Control Charts – Part 1
Data collection, chart ‘anatomy’, control
charts for variables
2
Data Collection
Recall the story about the drunk who loses his keys
and is seen near a street light looking for them.
Someone comes along and offers to help.
Helper – “You sure you dropped them near here?”
Drunk – “Nope…dropped them over there in the bushes.”
Helper – “Then why are we looking here???”
Drunk – “The light is better here.”
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Data Collection FAQs
Q: What should I measure?
A: Depends on what you want to do with the
information and on the process itself
Q: How many pieces should I measure (i.e.,
sample size)?
A: Depends on how accurate you want to be
and the amount of variability.
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Rational Subgrouping
Choices:
Subgroup size
Frequency
Goal:
minimize variability within the subgroup and
maximize (opportunity for variability) between
subgroups
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Rational Subgrouping
Which approach would best meet the goal:
A. Measure 140 pieces at noon each day
B. Measure 5 pieces every 20 minutes between 8
am and 5 pm
C. Measure 70 pieces at 8 am, 70 pieces at 5 pm
D. All of the above
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Anatomy of a Control Chart
Variables (X-bar/R) chart - one chart for
process centering, one chart for spread
Points are average (or range) for each subgroup
Centerline (the ‘grand average’)
Upper and lower control limits (UCL and LCL)
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Calculating Control Limits
To calculate the centerline and limits,
remember what we’re monitoring
For X-bar chart we’re monitoring the average and
the variability of the average
For R chart we’re monitoring the variability and
the variability of the variability (???)
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Setting Up an X Chart
Assumed – Estimates of µ and have
already been made from a large sample.
Draw centerline at X
Draw trial upper limit at X + 3 X
Draw trial lower limit at X - 3 X
But what is this?This we know. It’s the grand
average of all the X from the
samples
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
The distribution of X
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Sample 1
Sample 2
Sample 3
Sample 4
“Spread” of the X’s
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
The spread of the X’s
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
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1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
The spread of the X’s
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
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The Distribution of X
The spread of X ( X) is much less than
the spread of the individual data points.
In fact, statistical theory tells us that:
nX
This we know
This we have to estimate
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Q How can we estimate ?
A From our sample data, either s or R
Q Are these good estimates of ?
A Unfortunately, NO
How well s or R estimate depends on
the sample size. So we use correction
factors:
42
.ˆ
c
sor
d
Rest
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X Chart Limits
Therefore, the limits for the X chart are:
RAXn
dR
XXUCLXX 2
233
RAXn
dR
XXLCLXX 2
233
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Setting up R Charts
Draw centerline at R
Draw trial upper limit at D4R
Draw trial lower limit at D3R
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Control Charts – Let’s build one
Back to XYZ Forest Products
One recommendation based on DOE results was
to control/monitor MC at 6%
Easier said than done!
What does it mean to ‘control?’
How are things now?
How much variability is acceptable?
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Control ChartHands-On
XYZ decides to collect 5 handles from
production every 20 minutes.
They cut 2 inches from the tip of each handle
and measure MC using the ovendry test
Data are in X bar R data.xlsx
Let’s calculate the centerline and control
limits for both the X-bar and the R chart
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Interpreting the Chart
Start with R chart
If variability not ‘in control’, then limits are
meaningless
What are indicators of ‘out of control’?
Next, look at X-bar chart
Histogram
Capability – can we meet specs.?
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Process Capability AnalysisCan we meet the specifications?
Process capability - Measuring process potential by comparing the specification width to the variation of the process
or
How “wide” are the specs relative to the natural spread of the process?
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Capability Indices:
6
LSLUSLC p
ˆ3
ˆ LSLC pl
ˆ3
ˆ
USLC pu
pupl CC
pkC ,min
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A visual look at process variation
µLSL USL(Target) 22
It’s up, it’s… good???
23 ft. 4 inches
B
E
A
V
S
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What are the “specs?”
23.3011.65
B
E
A
V
S
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Kicker “Capability”
Kicker 1 – After 100 kicks, side-to-side
variation is 18 feet (± 9).
Kicker 2 – …variation is 8 feet (± 4).
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Kicker “Capability”
23.30 11.65
Kicker 1
Kicker 2
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Cp
A Simple Ratio
The Specifications - Each kicker has 23.3 total feet of “wiggle room”
“Process” Spread Kicker 1 - 18 feet
Kicker 2 - 8 feet
Their Cp is spec ÷ process Kicker 1 – 23.3/18 = 1.29
Kicker 2 – 23.3/8 = 2.91
Who is the better kicker?
