This document provides an overview of control charts for continuous data. It discusses control chart fundamentals like control limits and distinguishing between common and special cause variation. It introduces the X-bar and R chart used for variable data with subgroup sizes of 3-9. An example X-bar and R chart is presented using data on customer hold times. The document also covers the Individuals and Moving Range chart which can be used when the subgroup size is 1. Control chart assumptions and interpretation of control limits are explained.
Basic Civil Engineering first year Notes- Chapter 4 Building.pptx
NG BB 26 Control Charts
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National Guard
Black Belt Training
Module 26
Control Charts
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CPI Roadmap – Measure
8-STEP PROCESS
6. See
1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize
Counter-
the Performance Improvement Root Counter- Results Successful
Measures
Problem Gaps Targets Cause Measures & Process Processes
Through
Define Measure Analyze Improve Control
TOOLS
•Process Mapping
ACTIVITIES
• Map Current Process / Go & See •Process Cycle Efficiency/TOC
• Identify Key Input, Process, Output Metrics •Little’s Law
• Develop Operational Definitions •Operational Definitions
• Develop Data Collection Plan •Data Collection Plan
• Validate Measurement System •Statistical Sampling
• Collect Baseline Data •Measurement System Analysis
• Identify Performance Gaps •TPM
• Estimate Financial/Operational Benefits •Generic Pull
• Determine Process Stability/Capability •Setup Reduction
• Complete Measure Tollgate •Control Charts
•Histograms
•Constraint Identification
•Process Capability
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Learning Objectives
Control chart fundamentals
Use of control charts to identify Common Cause and
Special Cause variation
Factors to consider in constructing control charts
Variables control charts
Attribute control charts
Understand the interpretation and application of these
charts
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Control Chart Terms
Control Chart = a time plot showing process
performance, mean (average), and control limits
The Voice of the Process !!!
I-MR Chart of Pizza Preparation Time
1
20
U C L=18.48
Individual Value
15
_
10 X=10.58
5
LC L=2.67
3 6 9 12 15 18 21 24 27 30
Control charts measure the “health” of the process
O bse r v ation
10.0 U C L=9.71
7.5
Moving Range
5.0
__
M R=2.97
2.5
0.0 LC L=0
3 6 9 12 15 18 21 24 27 30
O bse r v ation
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Control Chart Terms
Control Limits = statistically calculated boundaries
within which a process in control should operate
These boundaries result from the process itself and are
NOT customer specifications
I-MR Chart of Pizza Preparation Time
1
20
U C L=18.48
Individual Value
15
_
10 X=10.58
5
LC L=2.67
3 6 9 12 15 18 21 24 27 30
O bse r v a tion
10.0 U C L=9.71
7.5
Moving Range
5.0
__
M R=2.97
2.5
0.0 LC L=0
3 6 9 12 15 18 21 24 27 30
O bse r v a tion
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Common vs. Special Cause
Measurements display variation
Variation is either:
Common Cause Variation
This is the consistent, stable, random variability within the process
We will have to make a fundamental improvement to reduce common
cause variation
Is usually harder to reduce
Special Cause Variation
This is due to a specific cause that we can isolate
Special cause variation can be detected by spotting outliers or
patterns in the data
Usually easier to eliminate
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Process Control
When a process is “in control”
This implies a stable, predictable amount of variation (common
cause variation)
This does not mean a "good" or desirable amount of variation
When a process is “out-of-control”
This implies an unstable, unpredictable amount of variation
It is subject to both common AND special causes of variation
A process can be in statistical control and not capable of consistently
producing good output within specification limits
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Types of Control Charts
The Control Chart family can be broken into two groups
based on the type of data we are charting:
Continuous/Variable
Attribute/Discrete
Since we “prefer” Continuous data we will study this group
of Control Charts first
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Continuous Data Control Charts
The theory of all Control Charts can be learned by studying
the Xbar (Average) and R (Range) chart for continuous
data
We will then explore the I-MR (Individuals - Moving Range)
Chart
Xbar-R Charts allow us to study:
Variation “within each subgroup” (precision) on the R chart
Variation “between each subgroup” (accuracy) on the Xbar chart
Note: Look at the R chart first, if it is in control, then look at the
Xbar chart
Examples of continuous data: width, diameter,
temperature, weight, cycle times, etc.
