
ever made exactly alike”
 Types
of variation
Within the piece
 Piece to piece
 Time to time

 Types
of variation causes:
Chance (Natural/Common) causes.
 Assignable (Special) cause.

Chance causes
 Natural
 Expected
 Numerous
 Small importance
 Difficult to detect or identify
 In a state of statistical control.

Assignable causes
 Large in magnitude
 Easy to detect or identify
 Out of control.

Control Charts Procedures:
1.
2.
3.
4.
5.
6.
Select quality characteristic
Choose rational subgroup
Collect data
Determine trial central line and control
limits
Establish the revised central line and
control limits
Achieve objective
X bar chart monitors the between sample
variability
 R chart monitors the within sample
variability.

 Guidelines on size:
 With larger subgroups, the control chart
becomes more sensitive to small variation
 With larger subgroups, the inspection cost
per subgroup increases
 If destructive testing is required, a minimal
number is beneficial
 Statistically, subgroups of 4 or more will
have their averages normally distributed
regardless of their population distribution
 Subgroup of 5 are widely used in industry
Control Limits for the X-bar chart
UCL  x  A 2 R
Center Line  x
LCL  x  A 2 R

A2 is found in Appendix VI for various values of
n.
Control Limits for the R chart
UCL  D4 R
Center Line  R
LCL  D3 R

D3 and D4 are found in Appendix VI for various
values of n.
Estimating the Process Standard Deviation
 The process standard deviation can be estimated
using a function of the sample average range.
R
 
d2

This is an unbiased estimator of 
Trial Control Limits
If the process is in control for the m samples
collected, then the system was in control in
the past.
 If all points plot inside the control limits
and no systematic behavior is identified,
then the process was in control in the past,
and the trial control limits are suitable for
controlling current or future production.

Control Limits, Specification Limits, and Natural
Tolerance Limits
Control limits are functions of the natural
variability of the process
 Natural tolerance limits represent the
natural variability of the process (usually set
at 3-sigma from the mean)
 Specification limits are determined by
developers/designers.

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