Six Sigma Process Capability Study (PCS) Training Module

1 April 9, 2016 – v3.0
Six Sigma Process Capability Study (PCS)
by Operational Excellence Consulting LLC

2 April 9, 2016 – v3.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Basic Statistics and Process Capability
Section 4: Introduction to Statistical Process Control
Section 5: Definitions of Process Capability Indices
Section 6: Non-Normal Distributed Processes
Process Capability Study – Table of Contents

3 April 9, 2016 – v3.0

4 April 9, 2016 – v3.0
How to get EVIDENCE to the following questions ?
 Case 1: RF Design specifies a critical capacitor in a design with (2 ± 0.1)F.
Is a particular supplier able to provide us with this type of component? How
well can the supplier meet the target/nominal value of 2F?
 Case 2: Mechanical Design specifies the distance between two holes by (32
± 0.5)mm. Is a particular supplier able to provide us with the required
accuracy? How well can the supplier meet the target/nominal value of 32mm?
 Case 3: The GSM Standard requires that the test “Peak TX Power” falls
between 28.3dB and 32.0dB. Is the new product able to fulfill this
requirement?
 Case 4: A fine pitch component requires a placement accuracy of ± 0.5mm. Is
the existing placement machine able to fulfill this requirement?

5 April 9, 2016 – v3.0
The process and quality control methods and techniques used today
got their start in the American Civil War at around 1789, when Eli
Whitney took a contract from the U.S. Army for the manufacture of
10,000 rifles at the unbelievably low price of $13.40 each.
At that time most of the products were handmade by small owner-
managed shops and product parts were thus not interchangeable.
The result of Whitney’s first mass production trail was that the rifles did
not work as well as the handmade rifles. In addition, the copied parts
did not fit as expected.
The History of Statistical and Process Thinking

6 April 9, 2016 – v3.0
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”.
The History of Statistical and Process Thinking

7 April 9, 2016 – v3.0
Process Inspection
Good
Bad
Repair
Scrap
+
Monitor/Adjust
The Traditional Production Concept
The Detection Control Scheme

8 April 9, 2016 – v3.0
• The traditional production concept does not help us to
produce only good products.
• Every product has to be inspected.
• Products have to be repaired or even scraped.
• With respect to productivity and efficiency every activity
after the actual production process is a non-value added
activity.

9 April 9, 2016 – v3.0
Prevention Control Scheme
Process Inspection
Good
Bad
Repair
Scrap
+

10 April 9, 2016 – v3.0
Prevention Control Scheme
Process Inspection
Good
Bad
Repair
Scrap
+
An Advanced Production Concept
Monitor/Adjust
Learn/Improve
Selective measurement
• Product
• Process

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Statistical Process Thinking - A Definition
All work is a series of
interconnected processes
All processes vary
Understanding, reducing and
controlling process variation
are keys to success
ASQ

12 April 9, 2016 – v3.0
Customer Satisfaction
or
Customer Dissatisfaction
Process/
System
Material
Machines Methods
Men
Environment
The Variation Management Approach

13 April 9, 2016 – v3.0
The Six Sigma Approach
Analytically speaking, this understanding may be expressed as
The process output “y” is a function of the process inputs “x1, x2, ... , xN”
where y is some product or process characteristic, also called the dependent
variable, and (x1, x2 ,..., xN) describes all the independent variables in the
cause system.
Thus, we may interpret this expression to mean the output variable (y) is a
function (f) of the input variables (x1, x2,..., xN) .
y = f(x1, x2, ... , xN)

14 April 9, 2016 – v3.0
Six Sigma - What is a Defect ?
A defect is any variation of a required
characteristic of the product or its part,
which is far enough removed from its
nominal value to prevent the product from
fulfilling the physical and functional
requirements of the customer.