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Control Chart Assumptions
Normally Distributed Data
Control limits approximate +/- 3 sigma from the mean
These control limits are based upon a normal distribution
If the distribution of the data is non-normal, you must use one of the
x-bar charts, because the x-bars are likely to be normally distributed
due to the effects of the Central Limit Theorem
Rule of thumb for x-bar charts is subgroups of at least 4. Rarely is
the underlying distribution so far from normal to require larger
subgroups to achieve normality in the x-bars.
Independent Data Points
“Independence” means the value of any given data point is not
influenced by the value of any other data point (it is random)
Violation of this assumption means the probability of any given data
value occurring is not determined by its distance from the mean, but
by its place in the sequence in a data series or pattern
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Continuous Data Control Charts
Measurement
(Continuous/Variable Data)
Subgroup Size of 1 Subgroup Size < 3-9 Subgroup Size > 9
I-MR Xbar-R Xbar-S
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Continuous Data Control Charts
Utilize probabilities and knowledge of the normal distribution
I-MR chart is used:
When you are learning about a process with few data points
When sampling is very expensive
When the sampling is by destructive testing and
When you are building data to begin another chart type
Xbar-R Chart is used with a sampling plan to monitor repetitive
processes. The sub-group sizes are from 3 to 9 items. Frequently
practitioners will choose subgroups of 5. All of the theory of Control
Charts can be applied with these charts
Xbar-S Chart is used with larger sample groups of 10 or more items.
Statisticians sometimes state that the standard deviation is only robust
when the subgroup size is greater than 9 (These charts are similar to
the Xbar-R Chart)
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Introduction to Xbar-R
Xbar-R Charts are a way of displaying variable data
Examples of variable data: width, diameter, temperature, weight,
time, etc.
R Chart: a look at “Precision”
Displays changes in the „within‟ subgroup dispersion of the
process. Often called “Short-Term Variation.”
Asks "Is the variation in the measurements „within‟ subgroups
consistent?”
Must be “in control” before we can build or use the Xbar chart
Xbar Chart: a look at “Accuracy”
Shows changes in the average value of the process and is a
visualization of the “Longer-Term Variation”
Asks "Is the variation „between‟ the averages of the subgroups
more than that predicted by the variation within the subgroups?“
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Mechanics of an Xbar-R Chart
Control charts track processes by plotting data over
time in the form:
Range Chart X Chart
Upper Control Limit Averages
Upper Control Limit
Chart = X Double Bar + A2 R Bar Upper Control Limit
Range Chart = D4Rbar Upper Control Limit
Center Line Averages Chart =
Average of the Subgroup Averages Center Line (X)
Center Line Range Chart =
Average of the Subgroup Ranges Center Line (R) Lower Control Limit Averages
Chart = X Double Bar - A2 R Bar Lower Control Limit
Lower Control Limit
Range Chart = D3Rbar
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Example: Xbar-R Chart
Stat > Control Charts > Variables Charts for Subgroups > Xbar-R
Open the worksheet data file called ORDER TAKING.MTW
In this file, orders are taken by order entry clerks. The data is the average hold
time a customer waits before speaking with a person to take their order.
The delays are a problem, as many customers give up and we have a dropped call
and lost order
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Example: Xbar-R Chart
Double click on
C-1 Ave Hold Time
This places it in the 5
Variables box
Type in 5 for your
Subgroup size
Our response is Ave. Hold Time and we choose
5 cells to represent our Subgroup size
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How Do We Interpret This Chart?
Xbar-R Chart of Ave. Hold Time
1
16
U C L=14.97
14
Sample M ean
Xbar Chart 12 _
_
X=10.88
10
8
LC L=6.79
1 2 3 4 5
Sample
16
U C L=15.01
12
Sample Range
R Chart 8 _
R=7.10
4
0 LC L=0
1 2 3 4 5
Sample
Always Look at the R Chart first !
Only if it is in control, is the Xbar chart usable !
