Six Sigma - Statistical Process Control (SPC)

1 March 30, 2020 – v6.0
Six Sigma Statistical Process Control
by Operational Excellence Consulting LLC

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Section 1: Statistical Process Thinking
Section 2: Basic Statistics
Section 3: Introduction to Statistical Process Control
Section 4: Statistical Process Control Charts
Section 5: Sample Size and Frequency
Section 6: Out-of-Control Action Plan
Section 7: Process Control Plan
Statistical Process Control (SPC) – Table of Contents
This document is a partial preview. Full document download can be found on Flevy:
https://flevy.com/browse/document/six-sigma--statistical-process-control-spc-604

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GO - Test
NO-GO - Test
The first time that one presented machine produced parts was 1851 at the
industry exhibition in the Crystal Palace in London. An American gun
smith took 10 working guns, took them apart, mixed all the parts in a box
and re-assembled them again. This was found a quite surprising
“experiment”.
Statistical Process Thinking – A little bit of History

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Statistical Process Thinking – Outputs and Inputs
• A basic statistical process thinking premise is that the process
output you are concerned about is depends on process
inputs
• This is expressed algebraically as
• Y is a measure or attribute of the process output
• X1, X2, etc. represent attributes of process inputs
We need to shift our thinking from managing results (Y) to
understanding and controlling the process inputs (Xs).
Y = f(X1, X2, X3, … , Xn)

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• The traditional process control concept does not help us
to produce or deliver only good products or services.
• Every process outcome, product or service, has to be
inspected.
• Products have to be repaired or even scraped.
• Rendered services result in customer dissatisfaction.
• With respect to productivity and efficiency every activity
after the actual process is a non-value adding activity.
The Traditional Process Control Concept

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• The advanced process control concept monitors one or
several critical process and product characteristics (Ys).
• The objective is to identify outliers, trends and shifts in
those characteristics, even prior to this causing defects in
the process outcome.
• The advanced process control concept enables
organizations to reduce inspection activities, rework and
scrap.
• The advanced process control concept enables
organizations to identify critical process inputs (Xs) that
impact critical process and product characteristics (Ys).
The Advanced Process Control Concept

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A defect is any variation of a required characteristic (Y) of the
product or service, which is far enough removed from its
target value to prevent the product or service from fulfilling
the physical and functional requirements of the customer.
Statistical Process Thinking – Defect Definition

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Remarks or Questions ?!?Remarks or Questions ?!?

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Basic Statistics – Overview
• Types of data
• Measures of the Center of a Data Sample
– Mean
– Median
• Measures of the Spread of a Data Sample
– Range
– Variance (s2)
– Standard Deviation (s)
• Properties of a Normal Distribution
• Binomial and Poisson Distribution

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Example: y1 = 5 y2 = 7 y3 = 4 y4 = 2 y5 = 6
Measures of Center – The Sample Average
Definition:
. . .
5 7 4 2 6
5
24
5
4.8

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Measures of Center – Mean versus Median
MedianMean
• Uses every value in the sample
• Influenced by outliers
• Involves more computation
• Only uses middle value(s)
• Not influenced by outliers
• Little mathematical calculation
14 16 18 20 22 24 26 28 48 50
| | | | | | | | | |
Median = 16
Mean = 21.14

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y3
y
average
_
y2
y1
y10
Measures of Variability – Sample Standard Deviation
Time
y6
-
-
…
10 1

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Example:
Measures of Variability – Sample Variance
. . .
1
          7.3
)15(
8.468.428.448.478.45
22222
2



s
Definition:
y1= 5 y2= 7 y3= 4 y4= 2 y5= 6

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Basic Statistics – How to create a Histogram
A histogram provides graphical presentation and a first estimation about
the location or center, spread and shape of the outcome or results of
the process.