15 April 9, 2016 – v3.0
The key to process control and continuous process
improvement is to understand the meaning and causes
of variation in the outcome of the process.
Variation Management – Continuous Improvement

16 April 9, 2016 – v3.0
Upper Specification Limit (USL)
Defect
nominal value
Process Capability and Process Control
Lower Specification Limit (LSL)
Defect
nominal value

17 April 9, 2016 – v3.0
Defect
nominal value
Defect
nominal value
process not capable
large variation & problem exist
root cause analysis
process improvement

18 April 9, 2016 – v3.0
process not capable
process out-of-control (trend)
corrective action
Defect
nominal value
Defect
nominal value
process not capable
large variation & problem exist
process improvement

19 April 9, 2016 – v3.0
Product/Process Quality and Variation
“Traditional”
Attitude
Nominal
value
LSL USL
100 %
0 %
 Inspection/Yield and
re-active Problem Solving
Quality
“Six Sigma”
Attitude
Nominal
value
LSL USL
100 %
0 %
 Process Capability &
Process Control with
pro-active Improvements
Quality

20 April 9, 2016 – v3.0
Supplier Selection based on Yield or Process Capability
125.08125.06125.04125.02125.00124.98124.96124.94124.92
Upper SpecLower Spec
s
Mean-3s
Mean+3s
Mean
n
k
LSL
USL
Targ
Cpm
Ppk
PPL
PPU
Pp
Long-Term Capability
0
24969
0
32255
0.00
2.50
0.00
3.23
Obs
PPM<LSL Exp
Obs
PPM>USL Exp
Obs
%<LSL Exp
Obs
%>USL Exp
0.026
124.923
125.080
125.001
782.000
0.029
124.950
125.050
*
*
0.62
0.65
0.62
0.63
Process Capability Analysis for Supplier 1
125.04125.02125.00124.98124.96
s
Mean-3s
Mean+3s
Mean
n
k
LSL
USL
Targ
Cpm
Ppk
PPL
PPU
Pp
Long-Term Capability
0
1
0
1
0.00
0.00
0.00
0.00
Obs
PPM<LSL Exp
Obs
PPM>USL Exp
Obs
%<LSL Exp
Obs
%>USL Exp
0.010
124.969
125.031
125.000
1000.00
0.00
124.95
125.05
*
*
1.62
1.63
1.62
1.62
Process Capability Analysis for Supplier 2
100 % Yield due to 100 % Inspection 100 % Yield due to a capable Process
Which Supplier would you select ???

21 April 9, 2016 – v3.0
Remarks or Questions ?!?

22 April 9, 2016 – v3.0

23 April 9, 2016 – v3.0
A histogram provides graphical presentation and a first estimation
about the location, spread and shape of the distribution of the process.
0 10 20 30 40 50
The Histogram

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Step 1: Collect at least 50 data points, but better 75 to 100 points, and organize
your data into a table. Sort the data points from smallest to largest and calculate
the range, means the difference between your largest and smallest data point, of
your data points.
The Histogram – How to create a Histogram?
Actual Measurements
Part Hole Size
1 2.6
2 2.3
3 3.1
4 2.7
5 2.1
6 2.5
7 2.4
8 2.5
9 2.8
10 2.6
Sorted Measurements
Part Hole Size
5 2.1
2 2.3
7 2.4
6 2.5
8 2.5
1 2.6
10 2.6
4 2.7
9 2.8
3 3.1
Minimum = 2.1
Maximum = 3.1
Range = 1.0

25 April 9, 2016 – v3.0
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.
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)

26 April 9, 2016 – v3.0
Step 3: Calculate the “start” and “end” point of each bar and count how many
data points fall between “start” and “end” point of each bar.
Start End
Bar 1 2.1 2.1 + 0.2 = 2.3
Bar 2 2.3 2.5
Bar 3 2.5 2.7
Bar 4 2.7 2.9
Bar 5 2.9 3.1
Minimum = 2.1
Maximum = 3.1
Range = 1.0
Bar Width = 0.2 (5 Bars)
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

27 April 9, 2016 – v3.0
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.
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

28 April 9, 2016 – v3.0
1. The bell-shaped distribution:
Symmetrical shape with a peak in the
middle of the range of the data.
While deviation from a bell shape should
be investigated, such deviation is not
necessarily bad.
The Histogram – Typical Patterns of Variation

29 April 9, 2016 – v3.0
2. The double-peaked distribution:
A distinct valley in the middle of the range
of the data with peaks on either side.
This pattern is usually a combination of
two bell-shaped distributions and suggests
that two distinct processes are at work.

30 April 9, 2016 – v3.0
3. The plateau distribution:
A flat top with no distinct peak and slight
tails on either sides.
This pattern is likely to be the result of
many different bell-shaped distribution
with centers spread evenly throughout the
range of data.

