Testing tools and AI - ideas what to try with some tool examples
Statistical process control
1. S TATISTICAL
PROCESS CONTROL
CUSTOMER & COMPETITIVE INTELLIGENCE FOR
PRODUCT, PROCESS, SYSTEMS & ENTERPRISE EXCELLENCE
DEPARTMENT OF STATISTICS
REDGEMAN@UIDAHO.EDU OFFICE: +1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT
3. Statistical Process Control
Statistical Process Control (SPC) can be
thought of as the application of statistical
methods for the purposes of quality control
and improvement.
Quality Improvement is perhaps foremost
among all areas in business for application
of statistical methods.
5. Control Charts:
Recognizing Sources of Variation
• Why Use a Control Chart?
– To monitor, control, and improve process performance over time by
studying variation and its source.
• What Does a Control Chart Do?
– Focuses attention on detecting and monitoring process variation over
time;
– Distinguishes special from common causes of variation, as a guide to
local or management action;
– Serves as a tool for ongoing control of a process;
– Helps improve a process to perform consistently and predictably for
higher quality, lower cost, and higher effective capacity;
– Provides a common language for discussing process performance.
6. Control Charts:
Recognizing Sources of Variation
• How Do I Use Control Charts?
– There are many types of control charts. The control
charts that you or your team decides to use should be
determined by the type of data that you have.
– Use the following tree diagram to determine which
chart will best fit your situation. Only the most
common types of charts are addressed.
7. Control Chart Selection: Variable Data
Measured & Plotted on a Continuous Scale such as Time,
Temperature, Cost, Figures.
n=1 2<n<9 n is ‘small’ n is ‘large’
median 3<n<5 n > 10
X & Rm X&R X&R X&S
8. Control Chart Selection: Attribute Data
Counted or Plotted as Discrete Events Such as Shipping
Errors, Waste or Absenteeism.
Defect or Defective Data
Nonconformity Data
Constant Variable Constant
Variable
sample size sample size n > 50 n > 50
c chart u chart p or np chart p chart
9. Control Chart
Construction
• Select the process to be charted;
• Determine sampling method and plan;
– How large a sample needs to be selected? Balance the time and cost to collect
a sample with the amount of information you will gather.
– As much as possible, obtain the samples under the same technical conditions:
the same machine, operator, lot, and so on.
– Frequency of sampling will depend on whether you are able to discern
patterns in the data. Consider hourly, daily, shifts, monthly, annually, lots,
and so on. Once the process is “in control”, you might consider reducing the
frequency with which you sample.
– Generally, collect 20-25 groups of samples before calculating the statistics
and control limits.
– Consider using historical data to establish a performance baseline.
10. Control Chart
Construction
• Initiate data collection:
– Run the process untouched, and gather sampled data.
– Record data on an appropriate Control Chart sheet or
other graph paper. Include any unusual events that occur.
• Calculate the appropriate statistics and control limits:
– Use the appropriate formulas.
• Construct the control chart(s) and plot the data.
13. Control Chart Interpretation:
Persistence Through Time
• A process can be characterized by:
– Examining its behavior during a sufficiently brief interlude of time
– Examining its behavior across a greater expanse of time.
• Stable process: one which performs with a high degree of
consistency at an essentially constant level for an extended
period of time
– “In-control”
• A process that is not stable is referred to as being in an out-
of-control state
14. Data Plot with PAT Zones
36.10 (A)
36
34.36 (B)
34
32.62 (C)
32
30.88
30
29.14 (C)
28 27.40 (B)
26 25.66 (A)
0 5 10 15 20 25 30
Item
15. Control Chart Interpretation:
Pattern Analysis Tests (PATs)
• PAT 1: One point plots beyond zone A on
either side of the mean
• PAT 2: Nine points in a row plot on the same
side of the mean
• PAT 3: Six consecutive points are strictly
increasing or strictly decreasing
• PAT 4: Fourteen consecutive points which
alternate up and down
16. Control Chart Interpretation:
Pattern Analysis Tests
• PAT 5: Two out of three consecutive points
plot in zone A or beyond, and all three points
plot on the same side of the mean
• PAT 6: Four out of five consecutive points
plot in zone B or beyond, and all five points
plot on the same side of the mean
17. Control Chart Interpretation:
Pattern Analysis Tests
• PAT 7: Fifteen consecutive points plot in
zones C, spanning both sides of the mean
• PAT 8: Eight consecutive points plot at more
than one standard deviation away from the
mean with some smaller than the mean and
some larger than the mean
18. Control Chart Interpretation:
Monitoring & Improving Processes
• The performance of every process will be
composed of two primary components:
– Controlled or guided performance which is
predictable in both an instantaneous and long-term
sense
– Uncontrolled variation
• Special or assignable causes
• Common causes
19. Control Chart Interpretation:
Monitoring & Improving Processes
• True process improvement is typically a
result of either:
– Breakthrough thinking
– Efforts to identify and reduce or eliminate common
causes of variation; methodical quantitatively oriented
tools which monitor a process over time --- the
approach taken generally by “control charts”.