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Control Chart Data Requirements
Data requirements for control chart applications:
Must be in time series order
Minimum of 25 consecutive (no time gaps) subgroups
or
Minimum of 100 consecutive observations
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I-MR Chart
The Individuals and Moving Range chart is also for
continuous data
It can be used for many transactional applications:
Revenue or cost tracking
Customer satisfaction
Call times
System response times
Wait times
Most common continuous measures – time and money!
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Individuals and Moving Range (I-MR) Chart
I-MR Chart of Pizza Preparation Time
1
20
U C L=18.48
Individual Value
15
_
10 X=10.58
5
LC L=2.67
3 6 9 12 15 18 21 24 27 30
O bse r v ation
10.0 U C L=9.71
7.5
Moving Range
5.0
__
M R=2.97
2.5
0.0 LC L=0
3 6 9 12 15 18 21 24 27 30
O bse r v ation
The top chart is a plot of individual pizza preparation times
The bottom chart is the Moving Range, in this case, the Range of two
adjacent pizza preparation times
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Control Limit Calculation
I Chart of Pizza Preparation Time
1
20
UCL=18.48
UCL
15
Individual Value
_ X
X=10.58
10
5
LCL
LCL=2.67
3 6 9 12 15 18 21 24 27 30
Observat ion
The UCL (Upper Control Limit) and the LCL (Lower Control Limit) are
calculated by Minitab using the sample/process data
The control limits approximate +/- 3 standard deviations (99+% of the data)
Here, 99+% of the pizzas are prepared between 2.6 and 18.7 minutes
Be careful not to confuse control limits and specification limits! If a data
point appears outside of the control limits, there is less than a 1% chance
that this was part of the normal process. Since it is very unlikely that this
value occurred by chance, it is called “Special Cause” variation.
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Control Limit Interpretation
I Chart of Pizza Prep Time 2
1
20.0
17.5 1
15.0 UCL=15.29
Individual Value
12.5
_
10.0 X=9.87
7.5 3 3
3 3 3
3 3 3
5.0
3 LCL=4.45
3 6 9 12 15 18 21 24 27
Observation
Another type of Special Cause variation occurs when there is a predictable
pattern in the data
The predictable pattern of the data means the data is not random and that
there is an underlying reason for this pattern – a Special Cause
The Western Electric rules are helpful in identifying patterns in the data
(these are in the appendix)
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Exercise: Begin Building an I-MR Chart
Let‟s begin building an I-MR chart for a Pizza Preparation process.
Begin by using the 5 Pizza Preparation Time measurements below
to start the calculations for a Control Chart on a flip-chart.
Individuals Chart
Plot each individual time measurement
Calculate the Centerline
The centerline on an Individuals chart is the overall average
Verify that the average is 9.6
The control limits will be calculated by a formula in Minitab. They
approximate +/- 3 standard deviations of the pizza prep times
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Pizza Exercise
Moving Range Chart
Calculate the ranges
The first range is between points 1 and 2
Range = Max - Min
12 - 7 = 5
The next range is between points 2 and 3
Range = Max - Min
11 - 7 = 4
Continue for the next 2 ranges
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Pizza Exercise (Cont.)
Moving Range Chart
Calculate the Centerline
The centerline is the average of the moving ranges, called R
For these 5 points (4 range calculations), verify that R = 3
The Control Limits will be calculated in Minitab. In this case
they approximate +/- 3 standard deviations of the range
values.
We expect the Control Limits to be tighter for the Moving
Range chart than for the Individuals chart
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Build I-MR Chart in Minitab
Let‟s continue with
our exercise:
1. Open the exercise
Exercise9.mtw
2. Choose:
Stat>
Control Charts>
Variables Charts for
Individuals>
I-MR
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I-MR Input Window
3. Double click on
C1 Pizza Preparation
Time. This places it in
the Variables box.
4. Click OK
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Individual and Moving Range (I-MR) Chart
I-MR Chart of Pizza Preparation Time
1
20
U C L=18.48
Individual Value
15
_
10 X=10.58
5
LC L=2.67
3 6 9 12 15 18 21 24 27 30
O bser vation
10.0 U C L=9.71
7.5
M oving Range
5.0
__
M R=2.97
2.5
0.0 LC L=0
3 6 9 12 15 18 21 24 27 30
O bser vation
Is our Pizza Prep process in statistical control?
Is the process likely to be acceptable to our customers?