31 April 9, 2016 – v3.0
4. The skewed distribution:
An asymmetrical shape in which the peak
is off-center in the range of the data and
the distribution tails off sharply on one
side and gently on the other.
This pattern typically occurs when a
practical limit, or a specification limit,
exists on one side and is relatively close
to the nominal value.

32 April 9, 2016 – v3.0
5. The truncated distribution:
An asymmetrical shape in which the peak
is at or near the edge of the range of the
data, and the distribution ends very
abruptly on one side and tails off gently on
the other.
This pattern often occurs if the process
includes a screening, 100 % inspection, or
a review process. Note that these
truncation efforts are an added cost and
are, therefore, good candidates for
removal.

33 April 9, 2016 – v3.0
The Histogram – The Bell-Shaped or Normal Distribution

34 April 9, 2016 – v3.0
The Histogram – Exercise 1
Distribution of Heights of U.S. Population:
Use the plot area below to construct a histogram from the
random sample of heights on the right:
59 66 63 70
60 66 69 70
65 62 71 72
68 65 67 69
65 66 70 68
64 64 73 73
63 67 71 68
63 68 70 68
65 67 64 71
61 64 70 72
70 63 68 68
68 63 66 66
64 63 67 74
63 62 66 68
62 62 67 70

35 April 9, 2016 – v3.0
Remarks or Questions ?!?

36 April 9, 2016 – v3.0

37 April 9, 2016 – v3.0
What are the Two Types of Product Characteristics ?
 Attribute: A characteristic that by comparison to some standard is judged
“Good” or “Bad” (free from scratches, fits, etc.)
 Variable: A characteristic measured in physical units (inches, volts, amps,
decibel, seconds, etc.)
 Variable characteristics are specified by designers as a nominal value
(target) and a tolerance around the target (variability).
 Manufacturing processes attempt to produce at the nominal value of the
characteristic.
 Since no process is perfect, variation of the characteristics will occur.
 Products with a product characteristic that falls inside the given
tolerances will be defined as “good” or “acceptable”.
 Products with a product characteristic that falls outside the given
tolerances will be defined as “bad” or “unacceptable”.

38 April 9, 2016 – v3.0
Relationship between Tolerance to Nominal Value
 The required value of a product or process characteristic is specified
as its nominal value.
 The maximum range of acceptable variation of the product or process
characteristic which will still work in the product determines the
tolerances about the nominal value.
Nominal Value
Specification
Upper
Spec. Limit (USL)
Lower
Spec. Limit (LSL) Tolerance

39 April 9, 2016 – v3.0
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
Measure of Location – The Sample Average
Definition:
N
xxx
x N

...21
8.4
5
24
5
62475


x

40 April 9, 2016 – v3.0
Example 1: x1 = 2 x2 = 5 x3 = 4
Definition: Order all data points from the smallest to largest. Then choose the
middle data point if the number of data points is odd, or the mean value of the
two middle data points if the number of data points is even.
Example 2: x1 = 5 x2 = 7 x3 = 4 x4 = 2
Example 3: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6 ?
Measure of Location – The Sample Median
2 – 5 – 4 2 – 4 – 5 median = 4
5 – 7 – 4 – 2 2 – 4 – 5 – 7 median = 4.5

41 April 9, 2016 – v3.0
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
Measure of Variability – The Sample Range
),...,,min(),...,,max( 2121 NN xxxxxxR 
Definition:
527)6,2,4,7,5min()6,2,4,7,5max( R

42 April 9, 2016 – v3.0
x3
x
average
_
x2
x1
x10
Measure of Variability – Sample Variance
     
9)110(
...
2
10
2
2
2
1
or
xxxxxx


Time
x6𝑥3 - 𝑥
𝑥2 - 𝑥

43 April 9, 2016 – v3.0
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
Measure of Variability – Sample Variance
     
)1(
...
22
2
2
12



N
xxxxxx
s N
          7.3
)15(
8.468.428.448.478.45
22222
2



s
Definition:

44 April 9, 2016 – v3.0
Example: x1 = 5 x2 = 7 x3 = 4 x4 = 2 x5 = 6
Measure of Variability – Sample Standard Deviation
     
)1(
...
22
2
2
1



N
xxxxxx
s N
LT
Definition:
          7.3
)15(
8.468.428.448.478.45
22222
2



LTs
92.17.3 LTs

45 April 9, 2016 – v3.0
Time t
Process
Characteristic
e.g. Hole Size
Process not in control
average
Subgroup size n = 5
Number of subgroups N = 7
Measure of Variability – The Principle of Subgrouping

46 April 9, 2016 – v3.0
Where
is the range of subgroup j, N the number of
subgroups, and d2 depends on the size n of a
subgroup (see handout).
sST , often notated as s or sigma, is another measure of
dispersion or variability and stands for “short-term
standard deviation”,
which measures the variability of a process or system
using “rational” subgrouping.
Measure of Variability – Standard Deviation sST
22
21 ...
dRd
N
RRR
s N
ST 


minmax XXRj 
n
2
3
4
5
6
7
8
9
10
d2
1.128
1.693
2.059
2.326
2.534
2.704
2.847
2.970
3.078

47 April 9, 2016 – v3.0
Time t
Process
Characteristic
e.g. Hole Size
Process not in control
average
Subgroup size n = 5
Number of subgroups N = 7
Measure of Variability – The Principle of Subgrouping
 sST stays the same, even if the process is not in control
 sLT increases over time because the process is not in control
 sST and sLT are identical if the process was in control

48 April 9, 2016 – v3.0
Long-term standard deviation:
Short-term standard deviation:
The difference between the standard deviations sLT and sST gives an
indication of how much better one can do when using appropriate
production control, like Statistical Process Control (SPC).
     
)1(
...
22
2
2
1



N
xxxxxx
s N
LT
Measure of Variability – Difference between sLT and sST
22
21 ...
dRd
N
RRR
s N
ST 



49 April 9, 2016 – v3.0
averageaverage
-1*s(igma)
average
-2*s(igma)
average
-3*s(igma)
average
+1*s(igma)
average
+2*s(igma)
average
+3*s(igma)
34.13 %34.13 %
13.60 % 13.60 %
2.14 %2.14 %
0.13 % 0.13 %
Measure of Variability – The Normal Distribution
If your process is under control, over 99.74% of your data points will fall between the
average ± 3s(igma) limits.
The distance between average ± 3s(igma) limits is called the width of the process or process capability.
Lower
Control Limit
Upper
Control Limit

50 April 9, 2016 – v3.0
Measure of Location and Variability – Exercise 2
Calculate the Mean Value or Average, Median, Range, and
long- and short-term Standard Deviation of the sample data.
You may copy the data into MS Excel and simplify the
calculations.
Group
1 59 66 63 62
2 60 66 69 65
3 65 62 71 72
4 68 65 67 69
5 65 66 70 68
6 64 64 73 73
7 63 67 71 68
8 63 68 65 68
MeasurementsOverall Mean Value =
Overall Median =
Subgroup Ranges =
Long-term Standard Deviation =
Short-term Standard Deviation =
Note: The Excel function for the Long-Term Standard Deviation is “stdev()”.

51 April 9, 2016 – v3.0
Measure of Location and Variability – Exercise 2 Results
Subgroup Median Range
1 59 66 63 62 63 7
2 60 66 69 65 66 9
3 65 62 71 72 68 10
4 68 65 67 69 68 4
5 65 66 70 68 67 5
6 64 64 73 73 69 9
7 63 67 71 68 68 8
8 63 68 65 68 67 5
Overall Range: 14
Overall Median: 66
Average Range: 7.1
Short-Term Standard Deviation: 3.46
Long-Term Standard Deviation: 3.55
Measurements

52 April 9, 2016 – v3.0

53 April 9, 2016 – v3.0
Attribute Data
(Count or Yes/No Data)
Variable Data
(Measurements)
Variable
subgroup
size
Subgroup
size
of 1
Fixed
subgroup
size
x 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
Process Control Charts – Types of Control Charts
Type of Data

54 April 9, 2016 – v3.0
 The x - 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.
This represents “long term” variation in the process.
 The second source of variation is the variation between
successive data points. This represents “short term” variation.
 Individual or x - charts should be used when there is only one data
point to represent a situation at a given time.
 To use the x - chart, the individual sample results should be
sufficient normally distributed. If not, the x - chart will give more
false signals.
Process Control Charts – The x - Chart

