Statistical Process Control
Douglas M. Stewart, Ph.D.
The Anderson Schools of Management
The University of New Mexico
Quality Control (QC)
Control – the activity of ensuring
conformance to requirements and taking
corrective action when necessary to
correct problems
Importance
Daily management of processes
Prerequisite to longer-term improvements
Designing the QC System
Quality Policy and Quality Manual
Contract management, design control and
purchasing
Process control, inspection and testing
Corrective action and continual improvement
Controlling inspection, measuring and test
equipment (metrology, measurement system analysis
and calibration)
Records, documentation and audits
Example of QC: HACCP System
1. Hazard analysis
2. Critical control points
3. Preventive measures with critical limits for
each control point
4. Procedures to monitor the critical control
points
5. Corrective actions when critical limits are
not met
6. Verification procedures
7. Effective record keeping and documentation
Inspection/Testing Points
Receiving inspection
In-process inspection
Final inspection
5
Receiving Inspection
Spot check procedures
100 percent inspection
Acceptance sampling
6
Acceptance Sampling
Lot received for inspection
Sample selected and analyzed
Results compared with acceptance criteria
Accept the lot
Send to production
or to customer
Reject the lot
Decide on disposition
7
Pros and Cons
of Acceptance Sampling
 Arguments for:
 Provides an assessment
of risk
 Inexpensive and suited
for destructive testing
 Requires less time than
other approaches
 Requires less handling
 Reduces inspector fatigue
Arguments against:
 Does not make sense for
stable processes
 Only detects poor quality;
does not help to prevent it
 Is non-value-added
 Does not help suppliers
improve
In-Process Inspection
What to inspect?
Key quality characteristics that are related
to cost or quality (customer requirements)
Where to inspect?
Key processes, especially high-cost and
value-added
How much to inspect?
All, nothing, or a sample
9
Economic Model
C1 = cost of inspection and removal of
nonconforming item
C2 = cost of repair
p = true fraction nonconforming
Breakeven Analysis: p*C2 = C1
If p > C1 / C2 , use 100% inspection
If p < C1 / C2 , do nothing
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Human Factors in Inspection
complexity
defect rate
repeated inspections
inspection rate
Inspection should never be a means of assuring
quality. The purpose of inspection should be to gather
information to understand and improve the processes
that produce products and services.
Gauges and
Measuring Instruments
Variable gauges
Fixed gauges
Coordinate measuring machine
Vision systems
12
Examples of Gauges
Metrology - Science of Measurement
Accuracy - closeness of agreement
between an observed value and a
standard
Precision - closeness of agreement
between randomly selected individual
measurements
Repeatability and
Reproducibility
Repeatability (equipment variation) –
variation in multiple measurements by an
individual using the same instrument.
Reproducibility (operator variation) variation in the same measuring
instrument used by different individuals
Repeatability and
Reproducibility Studies
Quantify and evaluate the capability of a
measurement system
Select m operators and n parts
Calibrate the measuring instrument
Randomly measure each part by each
operator for r trials
Compute key statistics to quantify
repeatability and reproducibility
Reliability and Reproducibility
Studies(2)
Measuremen t (M) made by
Operators (i from 1 to m) on
Parts (j from 1 to n) in
Trials (k from 1 to r)


  M ijk 


j
k

 average for each operator
xi 
nr
xD  max ( xi )  min ( xi ) difference (range) of operator averages
i
i
R ij  max ( M ijk )  min ( M ijk ) range for each part for each operator
k
k