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Western Electric Rules
Remember the tests that we used in Run Charts? These are used in
Control Charts as well.
The additional tests are called the “Western Electric Rules”
They can be found under Stat>Control Charts>Variables Charts for
Individuals>I-MR>I-MR Options>Tests
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Control Chart Tests
Upper Control Limit Point outside of the limit:
Control limits are calculated to measure the
Center Line
natural variability of a process. Any point on, or
outside, the limit is considered
Lower Control Limit abnormal and requires investigation.
Run:
Upper Control Limit A “run” is a series of points occurring
Center Line continually on one side of the center line. A
“run” of seven points is considered abnormal.
Lower Control Limit Also considered abnormal: 10 out of 11,12 of 14,
or 16 of 20 points on one
side of the center line.
Upper Control Limit
Trending:
Center Line Seven points in a continuous upward or
downward direction.
Lower Control Limit
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Control Chart Tests
Upper Control Limit Approaching the center line (hugging):
When most points lie within the center line and
1.5s it is not a controlled state and usually means
CL the mixing of data from different populations. This
Lower Control Limit makes the control limits too wide and
stratification of data is usually necessary.
Upper Control Limit
Cycling (periodicity):
CL Any repeated up and down trend is abnormal
and requires investigation.
Lower Control Limit
UCL
Approaching control limits:
CL 2 of 3 points lying outside the
2s line is considered abnormal.
LCL
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Special Causes Are Clues to the Process
A control chart is a guide to improving your process
Take advantage of every clue
Identify and investigate all special causes – they
teach us how things affect the process
Some special causes are good!
For example, in our pizza delivery case, a delivery time
out of control on the low side would be good. We
could investigate this case to try to discover a new best
practice.
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Process for Identifying Special Causes
Check all the W.E. rules each time you plot a point
Look across the entire chart
Circle all special causes
Investigate immediately – this is especially important.
Do not lose the opportunity to learn as much as possible
about the conditions that caused this special cause
variability.
Take notes on the investigation
You must investigate and eliminate the special cause!
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Next Steps
Identify assignable causes
Establish that the data are normally distributed
without the special cause data points
Circle the special causes
Eliminate special causes from the control limit
calculation
Recalculate control limits
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New Control Limits
I-MR Chart of Pizza Preparation Time
1
1
20
U C L=18.86
If you can investigate
Indiv idual V alue
15
and determine what 10
_
X=10.69
caused these
5
„Out of Control‟ LC L=2.52
points, you can then 0
3 6 9 12 15 18 21 24 27 30
delete them and
Observation
recalculate your 10.0
1
U C L=10.04
control chart limits
M ov ing Range
7.5
5.0
__
MR=3.07
2.5
0.0 LC L=0
3 6 9 12 15 18 21 24 27 30
Observation
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Attribute Data Charts
Two categories of attribute data:
Count data (outcomes: 0, 1, 2, 3, 4, 5, etc.)
Good/bad product data (only 2 possible outcomes)
Four common attribute charts:
C and U charts are used for count data of
Errors in the process, either a step in the process or the
overall process, or
Defects in the process‟ or steps‟ deliverables
NP and P charts are used for good/bad process,
service, or product data (items or process steps that
are defective or flawed)
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Which Chart to Use?
Count or Classification
(Discrete/Attribute Data)
Defects Defective Units
Fixed Variable Fixed Variable
sample sizes sample sizes sample sizes sample sizes
C Chart U Chart, NP Chart, P Chart,
“defect count“ “defects / unit” “no. defective” “proportion ”
Discrete/Attribute Data
To select an attribute chart, first choose between plotting defects or defective
units. Then decide between fixed or variable opportunity. The variable
opportunity charts are used more frequently.
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Deeper into Attribute Charts
Many transactional processes and manufacturing processes only
record data as to the service or the products being either bad
or good, defective or not defective
There are two sub-families in the Attribute control charts:
If we count defects (usually with any item having more
than one opportunity for a defect) we use the C or
Ucharts
If the sample size is always the same, use a C-chart. If the
sample size varies, use a U-chart.
If we count defective units instead of defects, we use
the NP or P charts
If the sample size is always the same, use a NP-chart. If the
sample size varies, use a P-chart.