55 April 9, 2016 – v3.0
Upper control limit =
Lower control limit =
The x- chart
The R- chart
,
where x1, x2, ..., xN are the measurements, N the number of measurements,
, and .
Process Control Limit – The x - Chart
RxRxdRx  66.2128.133 2
RxRxdRx  66.2128.133 2
RRD  267.34
003  RRD
N
xxx
x N

...21
1
...32



N
RRR
R N
1 iii xxR

56 April 9, 2016 – v3.0
Process Control Charts – x - Chart Example
1 0 .0
C HA
No k
F ran
to
nt h
OBSERVATIONS7
9
1 1
1 3
1 5
1 7
1 9
2
1
1
2
EEEEEEEEEEEEEEEEEEEEEEEE
A
L
U
RANGES0
1
2
3
4
5
6
L
U
R
01.01.8701.02.8701.03.8701.04.8701.05.8701.06.8701.07.8701.08.8701.09.8701.10.8701.11.8701.12.8701.01.8801.02.8801.03.8801.04.8801.05.8801.06.8801.07.8801.08.8801.09.8801.10.8801.11.8801.12.88
1 00
G ro
A uto
C L O
C urv
K -S
A V E
P R O
UC L
L C L
re e n
128.12 RdRsST 

57 April 9, 2016 – v3.0
Process Control Charts – The Central Limit Theorem
Regardless of the shape of the distribution of a population, the
distribution of average values, x-bar’s, of subgroups of size n drawn
from that population will tend toward a normal distribution as the
subgroup size n becomes large.
Laplace and Gauss
The standard deviation sx of the subgroup averages is smaller than the
standard deviation s of the individual measurements. The relationship
between these two standard deviation s and sx as follows, where n is
the nuymber of measurements in each subgroup:
_
_
nssx


58 April 9, 2016 – v3.0
Process Control Charts – Exercise 3
Throw the Dice:
Step 1: Throw the dice 30 times and record the results in the table on the right.
Step 2: Draw a Histogram #1 of the 30 data points in one of the spreadsheets below.
Step 3: Calculate the average to 2 consecutive throws and draw the histogram #2 of the resulting 15 data
points.
What do you see and why?
AverageResults
Histogram #1 Histogram #2

59 April 9, 2016 – v3.0
 The (x-bar / R) - chart should be used if
 the individual measurements are not normally distributed,
 one can rationally subgroup the data and is interested in
detecting differences between the subgroups over time.
 The (x-bar / R) - chart is a method of looking at two different
sources of variation. One source is the variation in subgroup
averages. The other source is the variation within a subgroup.
 The x-bar - chart shows variation over time or long-term variation
and the R - chart is a measure of the short-term variation in the
process.
Process Control Charts – The (x-bar/R) - Chart

60 April 9, 2016 – v3.0
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 Limit – The x-bar/R - Chart
  RAxndRx  223
RD 4
RD 3
N
xxx
x
N

...21
N
RRR
R N

...21minmax
iii xxR 
  RAxndRx  223

61 April 9, 2016 – v3.0
n
2
3
4
5
6
A2
1.880
1.023
0.729
0.577
0.483
D3
0
0
0
0
0
D4
3.267
2.574
2.282
2.114
2.004
d2
1.128
1.693
2.059
2.326
2.534
Factors for x- and x-bar/R - Charts

62 April 9, 2016 – v3.0
1 0 .0
C HA
No k
F ran
5 -
o f
AVERAGES.0
.0
.0
.0
.0
.0
2
1
1
2
AA*
A
L
U
RANGES0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
L
U
R
95.01.2095.01.2195.01.2395.01.2495.01.2595.01.2695.01.2795.01.2895.01.3095.01.3195.02.0195.02.0295.02.0395.02.0495.02.0695.02.0795.02.0895.02.0995.02.1095.02.1195.02.1395.02.2095.02.2195.02.22
INDIVIDUALS
6 .00 6 .0
G ro
A uto
C L O
C urv
K -S
A V E
P R O
UC L
L C L
e re e n
Process Control Charts – (x-bar/R) - Chart Example

