Spring-2024-Priesthoods of Augustus Yale Historical Review
Statistical Process Control
1. Statistical Process Control
Nicola Mezzetti, Ph.D.
Department of Information Engineering and Computer Science
University of Trento
nicola.mezzetti@gmail.com
A.A. 2014/2015
Nicola Mezzetti, Ph.D. Statistical Process Control
2. "How much variation should we leave to
chance?"
W. A. Shewhart
Nicola Mezzetti, Ph.D. Statistical Process Control
3. What is Statistical Process Control?
Statistical Process Control (SPC) is an industry standard
methodology for measuring and controlling quality during the
manufacturing process.
Attribute data (measurements) is collected from products as they
are being produced.
By establishing upper and lower limits, variations in the
processes are monitored before they result in a defective
product,
reducing the amount of material scrap along with direct and
indirect labor waste
eliminating the need for
6. History of Statistical Process Control
In 1924 Walter Shewhart developed a simple graphical method
for plotting collected data with predetermined control limits. This
was the
7. rst of a growing range of SPC charts, commissioned by
Bell Laboratories to improve the quality of telephones
manufactured.
Understanding the causes of variation within an industrial process
proved indispensable to identify actions to improve process and
output. In the 1950's, with the eective use of SPC, Deming
converted post war Japan into the world leader of manufacturing
excellence.
This approach is increasingly being applied in service industry by
thinking of systems as processes. As well as providing a basis for
quality improvement, SPC Charts also oer alternative methods of
displaying data.
Nicola Mezzetti, Ph.D. Statistical Process Control
8. When to use Statistical Process Control?
Are your quality costs really known?
Can current data be used to improve your processes, or is it
just data for the sake of data?
Are the right kinds of data being collected in the right areas?
Are decisions being made based on true data?
Can you easily determine the cause of quality issues?
Do you know when to perform preventative maintenance on
machines?
Can you accurately predict yields and output results?
Nicola Mezzetti, Ph.D. Statistical Process Control
9. About Statistical Process Control
Dr. W. Edwards Deming claimed that the majority of variation in a
process is due to operator over adjustment.
SPC gives operators a tool to determine when a statistically
signi
10. cant change has taken place in the process or when an
seemingly signi
11. cant change is just due to chance causes.
Nicola Mezzetti, Ph.D. Statistical Process Control
12. Why do Companies use SPC?
SPC itself will not make improvements.
SPC will give operating personnel a tool to identify when a
special cause of variation has entered the process so that the
special cause can be
eliminated (if the special cause has a negative impact on the
process), or
built into the process (if the special cause has a positive
impact on the process)
Moreover, SPC allows to
eliminate constant tweaking of the process
identify opportunities for improvement that can lead to
reduced variation and processes that are better aimed at their
target
Nicola Mezzetti, Ph.D. Statistical Process Control
13. Control Charts
Control chart is a tool used to study how a process changes
over time.
Measurements are plotted in time order. A control chart always
has
a central line for the average
an upper line for the upper control limit1
a lower line for the lower control limit
By comparing current data to these lines, you can draw
conclusions about whether the process variation is in control or
aected by special causes of variation.
Control charts for variable data are used in pairs:
The top chart monitors the average (x chart)
The bottom chart monitors the range (R chart)
1Control limits are determined by the capability of the process, whereas
speci
14. cation limits are determined by the customer's needs
Nicola Mezzetti, Ph.D. Statistical Process Control
15. Process Mean Chart
Center Line
x =
Pm
i=1
Pn
j=1 xij
mn
Control Limits
x 3
where 99.73% of all data
points should fall.
Plotted Statistics
xi =
Pn
j=1 xij
n
Nicola Mezzetti, Ph.D. Statistical Process Control
16. Process Variation Chart
Center Line
R
=
Pm
i=1 max(xij ) min(xij )
m
Upper Control Limit
D4R
Lower Control Limit
D3R
Plotted Statistics
Ri = max(xij ) min(xij )
Nicola Mezzetti, Ph.D. Statistical Process Control
17. How to use Control Charts?
Data is collected from the process, typically in subgroups of 3
to 5, and the subgroup mean and range is plotted on the charts.
Once a point is plotted the chart is interpreted to determine if
the process is staying in-control or if the process is out-of-control.
Data that falls within the control limits indicates that everything is
operating as expected.
Any variation within the control limits is likely due to a
common cause, the natural variation that is expected as part
of the process.