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Charts for Attribute Data
Most of the Attribute Control Charts are identical in
interpretation and very similar to create in Minitab
The equations used are slightly different, but still
based on the theory we learned with the Xbar Chart
One of the most commonly used attribute charts is
the P-Chart which plots Proportion Defective
If you calculated Proportion Defective as your
baseline capability metric – this chart is for you!
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P-Chart
P-Charts should be used whenever we are monitoring
proportion defective (percentage defective is just another
proportion)
Some uses of the P-Chart in transactional applications would
be:
Billing errors (proportion of total bills that had errors)
Defective loan applications
Proportion of invoices with errors
Proportion of missing reservations
Defective room service orders
Missing items
Proportion of customers who were dissatisfied with service
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P-Chart Pizza Exercise
Anthony's Pizza wishes to monitor defective pizzas
Each day for a month the cook keeps a count of the
number of defective pizzas for that day and also the
total number of pizzas that day
Let‟s use the first 5 days data below to start the P-
Chart on a flipchart
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P-Chart Pizza Exercise (Cont.)
Calculate the proportion defective
Recall the formula for proportion defective:
Number of Defective Units
Proportion Defective
Total Units
In this example:
Number of Defective Pizzas
Proportion Defective
Total Pizzas
For the first day:
9
Proportion Defective 0.021
420
Note: Percentage Defective, in this case,
would be 2.1% defective
Calculate the proportion defective for days 2-5
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P-Chart Pizza Exercise (Cont.)
Next, calculate the center line
The center line is the proportion of Total Defectives
(for all samples) to Total Units (for all samples)
Verify that this is 0.019
The Control Limits are calculated in Minitab
The equations are slightly different, but the Control Limits
are still calculated from the actual values, predicting the
range of 99% of the data
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P-Chart Input Window
3. Double click on
C-4 Defective Pizzas.
This places it in the
Variables box.
4. Place cursor in the
Subgroups sizes box
and then double click on
C-5 Number of Pizzas
to move it there
5. Click OK
Note: Minitab calculates the proportion defective for us. We enter the
defective units in the Variable box. Then we enter the total units over
that time period in the Subgroup sizes box.
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P-Chart
P Chart of Defective Pizzas
0.05 1
Note: Minitab recalculates
0.04 the control limits every
UCL=0.03689 time the subgroup size
0.03 changes.
Proportion
To get a straight line, you
_
0.02 P=0.01932 can enter a constant value
under “Subgroup size.”
0.01
In this example, the best
LCL=0.00174 constant would be the
0.00
average of the “Number of
3 6 9 12 15 18 21 24 27 30
Sample
Pizzas.”
Tests performed with unequal sample sizes
What are your thoughts around our defective pizzas?
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Exercise: Create a Control Chart
Now Anthony's Pizza wants to investigate sales history and
billing errors for the same month
In teams, continue with Exercise9.mtw. Use an I-
MR Chart to monitor sales for the month.
Use a P-Chart to observe the proportion of defective
bills
Prepare to teach back to the class on your findings
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Quick Review - Control Chart Reminders
There are several types of control charts:
Determine type of data: continuous or attribute
Be clear on the purpose and value you wish to gain
from the chart
Control limits are derived from process data
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Control Chart Uses and Benefits
Demonstrate stability and predictability of a process
over time
Range of variation within the “control limits”
Distinguish between common vs. special cause
variation
Provides more information than Run Charts
Can be used to demonstrate changes in performance
Provide a common language for process performance
Offer early warning of problems
BUT…..
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Control Chart Challenges
Must use correct type of chart for the data
Must meet normality and independence assumptions
Non-normal, continuous data must use x-bar chart to meet
normality requirement
Control limits vs. customer requirements
Remember that the control limits are providing the Voice of the
Process
We need to look at specification limits to see the Voice of the
Customer
A process “in control” may be ineffective, inefficient, or both!
Control charts require effective, ongoing data collection. To be
effective for determining root causes of special cause variation,
they must be reacted to immediately!
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Steps in Control Charting
Select process characteristic to control, the key x or Y
Collect data and calculate appropriate statistics
Assess data distribution normality
Construct preliminary control charts
Establish control (find and eliminate special causes)
Construct final control charts
Establish stability (find and reduce common causes)
Use for ongoing control purposes
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What Do Control Charts Tell Us?