63 April 9, 2016 – v3.0
 The (x-bar / s) - chart should be used instead the (x-bar / R) -
chart if the subgroup is larger than 10. In this case, the
standard deviation is a better measurement than the range for
the variation between individual measurements in a subgroup.
 The (x-bar / s) - chart can be used whenever one can use the
(x-bar / R) - chart.
 The (x-bar / s) - chart is a method of looking at sources of
variation. One chart looks at variation in the subgroup averages
x-bar. The other chart examines variation in the subgroups
standard deviation s.
Process Control Charts – The (x-bar/s) - Chart

64 April 9, 2016 – v3.0
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 a
subgroup, N the number of subgroups,
.
Process Control Limit – The x-bar/s - Chart
sAx  3
sAx  3
sB 4
sB 3
N
xxx
x
N

...21
N
sss
s N

...21

65 April 9, 2016 – v3.0
n
6
7
8
9
10
A3
1.287
1.182
1.099
1.032
0.975
B3
0.030
0.118
0.185
0.239
0.284
B4
1.970
1.882
1.815
1.761
1.716
c4
0.9515
0.9594
0.9650
0.9693
0.9727
Factors for x-bar/s - Charts

66 April 9, 2016 – v3.0
Common Causes: Causes that are implemented in the process due
to the design of the process, and affect all outcomes of the process.
Identifying these types of causes requires Design of Experiment
(DOE) methods.
Special Causes: Causes that are not present in the process all the
time and do not affect all outcomes, but arise because of specific
circumstances. Special causes can be identified using SPC.
Walter A. Shewhart (1931)
Out-of-Control Criteria – Two Causes of Variation

67 April 9, 2016 – v3.0
Unstable Process: A process in which variation is a result of both
common and special causes.
Stable Process: A process in which variation in outcomes arises
only from common causes.
Out-of-Control Criteria – Two Types of Processes

68 April 9, 2016 – v3.0
An out-of-control criteria is a signal of a special causes of
variation:
• Is a systematic pattern of the product or process
characteristic monitored and charted
• Has a low probability of occurring when the process is
stable and in control
SPC Out-of-Control Criteria – The Types of Signals

69 April 9, 2016 – v3.0
What is the “chance”
to loose the coin flip
11 times in a row?
1 =
2 =
…
…
…
11 =
11 times in a row?
1 = 50% or 0.50
2 =
…
…
…
11 =
11 times in a row?
1 = 50% or 0.50
2 = 25% or 0.50*0.50
…
…
…
11 =
11 times in a row?
1 = 50% or 0.50
2 = 25% or 0.50*0.50
…
…
…
11 = 0.049% or 0.5011

70 April 9, 2016 – v3.0
Out-of-Control Criteria – The Basic Idea
average average
+1*s(igma)
average
-1*s(igma)
average
+2*s(igma)
average
-2*s(igma)
average
-3*s(igma)
average
+3*s(igma)
34.13 %34.13 %
13.60 % 13.60 %
2.14 %2.14 %
0.13 % 0.13 %
If your process is under control, over 99.74% of your data points will fall between the
average ± 3s(sigma) limits and there is only a 0.13% that a measurement point would fall
outside these limits.
Lower
Control Limit
Upper
Control Limit

71 April 9, 2016 – v3.0
Process Out-of-Control Criteria
Below is a list of the most commonly used out-of-control criteria included
in Minitab 17 and as defined by Walter Shewhart in the 1920s.
Criteria 1: Outlier
Criteria 2 & 5 & 6: Process Shift
Criteria 3: Process Trend

72 April 9, 2016 – v3.0
SPC Criteria #1 – 1 Point above or below 3 Sigma
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.

73 April 9, 2016 – v3.0
SPC Criteria #2 – 9 Points on the same Side of the Average
9 consecutive points above or below
the process performance average
line often indicates a shift in process
performance.

74 April 9, 2016 – v3.0
SPC Criteria #3 – 6 Consecutive Points Increasing or Decreasing
6 consecutive points increasing or
decreasing often indicates a trend in
process performance due to a
special cause.

75 April 9, 2016 – v3.0
SPC Criteria #5 – 2 of 3 Points above or below 2 Sigma
2 of 3 consecutive points above or
below 2 Sigma line often indicates
a shift in process performance.