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Step 2: Determine the number of bars to be used to create the histogram
of the data points. Calculate the width of one bar by dividing the range
of your data by the number of bars selected.
Basic Statistics – How to create a Histogram?
Number of Bars:
less than 50
50 - 100
100 - 250
over 250
5 or 7
5, 7, 9 or 11
7 - 15
11 - 19
Number of Data Points:
Minimum = 2.1
Maximum = 3.1
Range = 1.0
Bar Width = 0.2 (5 Bars)

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Step 4: Draw the histogram indicating by the height of each bar the
number of data points that fall between the “start” and “end” point of
that bar.
Basic Statistics – How to create a Histogram?
Sorted Measurements
Part Hole Size Bar
5 2.1 1
2 2.3 2
7 2.4 2
6 2.5 3
8 2.5 3
1 2.6 3
10 2.6 3
4 2.7 4
9 2.8 4
3 3.1 5
0
1
2
3
4
5
NumberofDataPoints
2.1 2.3 2.5 2.7 2.9 3.1

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Basic Statistics – The Normal Distribution
average average
+1*s
average
-1*s
average
+2*s
average
-2*s
average
-3*s
average
+3*s
34.13 %34.13 %
13.60 % 13.60 %
2.14 %2.14 %
0.13 % 0.13 %
If your process Y creates a histogram with the shape of a normal distribution, about 99.74% of
your data points will fall between the average ± 3s limits.

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Basic Statistics – Test for Normal Distribution
The Normality Test from Anderson & Darling provides a method to determine if
your data comes from a process that creates normally distributed data.
The red line represents
the normal distribution.
If the all the individual data
points fall on the red line, the
sample data itself is perfectly
normally distributed.
As long as the p-value stays above
0.05, we can assume that the process
creates normally distributed data.This document is a partial preview. Full document download can be found on Flevy:

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Statistical Process Control – Definition
What is happening? What happened?

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Statistical Process Control – Types of Process Variation
• Processes experience two kinds of variation
– Common Cause
– Special Cause
• The types of variation observed in the data determine which
process improvement actions are to be taken

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Statistical Process Control – Special Causes
• Special causes, also known as assignable causes, often originate outside
the process and affect the regular, repeatable, and natural variation of
the process
• Special cause variation is not normally present in the process, but is an
irregular shock or upset to an X or a conversion activity
• Special causes create a change (e.g. outliers, trends, and shifts) in the
process and make it difficult to identify and analyze common causes
• A process that is regularly affected by Special cause variation can not
be considered stable
• It is ineffective to try to reduce Common cause variation or change the
process average until the process is stable
• While all processes have Common causes, not every process is affected
by Special causes

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Statistical Process Control – Control Charts
• Control charts are a useful tool to verify whether a process is in
control
• While a stable process is desirable, it is not a guarantee of
meeting specifications
• A process can be in statistical control and still produce out of
specification results
• The goal is for a process to be both stable and capable of
meeting customer requirements
DR. WALTER A. SHEWHART (1891–1967)

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The first use of control charts is to diagnose a process
• Many times the first step in improving a process is to study
and understand the current process
• The data for diagnostic control charting is collected, sometimes
off-line, by a process manager, engineer or improvement team
• The initial data collection often raises questions that require
additional data to identify sources of variation
• Many processes, especially new or modified ones, will have
special causes to resolve before we can start monitoring with
control charts
• 20 – 25 data points is normally a good starting point
Statistical Process Control – Purpose of Control Charts

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• Work to get very timely data
• Immediately search for the root cause when control chart gives
a “signal”
• Do not make fundamental changes in the process
• Seek ways to change some higher level or upstream process to
prevent that special cause from recurring
• Also consider whether a special cause could be originating
from within the process itself
• The Process FMEA can be a powerful tool to understand and
resolve the special cause as a failure mode
Statistical Process Control – Improving Special Causes

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1 s
2 s
3 s
1 s
2 s
3 s
%
of data points
UCL
LCL
Theitemweare
measuring
Statistical Process Control – Out-of-Control Rules
TIME
99-99.9%
90-98%
60-75%
Empirical
Rule:
Why 3s
is used.