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That depends…
23.30 11.65
Kicker 1
Kicker 2
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Cpk Accounts for Centering
Let’s say:
Kicker 1 centered at 11.65 feet (±9 feet)
Kicker 2 centered at 2 feet (±4 feet)
Cpk
Kicker 1 = MIN[(11.65-0)/9, (23.3-11.65)/9] = 1.29
Kicker 2 = MIN[(2-0)/4, (23.3-2)/4] = 0.5
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How about XYZ’s Process
Can it meet the specifications?
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Does out of control = out of spec?
No!
Control limits and specification limits are not
related
Control limits are based on the ‘voice of the
process’
Specification limits are based on the ‘voice of the
customer’
A process can be in control but out of spec.
A process can be out of control but in spec.
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Attributes Control Charts
So far, only discussed variables data –
measurements of length, width, density, etc.
What about other types of data:
Fraction nonconforming
Counts of nonconformities
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Attributes Control Charts
p charts – counts of fraction nonconforming in a
sample (sample size may vary)
np charts – counts of number of items
nonconforming (requires uniform sample size)
c charts – counts of nonconformities per unit
(sample size may be 1)
u charts – counts of nonconformities per unit area
Discrete data are things we count – require a
different method than continuous data
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Percent Nonconforming p charts
Advantages:
can be constructed
from existing
inspection data
may suggest areas for
use of variables control
charts
fairly simple arithmetic
(look-up tables
unnecessary)
Disadvantages:
don’t provide as much
information as
variables charts
may encourage
continued use of
product QC over
process QC
p charts
Data collection:
n = number of items inspected
D = number of nonconforming items
p = D÷n
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1. Select m preliminary samples, each of size n
2. pi = Di / n where i = 1, 2, …, m (m should = 20 to 25)
3. p = S pi
m
p estimates the unknown fraction nonconforming p
Constructing a p chart: estimating p
i=1
m
n
pppLCL
pCenterline
n
pppUCL
13
13
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p chart example
Let’s use data from XYZ Forest Products’
DOE
Open file pchart.xlsx
Enter data from Table 2, DOE publication
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Interpreting the chart
Is the p-chart in control?
Consider the process we’re analyzing
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Review
Voice of the Customer What’s the most important/costly quality problem?
(Checksheets, Pareto Charts)
Translation Where in our process might these problems occur?
(Flowcharts, C&E Diagrams)
Voice of the Process What variables impact the problem and how? (DOE)
How is our process currently operating? (Control Charts)
Once we ‘fix’ the problem, how can we ensure it stays fixed? (Control Charts)
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Continuous Improvement
40
A Model for Continuous ImprovementNot just problem-solving!
Voice of the Customer
What can we improve that would benefit our customers?
Translation
Where in the process should we make changes?
Voice of the Process
What specific changes should we make?
Once we ‘improve’ how can we ensure things don’t slip back?
Repeat!
41
Plant Optimization
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Q&A/Wrap-Up
43
Addendum
44
SPCStep-by-Step
Do
Plan
Check
Act
What to measure?
Where?
How?
How often?
Collect measurements
Calculate stats
Plot on chart
Interpret the charts
Compare to past data
Fix problems
Standardize processes
Monitor
Continuous improvement
Where does d2 come from?
The value d2 is a correction factor to
estimate from R. It is the average of the
w distribution, where:
Rw
• What does the w distribution look like?
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Example: The w distribution for n = 5
Source: Duncan, A.J. 1986. Quality Control and Industrial
Statistics: Fifth Edition
For n = 5,
d2 = 2.326
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To summarize -
the table values are:
d2 is the average of the w distribution – a
distribution that relates the sample range
R, to the population standard deviation
and
ndA
2
2
3
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Where do D4 and D3 come from?
Recap: To construct an R-chart, we need a
centerline (R) and a value that estimates R,
the variation of sample ranges (the “variation
of the variation”)
Again, there is no formula, however as with
estimating (using d2), using the w
distribution, we get a ratio - R/
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Another table value - d3
Recall that w is R/. The standard deviation
of w (w) is R/, also known as d3 and
2
3
d
RdR
2
3 ˆd
Rd RR
^
^
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And now, to get D4:
RR RUCL 3
2
33d
RdRUCLR
4
2
331 DRd
dRUCLR
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To get D3:
RR RLCL 3
2
33d
RdRLCLR
3
2
331 DRd
dRLCLR
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To summarize:
d3 is the standard deviation of the w
distribution – a distribution that relates the
sample range R, to the population standard
deviation and
2
343 31,
d
dDD
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Acknowledgements
SUSTAINABLE