76 April 9, 2016 – v3.0
SPC Criteria #6 – 4 of 5 Points above or below 1 Sigma
4 of 5 consecutive points above
or below the 2 Sigma line often
indicates a shift in process
performance.

77 April 9, 2016 – v3.0
Special Causes showing in the MR, R, or s Chart
1 data point above or below the 3
Sigma line is often the only
criteria used to identify special
causes in process performance.

78 April 9, 2016 – v3.0

79 April 9, 2016 – v3.0
Relationship between Tolerance and Process Capability
For any critical characteristic the Capability Index cp, also known as
the design margin, is quantified by :
ST
p
s
LSLUSL
LCL-UCL
USL - LSL
c




6
-
Variation)Process(TotalCapabilityorWidthProcess
Tolerance)(Designion WidthSpecificat
3
)-(
ST
p
s
TUSL
c

or
3
)-(
ST
p
s
LSLT
c

and with T = Target

80 April 9, 2016 – v3.0
The Capability Index cp
1.00 < cp < 1.33
Average
= Nominal Value or Target
LSL USLLCL UCL
average
+ 3*s (or sigma)
average
- 3*s (or sigma)
x

81 April 9, 2016 – v3.0
Process Performance Study
In some situations (e.g. order of production unknown, only small
sample produced) the short-term standard deviation sST cannot be
calculated and therefore the long-term standard deviation sLT has to
used. This means the actual process performance is evaluated, rather
than the potential process capability. To address this difference the
Performance Index pp is used instead of the Capability Index cp.
LT
p
s
LSLUSL
p


6
-
or
3
)-(
LT
p
s
TUSL
p


3
)-(
LT
p
s
LSLT
p

and with T = Target

82 April 9, 2016 – v3.0
Actual Process Performance
Cpk is the Capability Index adjusted for k, which considers actual
operation of the process and takes into account any difference
between the design nominal and the actual process mean value x-
bar.
)
3
-
,
3
-
(min
STST
pk
s
LSLx
s
xUSL
c


where is the average of all measurements or data points.x

83 April 9, 2016 – v3.0
The Capability Index cpk
nominal
value
LSL USLLCL UCL
average
1.33 < cp < 2.00, but 1.00 < cpk < 1.33
average
- 3*s(igma)
average
+ 3*s(igma)
x

84 April 9, 2016 – v3.0
In some situations (e.g. order of production unknown, only small sample
produced) the short-term standard deviation sST can not be calculated
and therefore the long-term standard deviation sLT has to used. This
means the actual process performance is evaluated, rather than the
potential process capability. To address this difference the Performance
Index ppk is used instead of the Capability Index cpk.
)
3
-
,
3
-
(min
LTLT
pk
s
LSLx
s
xUSL
p



85 April 9, 2016 – v3.0
ppk ccTx 
The Capability Index cpk
 Note, the cpk as defined above can also be used in the case of
unilateral specification, means that there exist only an upper or
lower specification limit and no target value.
Continue:
 Note, if the process average (x-bar) is equal to the midpoint
(nominal value) of the specification interval, means if

86 April 9, 2016 – v3.0
Quantifying Actual Process Performance
In some cases, the target value (T) of the process is not the
midpoint of the specification interval. For example the lower
specification limit may be the best value for the quality
characteristic. The Taguchi Capability Index cpm is a measure for
the difference between the average of the process and its target
value.
 22
6 Txs
LSLUSL
c
ST
pm



Note, if
ppm ccTx 

87 April 9, 2016 – v3.0
The Capability Index cpm
Nominal Value
or
Target
LSL USLLCL UCL
average
1.50 < cp < 2.00, 1.33 < cpk < 1.5, but 0.50 < cpm < 1.00
average
- 3*s(igma)
average
+ 3*s(igma)
x

88 April 9, 2016 – v3.0
In some situations (e.g. order of production unknown, only small
sample produced) the short-term standard deviation sST can not be
calculated and therefore the long-term standard deviation sLT has to
used. This means the actual process performance is evaluated,
rather than the potential process capability. To address this
difference the Performance Index ppm is used instead of the
Capability Index cpm.
 22
6 Txs
LSLUSL
p
LT
pm