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Statistical Process Control – Control Chart Rules
Below is a list of the most commonly used out-of-control criteria included
in Minitab and as initially defined by Walter Shewhart in the 1920s.
Criteria 1: Outlier
Criteria 2 & 5 & 6: Process Shift
Criteria 3: Process Trend

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Process Control Charts – Types of Control Charts
Discrete – Attribute Data
(Count or Yes/No Data)
Variable – Continuous Data
(Measurements)
Subgroup
size
of > 10
Subgroup
size
of 1
Subgroup
size
of <= 10
I / MR
- Chart
x-bar / R
- Chart
x-bar / s
- Chart
Count
Incidences or
nonconformities
Fixed
oppor-
tunity
Variable
oppor-
tunity
c
- Chart
u
- Chart
Yes/No Data
Defectives or
nonconforming units
Fixed
subgroup
size
Variable
subgroup
size
np
- Chart
p
- Chart
Type of Data

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• The I-MR (or Individuals – Moving Range) chart is a method of
looking at variation in a variable data or measurement.
• One source is the variation in the individual data points over time
(Individuals chart). This represents “long term” variation in the
process Y.
• The second source of variation is the variation between successive Y
data points (Moving Range chart). This represents “short term”
variation.
• I-MR charts should be used when there is only one data point to
represent a situation at a given time.
• To use the I-MR chart, the individual sample results should be
“sufficient” normally distributed. If not, the I-MR chart will give
more false signals, i.e. special causes.
Process Control Charts – The I-MR Chart

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Process Control Charts – I-MR Chart Example
Minitab: Stat > Control Charts > Variable Charts for Individuals > I-MR
This is a list of the most commonly used
out-of-control criteria included in Minitab
and as initially defined by Walter Shewhart
in the 1920s.

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Special Cause: 4 out of 5 points > 1 standard
deviation from center line (same side)

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Long-term standard deviation:
Short-term standard deviation:
If the process is stable and in statistical control, then sLT = sST. However, if
the process is not stable then sLT > sST .
The difference between sLT and sST gives an indication of how much better
one can do with respect to process variation when using appropriate
process control, like Statistical Process Control (SPC).
Process Control Charts – Difference between sLT and sST
!
" . . .
1
1.128$ ) 1.128⁄
. . .
1

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1. The process performance data indicates one
special cause in process. However, the special cause
show also in the MR chart, increasing the average
MR and therefore the short-term standard deviation
used to calculate the control limits for the I chart.
2. Special causes that show also in the MR chart
need to be excluded from the data to ensure that
all special causes in the I-chart can be identified.

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Process Control Charts – Control Chart Rule #1
All SPC Out-of-Control Criteria have
about a 1 in 1,000 chance to occur in
a process without a special cause.
Therefore, they are strong evidence
for the presence of a special cause.

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6 consecutive points increasing or
decreasing often indicates a trend in
process performance due to a special
cause.

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4 of 5 consecutive points above or below the
2 standard deviation line often indicates a
shift in process performance.

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Process Control Charts – Control Chart with Stages
The control chart indicates a shift in the Y
between the 15th and 16th data sample. This
could be caused by a change in the method,
operators, raw material batch, … .

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Process Control Charts – Control Chart with Stages
Using the “Stages” option supports the hypothesis
that the average of the Process Y and the standard
deviation of the Process Y may have changed after
the 15th data point. The process before and after
the change was stable and in control.

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Time t
Process
Characteristic
e.g. Hole Size
average
Subgroup size n = 5
Number of subgroups N = 7
Process Control Charts – The Principle of Subgrouping

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Process Control Charts – x-bar/R Chart Example
Minitab: Stat > Control Charts > Variable Charts for Subgroups > Xbar-R

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Upper control limit =
Lower control limit =
The R Chart
The x-bar Chart
where x-bar1, x-bar2, ..., x-barN are the averages of each subgroup, n the
number of items in a subgroup, N the number of subgroups,
., and
Process Control Charts – x-bar/R Chart Control Limits
- 3 ⋅ 1.128 ⋅ .$
- 3 ⋅ 1.128 ⋅ .$
RD 4
0
N
xxx
x
N

...21
N
RRR
R N

...21minmax
iii xxR 
, D4 depends on the subgroup size

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Process Control Charts – x-bar/s Chart Example
Minitab: Stat > Control Charts > Variable Charts for Subgroups > Xbar-S