  Rij 


j

 average range for each operator
Ri 
n


  Ri 
 average range of all
R  i
m
Reliability and Reproducibility
Studies(3)
Control limit of ranges Rij  D4  R
Use number tri als (r) for n in table. Check
for randomness of errors.
Repeatabil ity or Equipment Variation
EV  K1  R
K1 is a constant t ied to # of trials
Reproducib ility or operator (appraisal ) variation
 EV 2 
 K 2 is a constant t ied to # of operators
AV  K 2  xD   
n

r


Repeatabil ity and Reproducib ility
2
R&R 
EV 2   AV 2
Results are in actual units measured. Customary to express
as percentage s.
Under 10% - Acceptable
10 - 30% - ? based on importance and repair cost
Over 30% - Unaccepta ble
R&R Constants
Number of
Trials
K1
Number of
Operators
K2
2
3
4
5
4.56 3.05 2.50 2.21
2
3
4
5
3.65 2.70 2.30 2.08
R&R Evaluation
Under 10% error - OK
10-30% error - may be OK
over 30% error - unacceptable
R&R Example
 R&R Study is to be conducted on a gauge being used to
measure the thickness of a gasket having specification
of 0.50 to 1.00 mm. We have three operators, each
taking measurement on 10 parts in 2 separate trials.
x1  0.830
x2  0.774
x3  0.829
R1  0.037
R2  0.034
R3  0.017
Calibration
Calibration - comparing a measurement
device or system to one having a known
relationship to national standards
Traceability to national standards
maintained by NIST, National Institute of
Standards and Technology
Statistical Process Control (SPC)
A methodology for monitoring a process to
identify special causes of variation and
signal the need to take corrective action
when appropriate
SPC relies on control charts
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Common
Causes
Special
Causes
Histograms do not
take into account
changes over time.
Control charts
can tell us
when a process
changes
Control Chart Applications
Establish state of statistical
control
Monitor a process and signal
when it goes out of control
Determine process capability
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Commonly Used Control Charts
Variables 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)
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Developing Control Charts
1. Prepare
 Choose measurement
 Determine how to collect data, sample size,
and frequency of sampling
 Set up an initial control chart
2. Collect Data
 Record data
 Calculate appropriate statistics
 Plot statistics on chart
Next Steps
3. Determine trial control limits
 Center line (process average)
 Compute UCL, LCL
4. Analyze and interpret results
 Determine if in control
 Eliminate out-of-control points
 Recompute control limits as
necessary
Typical Out-of-Control Patterns
Point outside control limits
Sudden shift in process average
Cycles
Trends
Hugging the center line
Hugging the control limits
Instability
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Shift in Process Average
Identifying Potential Shifts
Cycles
Trend
Final Steps
5. Use as a problem-solving tool
 Continue to collect and plot data
 Take corrective action when
necessary
6. Compute process capability
Process Capability
 Capability Indices
UTL  LTL
6
if C p  1 is defined as capable (1.5 more often the minimum)
Cp 
Example : Part specificat ion is 10.75mm  .25mm   0.0868mm
11.00  10.50
Cp 
 0.96
6  0.0868
Process Capability (2)
UTL  
3
  LTL
C pl 
3
C pk  min C pl , C pu 
C pu 
C pu 
C pl 
C pk  C p 1  K  where K 
11.0  10.7171
 1.086
3  0.0868
10.7171  10.5
 0.834
3  0.0868
2  T
Tolerance
Example : same as above, but assume process is centered at 10.7171mm
Cp
C pm 
1
  T 
2
T is the Target
2
0.960
C pm 
10.7171  10.75
2
1
0.8682
 0.8977
Capability Versus Control
Control
Capability
Capable
In Control
Out of Control
IDEAL
Not Capable
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Process Capability Calculations
Excel Template
Special Variables Control Charts
x-bar and s charts
x-chart for individuals
Charts for Attributes
Fraction nonconforming (p-chart)
Fixed sample size
Variable sample size
np-chart for number nonconforming
Charts for defects
c-chart
u-chart
Control Chart Selection
Quality Characteristic
variable
attribute
defective
n>1?
no
x and MR
yes
n>=10 or no
computer?
yes
x and s
defect
x and R
constant
sample
size?
yes
no
p-chart with
variable sample
size
constant
sampling
unit?
p or
np
yes
no
c
u
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Control Chart Design Issues
Basis for sampling
Sample size
Frequency of sampling
Location of control limits
65
Pre-Control
LTL
Red
Zone
UTL
Green Zone
Red
Zone
nominal
value
Yellow Zones
67
SPC Implementation
Requirements
Top management commitment
Project champion
Initial workable project
Employee education and training
Accurate measurement system
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