When the process mean has shifted I-MR Chart of Pizza Preparation Time
1
1
20
U C L=18.86
When process variability has changed
Individual V alue
15
_
10 X=10.69
When special causes are present
5
LC L=2.52
0
3 6 9 12 15 18 21 24 27 30
Process is not predictable
O bser vation
1
10.0 U C L=10.04
Opportunity to learn about the process
M oving Range
7.5
5.0
When no special causes are present
__
M R=3.07
2.5
0.0 LC L=0
Process is predictable
3 6 9 12 15 18 21 24 27 30
O bser vation
No clues to improvement available; may need to introduce a special cause
in order to understand cause and effect, and then to effect a change
Control charts tell you when, not why!!
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Process Control Chart Template
I-MR Chart of Delivery Time
The current baseline
40
delivery time is stable UC L=37.70
over time with both
Indiv idual V alue
35
the Moving Range 30 _
X=29.13
(3.22 days) and 25
Individual Average
LC L=20.56
(29.13 days) 20
1 28 55 82 109 136 163 190 217 244
experiencing common Observation
cause variation 10.0
UC L=10.53
255 data points
M ov ing Range
7.5
collected with zero 5.0
subgroups, thus the
__
MR=3.22
2.5
I&MR control chart
0.0 LC L=0
selected 1 28 55 82 109 136 163 190 217 244
Observation
- Example - Required As Applicable
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Exercise: Prepare a Control Chart
Objective
Create control charts for the GGA's Budget Department
Instructions
Identify Primary Y metric
Determine best control charts to use
Run proper control chart for that data
Time = 15 Minutes
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Takeaways
Control limits are calculated from a time series of the characteristic we are
measuring
Different formulas are available, depending on the type of data
Control limits should not be recalculated each time data are collected
The control limits are a function of the sampling and subgrouping plan
Variation due to "assignable cause" is often the easiest variation to
reduce/eliminate
Control limits are not related to standards! Nor are they specifications!
Control limits are a measure of what the process is doing/has done. It is the
present/past tense, not the future (what we want the process to do or what it
has the potential to do)
UNCLASSIFIED / FOUO 56
57. UNCLASSIFIED / FOUO
What other comments or questions
do you have?
UNCLASSIFIED / FOUO
58. UNCLASSIFIED / FOUO
References
Wheeler, Donald J. & Chambers, David S.,
Understanding Statistical Process Control, Second
Edition, SPC Press, Knoxville Tennessee, 1992
Pruit, James M. & Snyder, Helmut, Essentials of SPC
in the Process Industries, Instrument Society of
America, 1996
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60. UNCLASSIFIED / FOUO
Western Electric Rules
1. One point beyond Zone A
Detects a shift in the mean, an increase in the
standard deviation, or a single aberration in the
process. For interpreting Test 1, the R chart can be
used to rule out increases in variation.
2. Nine points in a row in Zone C or beyond
Detects a shift in the process mean
3. Six points in a row steadily increasing or
decreasing
Detects a trend or drift in the process mean. Small
trends will be signaled by this test before Test 1.
UNCLASSIFIED / FOUO 60
61. UNCLASSIFIED / FOUO
Western Electric Rules (Cont.)
4. Fourteen points in a row alternating up and down
Detects systematic effects, such as two alternately used
machines, vendors, or operators
5. Two out of three points in a row in Zone A or
beyond
Detects a shift in the process average or increase in the
standard deviation. Any two out of three points provide a
positive test.
6. Four out of five points in Zone B or beyond
Detects a shift in the process mean. Any four out of five
points provide a positive test.
UNCLASSIFIED / FOUO 61
62. UNCLASSIFIED / FOUO
Western Electric Rules (Cont.)
7. Fifteen points in a row in Zone C, above and
below the center line
Detects stratification of subgroups when the
observations in a single subgroup come from various
sources with different means
8. Eight points in a row on both sides of the
center line with none in Zone C
Detects stratification of subgroups when the
observations in one subgroup come from a single
source, but subgroups come from different sources
with different means
UNCLASSIFIED / FOUO 62