20. Control Chart Interpretation
• The vertical axis coordinate of a point
plotted on the chart corresponding to the
value of an appropriate PPM and the
horizontal axis coordinate of a point plotted
on the chart corresponding to the time in
sequence at which the observation was
made with the time between observations
divided into equal increments.
21. Control Charts: Colors Used
UCL
A
* U2SWL
B *
* U1SL
** C
* * * * CL
* *
C *
* L1SL
* B *
L2SWL
A
LCL
22. P Charts for the Process Proportion
Based on m preliminary samples from the process. While the
number of items, n, may vary from sample to sample, it is
customary for each of the samples in a given application to
include the same number of items, n. For the ith of these m
samples, let
Yi = number of defective units in the sample
Then the proportion defective for the ith sample is:
pi = Yi / ni
23. Control Chart Interpretation
• Center line (CL) positioned at the estimated mean
• Upper and lower one standard deviation lines (U1SL and
L1SL) positioned one standard deviation above and
below the mean.
• Upper and lower two standard deviation warning lines
(U2SWL and L2SWL) positioned at two standard
deviations above and below the mean.
• Upper and lower control lines (UCL and LCL) positioned
at three standard deviations above and below the mean.
24. P Charts for the Proportion
An estimate of the overall process proportion defective is
p = (Y1+Y2+...+ Ym) / (n1+n2+...+ nm)
= (total defectives) / (total items)
When all samples have n items each then p = (p1 + p2 + ... + pm)/m
The estimated standard deviation of the process proportion
defective is
Sp = √ p (1-p)/ ni
25. P Chart Control Lines & Limits
The coordinates for the seven lines on the P
chart are positioned at:
CL = p
U1SL = p + Sp L1SL = p - Sp
U2SWL = p + 2Sp L2SWL = p - 2Sp
UCL = p + 3Sp LCL = p - 3Sp
26. South of the Borders, Inc.
Custom Wallpapers & Borders
Free Estimates
(013) 555-9944
27. South of the Borders, Inc.
South of the Borders, Inc. is a custom wallpapers and borders
manufacturer. While their products vary in visual design, the
manufacturing process for each of the products is similar.
Each day a sample of 100 rolls of wallpaper border is
sampled and the number of defective rolls in the sample is
noted.
The number of defective rolls in samples from 25 consecutive
production days follows.
Determine all coordinates; construct & interpret the p chart.
PATs 1, 2, 3 and 4 apply to p charts.
31. P Chart for Defective Wallpaper Rolls
3.0SL=0.1590
0.15
2.0SL=0.1322
Proportion
1.0SL=0.1053
0.10
P=0.07840
0.05 -1.0SL=0.05152
-2.0SL=0.02464
0.00 -3.0SL=0.000
Subgroup 0 5 10 15 20 25
Rolls 8 6 9 11 10
Proportion of Defective Rolls Received
32. South of the Borders, Inc.
P Chart Interpretation
• No violations of PATs one through four are apparent.
This implies that the process is “in a state of statistical
control”.
• It does not indicate that we are satisfied with the
performance of the process.
• It does, however, indicate that the process is stable
enough in its performance that we may seriously
engage in PDCA for the purpose of long-term process
improvement.
33. C and U Charts for
Nonconformities
• When data originates from a Poisson
process, it is customary to monitor output
from the process with a defects or C chart
• Recall the Poisson Distribution with mean =
c and standard deviation = √c
• P(y) = cye-c/y!
