1. Statistical Process Control (SPC)
Graduate School of Business
University of Colorado
Boulder, CO
Professor Stephen Lawrence

2. Process Control Tools
Process tools assess conditions in existing processes to detect problems that require intervention in order to regain lost control.
Check sheets Pareto analysis
Cause & effect diagrams Scatterplots
Histograms Control charts

3. Check Sheets 27 15 19 20 28 Check sheets explore what and where
an event of interest is occurring.
Attribute Check Sheet
Order Types am-9am 9am-11am 11am-1pm pm-3pm 3pm-5-pm
Emergency
Nonemergency
Rework
Safety Stock
Prototype Order
Other

4. Run Charts
measurement
time

5. SCATTERPLOTS Variable A Variable B x x x x x x xx x x x x xx x x x

6. A B C D E F G PARETO ANALYSIS H Percentage of Occurrences
A method for identifying and separating
the vital few from the trivial many. A B
Percentage of Occurrences C D E F G H I J
Factor

7. CAUSE & EFFECT DIAGRAMS
Classification
Error
Inspection
Pins not
Assigned
Defective
Pins
Received
Damaged
in storage
CPU Chip
BAD
CPU
Maintenance
Equipment
Condition
Employees
Procedures
and Methods
Training
Speed

8. Frequency of Occurrences
HISTOGRAMS
A statistical tool used to show the extent
and type of variance within the system.
Frequency of Occurrences
Outcome

9. Deming’s Theory of Variance
Variation causes many problems for most processes
Causes of variation are either “common” or “special”
Variation can be either “controlled” or “uncontrolled”
Management is responsible for most variation
Categories of Variation
Management
Employee
Controlled Variation
Uncontrolled Variation
Common Cause
Special Cause

10. Causes of Variation What prevents perfection? Process variation...
Natural Causes
Assignable Causes
Inherent to process
Random
Cannot be controlled
Cannot be prevented
Examples
weather
accuracy of measurements
capability of machine
Exogenous to process
Not random
Controllable
Preventable
Examples
tool wear
“Monday” effect
poor maintenance
Common causes: * Inherit to the process
* Random
* Not controllable by operators
* Management is responsible
(e.g., a filling cereal machine may be replaced by a better one)
Assignable causes: * Not part of the process
* Not random
* Operators have control
* Management is responsible for training operators.
(e.g., a machine is not properly set)

11. Product Specification and Process Variation
desired range of product attribute
part of product design
length, weight, thickness, color, ...
nominal specification
upper and lower specification limits
Process variability
inherent variation in processes
limits what can actually be achieved
defines and limits process capability
Process may not be capable of meeting specification!

12. Process Capability LSL Spec USL
Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned.
Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3.
Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target.
Capability - It is the ability to meet customer’s specifications.
In control - A process that does not show signs of assignable causes of variation.

13. Process Capability
Measure of capability of process to meet (fall within) specification limits
Take “width” of process variation as 6
If 6 < (USL - LSL), then at least 99.7% of output of process will fall within specification limits
LSL
Spec
USL

14. Process Capability Ratio
Define Process Capability Ratio Cp as
If Cp > 1.0, process is... capable
If Cp < 1.0, process is... not capable

15. Process Capability -- Example
A manufacturer of granola bars has a weight specification
2 ounces plus or minus 0.05 ounces. If the standard deviation
of the bar-making machine is 0.02 ounces, is the process capable?
USL = = 2.05 ounces
LSL = = 1.95 ounces
Cp = (USL - LSL) / 6
= ( ) / 6(0.02)
= / 0.12 =

16. Process Centering LSL Spec USL
Out of control - The process is out of control because the distribution is not centered around the target value. The assignable cause may be a wrong setting for the particularned.
Not capable - The process is in control but is not capable. This is determined by the large probability of producing parts that do not meet the engineering index (called Cp) is less than 1. Cp has a value of one when the range of the distribution (measure by standard deviations from the mean) equals the range of the tolerances. Capable process have a Cp value of at least 1.3.
Capable - The process is now capable because the entire distribution falls within the tolerance limits. Improvements occur when the range of the distribution is reduce, increasing the probability for producing on target.
Capability - It is the ability to meet customer’s specifications.
In control - A process that does not show signs of assignable causes of variation.