If data falls outside of the control limits, this indicates that an
assignable cause is likely the source of the product variation
something within the process should be changed to
18. x the
issue before defects occur.
Nicola Mezzetti, Ph.D. Statistical Process Control
19. Interpreting Control Charts
The most common patterns to watch out for are:
One point outside of the control limits
Eight points in a row on either side of the center line
Eight points in a row trending in the same direction
Cycles or recurring trends
Nicola Mezzetti, Ph.D. Statistical Process Control
20. Combining Variability and Mean Charts
The R chart is examined before the x chart:
if the R chart indicates the sample variability is in statistical
control, then the x chart is examined to determine if the
sample mean is also in statistical control
if the sample variability is not in statistical control, then the
entire process is judged to be not in statistical control
Nicola Mezzetti, Ph.D. Statistical Process Control
21. Control Points
Before initiating any SPC program it is necessary to identify what
to count, that is control points. Control points can be related to
Process
Product
Financials
Nicola Mezzetti, Ph.D. Statistical Process Control
22. When to Use a Control Chart
Putting spec limits on control charts
Using control charts only to satisfy customer needs
Plotting data after the process has already been run
Using the wrong type of control chart for the process
Not reviewing control charts and how they are used on a
regular basis
Not
23. rst conducting a process capability study
Not taking random samples from the process, or not using a
sampling frequency or sample size that captures the variation
in the process
Nicola Mezzetti, Ph.D. Statistical Process Control
24. When to Use a Control Chart
When controlling ongoing processes by
25. nding and correcting
problems as they occur
When predicting the expected range of outcomes from a
process
When determining whether a process is stable (in statistical
control)
When analyzing patterns of process variation from special
causes (non-routine events) or common causes (built into the
process)
When determining whether your quality improvement project
should aim to prevent speci
26. c problems or to make
fundamental changes to the process
Nicola Mezzetti, Ph.D. Statistical Process Control
27. The Seven Step Process
The use of a Seven Step Process improves statistical process
control. Proper application of SPC will improve process, product
and
29. cation of appropriate measurable variables
Estimation of available resources and project cost
Estimation of project time line
Application of appropriate statistical techniques
Implementation of corrective action
Statistical monitoring of identi
32. ne it as the
six sigma spread.
The term Process Capability means the ability of the process
spread to
33. t within the tolerance spread.
For the comparison of two or more processes, we'll need some kind
of index number to help us compare apples to apples.
Nicola Mezzetti, Ph.D. Statistical Process Control
34. Basic Capability Indices
Pp = Process Performance, a simple and straightforward
indicator of process performance.
Ppk = Process Performance Index, adjustment of Pp for the
eect of non-centered distribution.
Cp = Process Capability, a simple and straightforward indicator
of process capability.
Cpk = Process Capability Index, adjustment of Cp for the eect
of non-centered distribution.
Nicola Mezzetti, Ph.D. Statistical Process Control
35. Process Performance (Pp)
For the Pp index we take a sampling from the process, measure the
characteristic in question, and calculate the average and standard
deviation using the standard formulas.
The average 3 will account for 99.73% of the entire population.
So 6 will essentially represent all of the product.
For the Pp index, we want to see how well this 6 spread could
36. t
into the tolerance spread.
Let's suppose our tolerance is 5 units and our is 1. The
tolerance spread is 10 and the process spread is 6; if we divide
the tolerance by the process we get 1.67 Pp.
Since the process spread is the denominator in this equation, any
number greater than 1 is good and any number less than 1 is poor.
Nicola Mezzetti, Ph.D. Statistical Process Control
37. Considerations on Pp
But is the process centered in the tolerance zone?
Since there is more of the population closer to the average in
the normal distribution, it is important to have the average in
the middle of the tolerance.
The formula for the Pp index does not consider this in any
way; in theory, you could have a good Pp and run 100% scrap.
Think of this index as the potential capability of the process.
If we can center the process in the tolerance zone perfectly, it
will achieve the quality represented by the Pp index.
Nicola Mezzetti, Ph.D. Statistical Process Control
38. Process Performance Index Ppk
To keep our values consistent with the Pp values, we'll only use the
smallest (or minimum) of these two indices for the Ppk value.
And we have now arrived at our textbook formula for Ppk
Ppk =
min(USL X
; X
LSL)
3
Nicola Mezzetti, Ph.D. Statistical Process Control
39. Process Capability: Cp and Cpk
Process Capability is an indicator of the process' stability.
if a process was stable2 then we can trust it to maintain the
Pp (or Ppk ) value for a longer period of time.