34. C & U Charts for Nonconformities
• C represents the average number of defects
(nonconformities) per measured unit with all units
assumed to be of the same “size” and all samples are
assumed to have the same number of units
• m = 20 to 40 initial samples
• C = (number of defects in the m samples) / m
• Estimated standard deviation= √C
35. C Control Chart
Coordinates
• CL = C
• UCL = C+3 C and LCL = C-3 C
• U2SWL= C+2 C and L2SWL = C- 2 C
• U1SL = C+ C and L1SL = C- C
37. Scientific & Technical Materials, Inc.
• Scientific & Technical Materials, Inc. produces
material for use as gaskets in scientific, medical,
and engineering equipment. Scarred material can
adversely affect the ability of the material to fulfill
its intended use.
• A sample of 40 pieces of material, taken at a rate
of 1 per each 25 pieces of material produced gave
the results on the following slide. Use this
information to construct and interpret a C chart.
40. Scientific & Technical Materials, Inc.
C Chart for Gasket Material Data
10
9
8
7 UCL
6
U2SWL
5
4
U1SL
3
CL
2
1 L1SL
0
1
3
5
7
9
25
11
13
15
17
19
21
23
27
29
31
33
35
37
39
41. Scientific & Technical Materials, Inc.
C Chart Interpretation
• Application of PATs one through four indicates a
violation of PAT 1 at sample number 39 where 9
scars appear on the surface of the sampled material.
• Corrective measures would be identified and
implemented.
• After process stability was (re) assured, we would
move into PDCA mode.
42. Variation of the C chart
where Sample size may vary
U = (u1+u2+...+um) / (n1+n2+...+nm)
= (total # of defects) / (total # of units in the m samples)
• CL = U
• UCL = U+ 3 U/ni, LCL= U-3 U/ni
• U2SWL= U+ 2 U/ni, L2SWL= U- 2 U/ni
• U1Sl= U+ U/ni, L1SL= U- U/ni
U Chart
43. Control Charts for the
Process Mean and Dispersion
‘X bar’ Chart
Typically used to monitor process centrality (or location)
Limits depend on the measure is used to monitor process dispersion
(R or S may be used).
‘S’ or ‘Standard Deviation’ Chart:
Used to monitor process dispersion
‘R’ or ‘Range’ Chart:
Also used to monitor process dispersion
44. Sample Summary Information
• m = 20 to 40 initial samples of n observations each.
• Xi = mean of ith sample
• Si = standard deviation of ith sample
• Ri = range of ith sample
X = (X1 + X2 +... + Xm) / m
R = (R1 + R2 + ... +Rm)/m
S = (S1 + S2 + ... + Sm)/m
σ = R/d2 where d2 depends only on n
45. Coordinates for the X-bar Control Chart: “R”
• CL= X,
• UCL= X+ A2R,
• UCL= X- A2R
• U2SWL= X+ 2A2R/3
• L2SWL= X- 2A2R/3
• U1SL= X+ A2R/3
• L1SL= X- A2R/3
A2 is a constant that depends only on n.
46. Coordinates for an
R Control Chart
• CL= R
• UCL= D4R
• LCL= D3R
• U2SWL= R+ 2(D4-1)R/3
• L2SWL= R- 2(D4-1)R/3
• U1SL= R+ (D4-1)R/3
• L1SL= R- (D4-1)R/3
• where D3 and D4 depend only on n
48. Championship Card Company
Championship Card Company (CCC) produces collectible
sports cards of college and professional athletes.
CCCs card-front design uses a picture of the athlete, bordered
all-the-way-around with one-eighth inch gold foil. However,
the process used to center an athlete’s picture does not function
perfectly.
Five cards are randomly selected from each 1000 cards produced
and measured to determine the degree of off-centeredness of each
card’s picture. The measurement taken represents percentage
of total margin (.25”) that is on the left edge of a card. Data
from 30 consecutive samples is included with your materials,
and summarized on the following slides.