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The s Chart
The x-bar Chart
, and
where x-bar1, x-bar2, ..., x-barN are the averages of each subgroup, s1, s2, ..., sN
are the standard deviations of each subgroup, n the number of items in each
subgroup, N the number of subgroups,
.
Process Control Charts – x-bar/s Chart Control Limits
sAx  3
sAx  3
sB 4
sB 3
N
xxx
x
N

...21
N
sss
s N

...21
, B4 depends on the subgroup size
, B3 depends on the subgroup size
, A3 depends on the subgroup size

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Process Control Charts – The Binomial Distribution
The histogram above shows data from a
process that in average creates 30 defective
items in a sample of 100, i.e. a 30% Defect
Rate.
The Binomial distribution is very similar to a
Normal distribution.
The histogram above shows data from a
process that in average creates 5 defective
items in a sample of 100, i.e. a 5% Defect
Rate.
The Binomial Distribution is asymmetric due
to the lower boundary of 0.
Process data monitoring the number of defective items creates a
Binomial Distribution.

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Process Control Charts – np - Chart Example
Minitab: Stat > Control Charts > Attributes Charts > NP
Special Cause criteria for attribute
charts (outlier – trend – shift).

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with
where np1, np2, ..., npN are the number of defective items in each
subgroup of constant size n, and N the number of subgroups.
13)
2
1
( 






n
pn
pnpn
13)
2
1
( 






n
pn
pnpn
np
np np np
N
N

  
1 2
3
...
Process Control Limits – The np - Chart
Standard Deviation of the
Binomial Distribution.

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• The p - chart is used to look at variation in yes/no attribute data. It
can for example be used to monitor the percentages or proportions p
of defective items in a group of items.
• The number n of items in each group has not to be constant, but
should not vary more than 25 %.
• Operational definitions must be used to determine what constitutes a
defective item.
• The standard deviation of a Binomial distribution is
Process Control Charts – The p - Chart
where is the average proportion of defective items based on all
subgroups and n is the average subgroup size.
/̄
/̄ ⋅ 1 /̄ .̄$

97 March 30, 2020 – v6.0
Process Control Charts – The p – Chart Example
Proportion of defects in each subgroup.
In this case the subgroup size varied
between 95 and 105. Average defect
rate is 8.12%.
The standard deviation is depended
on the subgroup size. As the subgroup
sizes vary, the standard deviation and
control limits vary.
1 point > 3 standard deviations
from the center line indicates an
outlier in process performance.

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The number of wrong assembled components in 20 units were
1 - 20: 10 wrong assembled components
161 - 180: 2 wrong assembled components. Something changed ???
Process Control Charts – The Attribute “Count” Data

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• The c - chart is used to look at variation in counting-type attributes
data. It is used to determine the variation in the number of defects in
a constant subgroup size.
• For example, a c - chart can be used for example to monitor the
number on injuries in a plant for a specific time period. In this case,
the plant is the subgroup.
• To use the c - chart, the opportunities for incidences to occur in the
subgroup must be very large, but the number that actually occur must
be small.
• The standard deviation of a Poisson distribution is
Process Control Charts – The c – Chart
where is the average number of defects or occurrences based on all
subgroups.
1̄
1

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Process Control Charts – The c - Chart Example
Number of incidences in a specific
time period or samples over time.
In this case the average number of
incidences is 14.72.

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The number of invoices typing errors per week were
# of Error # of Invoices
Week 1: 7 95
Week 2: 5 90
Week 3: 9 100
Week 4: 12 125
Week 5: 8 95
Week 6: 4 50
Week 7: 6 55
Week 8: 9 80
Week 9: 15 125 → Something changed ???
Process Control Charts – The Attribute “Count” Data

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Process Control Charts – u - Chart Example
Minitab: Stat > Control Charts > Attributes Charts > U
Special Cause criteria for attribute
charts (outlier – trend – shift).