17. Process Centering -- Example
For the granola bar manufacturer, if the process is
incorrectly centered at 2.05 instead of 2.00 ounces, what
fraction of bars will be out of specification?
2.0
LSL=1.95
USL=2.05
Out of spec!
_____ of production will be out of specification!

18. Process Capability Index Cpk
Std dev 
Mean m
If Cpk > 1.0, process is... Centered & capable
If Cpk < 1.0, process is... Not centered &/or not capable

19. Process Capability Index -- Example
A manufacturer of granola bars has a weight specification
2 ounces plus or minus 0.05 ounces. If the standard deviation
of the bar-making machine is s = 0.02 ounces and the process mean is m = 2.01, what is the process capability index?
USL = 2.05 oz LSL = 1.95 ounces
Cpk = min[(m -LSL) / 3 , (USL- m) / 3 ]
= min[(2.01–1.95) / 0.06 , (2.05 – 2.01) / 0.06 ]
= min[1.0 , 0.67 ]
= 0.67
Therefore, the process is not capable and/or not centered !

20. Process Control Charts
Statistical technique for tracking a process and
determining if it is going “out to control”
Establish capability of process under normal conditions
Use normal process as benchmark to statistically identify abnormal process behavior
Correct process when signs of abnormal performance first begin to appear
Control the process rather than inspect the product!

21. Process Control Charts
Upper Spec Limit
Upper Control Limit 6
Target Spec 3
Lower Control Limit
Lower Spec Limit

22. Process Control Charts
UCL
Target
LCL
Time

23. When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean
Five or more points trending toward either limit
A sharp change in level
Other erratic behavior

24. Types of Control Charts
Attribute control charts
monitors frequency (proportion) of defectives
p - charts
Defects control charts
monitors number (count) of defects per unit
c – charts
Variable control charts
monitors continuous variables
x-bar and R charts

25. 1. Attribute Control Charts
p - charts
Estimate and control the frequency of defects in a population
Examples
Invoices with error s (accounting)
Incorrect account numbers (banking)
Mal-shaped pretzels (food processing)
Defective components (electronics)
Any product with “good/not good” distinctions

26. Using p-charts
Find long-run proportion defective (p-bar) when the process is in control.
Select a standard sample size n
Determine control limits

27. 2. Defect Control Charts c-charts
Estimate & control the number of defects per unit
Examples
Defects per square yard of fabric
Crimes in a neighborhood
Potholes per mile of road
Bad bytes per packet
Most often used with continuous process (vs. batch)

28. Using c-charts
Find long-run proportion defective (c-bar) when the process is in control.
Determine control limits

29. 3. Control Charts for Variables
x-bar and R charts
Monitor the condition or state of continuously variable processes
Use to control continuous variables
Length, weight, hardness, acidity, electrical resistance
Examples
Weight of a box of corn flakes (food processing)
Departmental budget variances (accounting
Length of wait for service (retailing)
Thickness of paper leaving a paper-making machine

30. x-bar and R charts Two things can go wrong Two control solutions
process mean goes out of control
process variability goes out of control
Two control solutions
X-bar charts for mean
R charts for variability

31. Problems with Continuous Variables
“Natural”
Process
Distribution
Mean not
Centered
Increased
Variability
Target

32. Range (R) Chart Choose sample size n
Determine average in-control sample ranges R-bar where R=max-min
Construct R-chart with limits:

33. Mean (x-bar) Chart Choose sample size n (same as for R-charts)
Determine average of in-control sample means (x-double-bar)
x-bar = sample mean
k = number of observations of n samples
Construct x-bar-chart with limits:

34. x & R Chart Parameters

35. When to Take Action
A single point goes beyond control limits (above or below)
Two consecutive points are near the same limit (above or below)
A run of 5 points above or below the process mean
Five or more points trending toward either limit
A sharp change in level
Other erratic behavior

36. Control Chart Error Trade-offs
Setting control limits too tight (e.g.,  ± 2) means that normal variation will often be mistaken as an out-of-control condition (Type I error).
Setting control limits too loose (e.g.,  ± 4) means that an out-of-control condition will be mistaken as normal variation (Type II error).
Using control limits works well to balance Type I and Type II errors in many circumstances.