50. Championship Card Company
Summary Information
n=5 A3 = 1.427
X = 49.63 B3 = NA
S = 7.42 B4 = 2.089
R = 18.63 D3 = NA
d2 = 2.326
D4 = 2.115
A2 = 0.577
σ = R/d2 = 8.01
51. Championship Card Company
X-bar and R Control Chart Limits
X based on R R
UCL 60.38 39.40
U2SWL 56.80 32.48
U1SL 53.22 25.55
CL 49.63 18.63
L1SL 46.05 11.71
L2SWL 42.47 4.79
LCL 38.89 ------
52. Championship Card Company
X Bar Chart for Sports Cards Centering Values
Limits Based on R
1
60 3.0SL=60.38
2.0SL=56.80
S m Me n
a ple a
1.0SL=53.22
50 X=49.63
- 1.0SL=46.05
- 2.0SL=42.47
40
- 3.0SL=38.89
0 10 20 30
Sample Number
Samples of 5 from each 1000 Cards Printed
53. Championship Card Company
R Chart for Sports Card Centering
40 3.0SL=39.40
2.0SL=32.48
30
a ple a ge
S m Rn
1.0SL=25.55
20
R=18.63
- 1.0SL=11.71
10
- 2.0SL=4.791
0 - 3.0SL=0.000
0 10 20 30
Sample Number
Samples of 5 Cards from each 1000 Produced
54. Championship Card Company
X-bar & R Chart Interpretation
• Application of all eight PATs to the X-bar chart indicated a
violation of PAT 1 (one point plotting above the UCL) at sample
2. Apparently, a successful process adjustment was made, as
suggested by examination of the remainder of the chart.
• Application of PATs one through four to the R chart indicated a
violation of PAT 1 at sample 29. Measures would be investigated
to reduce process variation at that point. The violation was a
“close call” and was out of character with the remainder of the
data.
• We are close to being able to apply PDCA to the process for the
purpose of achieving lasting process improvements.
55. Coordinates for the X bar Control Chart: “S”
• CL= X
• UCL= X= A3S
• LCL= X- A3S
• U2SWL= X+ 2A3S/3
• L2SWL= X- 2A3S/3
• U1SL= X+ A3S/3
• L1SL= X- A3S/3
• where A3 depends only on n
56. Coordinates on an S Control Chart
• CL= S
• UCL= B4S
• LCL= B3S
• U2SWL= S+ 2(B4-1)S/3
• L2SWL= S- 2(B4-1)S/3
• U1SL= S+ (B4-1)S/3
• L1SL= S- (B4-1)S/3
• where B3 and B4 depend only on n
58. Championship Card Company
X-bar and S Chart Limits
X based on S S
UCL 60.22 15.49
U2SWL 56.69 12.80
U1SL 53.16 10.11
CL 49.63 7.42
L1SL 46.11 4.72
L2SWL 42.58 2.03
LCL 39.05 ------
59. Championship Card Company
X Bar Chart for Sports Cards Centering Values
1 Limits Based on S
60 3.0SL=60.22
2.0SL=56.69
S m Me n
a ple a
1.0SL=53.16
50 X=49.63
- 1.0SL=46.11
- 2.0SL=42.58
40
- 3.0SL=39.05
0 10 20 30
Sample Number
Samples of 5 from each 1000 Cards Printed
60. Championship Card Company
S Chart for Sports Card Centering Values
15 3.0SL=15.49
2.0SL=12.80
Sm S v
a ple tde
10 1.0SL=10.11
S=7.416
5 - 1.0SL=4.724
- 2.0SL=2.032
0 - 3.0SL=0.000
0 10 20 30
Sample Number
5 Cards Sampled from each 1000 Cards Produced
61. Championship Card Company
X-bar & S Chart Interpretation
• Application of all eight PATs to the X-bar chart indicates a
violation of PAT 1 (one pt. above the UCL) at sample 2.
Judging from the remainder of the chart, the process was
successfully adjusted.
• Application of the first four PATs to the S chart indicates no
violations.
• In summary, the process appears to have been temporarily
“out-of-control” w.r.t. its mean at sample 2. The process was
successfully adjusted and may now be subjected to PDCA for
permanent improvement purposes.
62. Common Questions for Investigating an
Out-of-Control Process
• Are there differences in the measurement accuracy of instruments /
methods used?
• Are there differences in the methods used by different personnel?
• Is the process affected by the environment, e.g.
temperature/humidity?
• Has there been a significant change in the environment?
• Is the process affected by predictable conditons such as tool wear?
• Were any untrained personnel involved in the process at the time?
• Has there been a change in the source for input to the process such as
a new supplier or information?
• Is the process affected by employee fatigue?