We can think of the Pp (or Ppk ) as a snapshot of the process
capability at a given moment (short term capability indices).
If we want to know the capability of a process over the long term,
we'd like to know how stable that process is.
The classic test for stability is the control chart.
2A process is stable if it will stay at the same average and standard
deviation for a reasonable period.
Nicola Mezzetti, Ph.D. Statistical Process Control
40. Computing Cp and Cpk
For Pp (or Ppk ) we estimated the standard deviation using the
mathematical formula:
s =
sPn
i=1
Pm
j=1 (Xij X
)2
nm 1
The only dierence mathematically between the Cp/Cpk and the
Pp/Ppk is how you estimate the standard deviation.
you use the average Range to estimate the standard deviation
by dividing it by the d2 constant factor3
R
=d2
=
R
d2
Cpk would be the long term performance capability index.
3d2 = 2:059
Nicola Mezzetti, Ph.D. Statistical Process Control
41. Potential and Performance Capability Indices
Many people consider the Pp and Cp indices the potential
capability of the process and the Ppk and Cpk the performance
capability indices, so
Pp would be the short term potential capability index
Cpk would be the long term performance capability index
But... why does anyone use the Ppk index anymore?
On a new production part, during the initial phases of
production, you have yet to get the control chart established
enough to enforce stability: the only choice you have is the
Ppk index based upon the small sample you have at this time!
Nicola Mezzetti, Ph.D. Statistical Process Control
42. The ppm equivalent Capability Index (Cpppm)
There are numerous other indices.
One that is actually quite useful is the Cpppm
4 which has the
advantage of being comparable in application to the Cpk (or
Ppk ) index.
4the ppm superscript stands for parts per million.
Nicola Mezzetti, Ph.D. Statistical Process Control
43. The ppm equivalent Capability Index: Examples
Let's make an example! What does having Cpk = 1:00 represent in
terms of parts per million rejected?
We know that X
3 corresponds to 99:73% of the
population
In a million parts, this would equal 997:300 parts, leaving
2:700 parts rejected
An equivalent Cpppm of 1.00 should have 2:700 ppm rejected
Having a Cpk = 1:33 is the same as a process spread of 4
We know that X
4 corresponds to 99:9937% of the
population
In a million parts, this would equal 999:937 parts, leaving 63
parts rejected
An equivalent Cpppm of 1.33 should have 63 ppm rejected
Nicola Mezzetti, Ph.D. Statistical Process Control
44. More on the ppm equivalent Capability Index
Suppose your customer wants a capability index Ppk (or Cpk ) equal
to 1:33 but you only have attribute data. You've sorted 55:000
parts and found 3 parts defective.
Can you tell your customer you are supplying parts at Cpk of 1:33?
We know that a Cpk of 1:33 is equivalent to 64ppm rejected. This
means that in 55:000 parts at a Cpk of 1:33 we should
45. nd 3; 52
defects.
We found 3, so it sounds like we are doing better than an
equivalent Cpk of 1.33 and there is good basis for declaring we are
satisfying the requirement, BUT...
... this is true only if your data is normally distributed5
5All of the capability indices are based upon the normal distribution. If your
data isn't normal, then your capability is in doubt.
Nicola Mezzetti, Ph.D. Statistical Process Control
47. ts of Statistical Process Control
With real-time SPC you can:
Dramatically reduce variability and scrap
Scienti
48. cally improve productivity
Reduce costs
Uncover hidden process personalities
Instant reaction to process changes
Make real-time decisions on the shop
oor
Nicola Mezzetti, Ph.D. Statistical Process Control
49. Measuring the ROI on Statistical Process Control
To measure the ROI on your SPC investment, start by identifying
the main areas of waste and ineciency at your facility. Common
areas of waste include:
scrap
rework
over inspection
inecient data collection
incapable machines and/or processes
paper based quality systems
inecient production lines
Nicola Mezzetti, Ph.D. Statistical Process Control
50. References
Montgomery, D. C., Introduction to Statistical Process
Control, 6th ed., Wiley and sons, 2009.
Steiner, S., Abraham, B., MacKay, J., Understanding Process
Capability Indices, Institute for Improvement of Quality and
Productivity, Department of Statistics and Actuarial Science,
Universty of Waterloo.
Nicola Mezzetti, Ph.D. Statistical Process Control