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u u n 3
with andu
c c c
n n n
N
N

  
  
1 2
1 2
...
..
, n
n n n
N
N

  ( ... )1 2
where c1, c2, ..., cN are the number of occurrences in each subgroup and
n1, n2, ..., nN are the number of items or units in each of the N subgroups.
Note: The subgroup sizes should not vary more than 25% around the
average subgroup size.
 0,3max nuu 
Process Control Limits – The u - Chart

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113 March 30, 2020 – v6.0
avg
Sample Size & Frequency – Subgroup Size & Sensitivity
USL
avg + STs3
UCLLCL
avg - STs3 avg + E
Shift of Y by E
Defects

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• The frequency of sampling of two consecutive individual data
points or subgroups of data points can be determined by
dividing the average time period between two out-of-control
situations by at least 3 but not more than 6.
Example: If experience shows that your process produces
defects or goes out-of-control once every 12-hour shift, you
should start with collect measurements from your process Y
every 2 to 4 hours.
Sample Size & Frequency – Sample Frequency
• However, no general rule can be defined about
which time interval works best. You have to start
with a good (conservative) guess and refine the
time interval if or as necessary.

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119 March 30, 2020 – v6.0
Start
Checkpoints
Activators
Corrective ActionsNo
No
No
Yes
Yes
Yes
Yes
Yes
Yes
End
No
No
Out-of-Control-Action-Plans (OCAP)

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The checkpoints instruct the process operator or owner to
investigate specific items as possible assignable causes for
the out-of-control situation.
Once a checkpoint has identified a probable assignable
cause for the out-of-control situation, the OCAP will flow
into a terminator or corrective action.
Out-of-Control-Action-Plans – Checkpoints

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... typically generate one or more of the following actions:
 Eliminate the most common assignable causes
 Analyze the activators
 Revise the order of the checkpoints and terminators
 Train the operators or owner to perform more of the
corrective actions included into the OCAP to resolve
out-of-control situations quickly
An Analysis of Out-of-Control-Action-Plans ...

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Remarks or Questions ?!?Remarks or Questions ?!?

127 March 30, 2020 – v6.0
Process Control Plan – Objective
A Control Plan is a written statement of an organization’s quality
planning actions for a specific process, product, or service.
The Objective of an effective Process Control Plan is to
• operate processes consistently on target with minimum
variation, which results in minimum waste and rework
• assure that product and process improvements that have been
identified and implemented become institutionalized
• provide for adequate training in all standard operating
procedures, work instructions and tools
Customer
Requirements
Product & Part
Characteristics
Process
Input & Output
Characteristics
Process
Controls
Process
Control
Plan

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• Process: Name of the process to be controlled
• Process Step: The process steps of the process to be controlled
• Characteristic (Product/Process): Name of the characteristic of a process step or
a product, which will actually be controlled.
• Specification: Actual specification, which has been set for the characteristic to
be controlled. This may be verified e.g. in standards, drawings, requirements
or product requirement documents.
• Control Limits: Control limits are specified for characteristics that are
quantifiable and selected for trend analysis (x-bar/R, x/mR, p charts). When
the process exceeds these limits, corrective actions are required.
• Measurement System: Method used to evaluate or measure the characteristic.
This may include e.g. gages, tools, jigs and test equipment or work methods.
An analysis of the repeatability and the reproducibility of the measurement
system must first be carried out (e.g. Gage R&R Study).
Process Control Plan – Template

131 March 30, 2020 – v6.0
 Process maps detail manufacturing
steps, material flow and important
variables
 Key product variables identified with
importance to customer, desired target
value and specification range defined
 Key and critical process input variables
identified with targets, statistically
determined control limits & control
strategies defined
 Measurement systems are capable with
calibration requirements specified
 Sampling, inspection and testing plans
include how often, where and to
whom results are reported
 Reaction plan in place for out-of-spec
conditions and material
 Operating procedures identify actions,
responsibilities, maintenance schedule
and product segregation requirements
 Training materials describe all aspects
of process operation and responsibili-
ties
 Process improvement efforts fully
documented and available for refe-
rence
 Control plan is reviewed and updated
quarterly and resides in the operating
area
Process Control Plan – Check List

133 March 30, 2020 – v6.0

135 March 30, 2020 – v6.0
The End …
“Perfectionis not attainable, but if we chase perfectionwe can catch
excellence.” - Vince Lombardi

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