63. Common Questions for Investigating an
Out-of-Control Process
• Has there been a change in policies or procedures such as
maintenance procedures?
• Is the process frequently adjusted?
• Did the samples come from different parts of the process? Shifts?
Individuals?
• Are employees afraid to report “bad news”?
64. Process Capability:
The Control Chart Method for Variables Data
1. Construct the control chart and remove all special causes.
NOTE: special causes are “special” only in that they come and go,
not because their impact is either “good” or “bad”.
3. Estimate the standard deviation. The approach used depends on
whether a R or S chart is used to monitor process variability.
^ _ ^ _
σ = R / d2 σ = S / c4
Several capability indices are provided on the following slide.
65. Process Capability Indices: Variables Data
^ ^
CP = (engineering tolerance)/6σ = (USL – LSL) / 6σ
This index is generally used to evaluate machine capability.
tolerance to the engineering requirements. Assuming that
the process is (approximately) normally distributed and
that the process average is centered between the
specifications, an index value of “1” is considered to
represent a “minimally capable” process. HOWEVER …
allowing for a drift, a minimum value of 1.33 is ordinarily
sought … bigger is better. A true “Six Sigma” process that
allows for a 1.5 σ shift will have Cp = 2.
66. Process Capability Indices: Variables Data
^ ^
CR = 100*6σ / (Engineering Tolerance) = 100* 6 σ /(USL –LSL)
This is called the “capability ration”. Effectively this
is the reciprocal of Cp so that a value of less than
75% is generally needed and a Six Sigma process
(with a 1.5σ shift) will lead to a CR of 50%.
67. Process Capability Indices: Variables Data
^ ^
CM = (engineering tolerance)/8σ = (USL – LSL) / 8σ
This index is generally used to evaluate machine capability.
Note … this is only MACHINE capability and NOT the
capability of the full process. Given that there will be
additional sources of variation (tooling, fixtures, materials,
etc.) CM uses an 8σ spread, rather than 6σ. For a machine
to be used on a Six Sigma process, a 10σ spread would be
used.
68. Process Capability Indices: Variables Data
= ^ = ^
ZU = (USL – X) / σ ZL = (X – LSL) / σ
Zmin = Minimum (ZL , ZU)
Cpk = Zmin / 3
This index DOES take into account how well or how poorly
centered a process is. A value of at least +1 is required with a
value of at least +1.33 being preferred.
Cp and Cpk are closely related. In some sense Cpk represents the
current capability of the process whereas Cp represents the
potential gain to be had from perfectly centering the process
69. Process Capability: Example
Assume that we have conducted a capability analysis using X-bar and R
charts with subgroups of size n = 5. Also assume the process is in
statistical control with an average of 0.99832 and an average range of
0.02205. A table of d2 values gives d2 = 2.326 (for n = 5). Suppose LSL =
0.9800 and USL = 1.0200
^ _
σ = R / d2 = 0.02205/2.326 = 0.00948
Cp = (1.0200 – 0.9800) / 6(.00948) = 0.703
CR = 100*(6*0.00948) / (1.0200 – 0.9800) = 142.2%
CM = (1.0200 – 0.9800) / (8*(0.00948)) = 0.527
ZL = (.99832 - .98000)/(.00948) = 1.9
ZU = (1.02000 – .99832)/(.00948) = 2.3 so that Zmin = 1.9
70. Process Capability: Interpretation
Cp = 0.703 … since this is less than 1, the process is not regarded as being
capable.
CR = 142.2% implies that the “natural tolerance” consumes 142% of the
specifications (not a good situation at all).
CM = 0.527 = Being less than 1.33, this implies that – if we were dealing with a
machine, that it would be incapable of meeting requirements.
ZL = 1.9 … This should be at least +3 and this value indicates that
approximately 2.9% of product will be undersized.
ZU = 2.3 should be at least +3 and this value indicates that approximately 1.1%
of product will be oversized.
Cpk = 0.63 … since this is only slightly less that the value of Cp the indication is
that there is little to be gained by centering and that the need is to reduce
71. S TATISTICAL
PROCESS CONTROL
End of Session
DEPARTMENT OF STATISTICS
REDGEMAN@UIDAHO.EDU OFFICE: +1-208-885-4410
DR. RICK EDGEMAN, PROFESSOR & CHAIR – SIX SIGMA BLACK BELT