Note, if
ppm ppTx 

89 April 9, 2016 – v3.0
Why are cp and cpk useful and significant Measures?
 To maximize cp requires the joint and concurrent effort of both product and process
designers.
 Product Design has the goal of increasing the allowable tolerance to the maximum
which will still permit successful function of the product.
 Process Design has the goal of minimizing the variability of the process which
reproduces the characteristic required for successful function of the product, and for
centering the process on target (nominal) value of the characteristic.
 A high cp Index indicates that the process is capable of reproducing the
characteristic. (It makes no statement about the centering of the process.)
 A high cpk Index indicates that the process is actually reproducing the characteristic
within the desired limits. (It makes no statement about the inherent capability, other
than its minimum value.)
 cpk < 1.00 means “Process not capable” 1.00
< cpk < 1.33 means “Process is marginal” cpk >
1.33 means “Process is capable”

90 April 9, 2016 – v3.0
Confidence Limits for the Capability Indices
The confidence intervals for the capability indices can be defined as
follows.
where m= n·N is number of measurements and (1-100% the required
confidence (e.g. for a confidence of 95 % is z/2 = 1.96).
   12
ˆ
ˆ
12
ˆ
ˆ 22




m
c
zcc
m
c
zc
p
pp
p
p 
   
2
2
2
2
2
22
2
2
2
2
2
1
2
1
ˆ
ˆ
1
2
1
ˆ
ˆ





 




 







 




 


s
Tx
s
Tx
m
c
zcc
s
Tx
s
Tx
m
c
zc
pm
pmpm
pm
pm 
    mm
c
zcc
mm
c
zc
pk
pkpk
pk
pk








9
1
12
ˆ
ˆ
9
1
12
ˆ
ˆ
2
2
2
2 

91 April 9, 2016 – v3.0

92 April 9, 2016 – v3.0

93 April 9, 2016 – v3.0
Confidence Limits for the Capability Index cp
95% confidence interval
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.4000
1.6000
0 100 200 300 400 500 600
m=N*n
delta
cp=0.5
cp=1.0
cp=1.5
cp=2.0
N = # of Subgroups
n = Subgroup Size

94 April 9, 2016 – v3.0
Confidence Limits for the Capability Index cpk
95% confidence interval
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.4000
1.6000
0 100 200 300 400 500 600
m=N*n
delta
cpk=0.5
cpk=1.0
cpk=1.5
cpk=2.0
N = # of Subgroups
n = Subgroup Size

95 April 9, 2016 – v3.0

96 April 9, 2016 – v3.0
Treatment of Non-Normality - Skewness
Transformation of the original measurements using
“ The Power Ladder ”.

originaltransform xx 
This method works best when the ratio of the largest to smallest
measurement value is greater than zero. If this is not the case, a simple
transformation like
with a appropriate constant “c” should work.
cxx originaltransform 

97 April 9, 2016 – v3.0
6543210-1
s
Mean-3s
Mean+3s
Mean
n
k
LSL
USL
Targ
Cpm
Cpk
CPL
CPU
Cp
Short-Term Capability
0
94876
30000
7429
0.00
9.49
3.00
0.74
Obs
PPM<LSL Exp
Obs
PPM>U SL Exp
Obs
%<LSL Exp
Obs
%>USL Exp
0.80061
-1.35198
3.45169
1.04985
100.000
0.300
0.000
3.000
*
*
0.44
0.44
0.81
0.62
Process Capability Analysis for Original
20100
2.4
1.6
0.8
0.0
Xbar and R Chart
Subgr
Means
X=1.050
3.0SL=2.124
-3.0SL=-0.02428
6
4
2
0
Ranges
1
R=1.862
3.0SL=3.938
-3.0SL=0.000
20100
Last 20 Subgroups
6
4
2
0
SubgroupNumber
Values
30
3.45169-1.35198
Cp: 0.62
CPU: 0.81
CPL: 0.44
Cpk: 0.44
Capability Plot
Process Tolerance
Specifications
StDev: 0.800612
III
III
5.02.50.0
Normal Prob Plot
5.02.50.0
Capability Histogram
Process Capability Sixpack for Original
“The Original Measurements“
The Capability Study indicates
almost 100 000 PPM below the
lower specification limit.

98 April 9, 2016 – v3.0
3.02.52.01.51.00.50.0-0.5-1.0
8
7
6
5
4
3
2
1
0
95% Confidence Interval
StDev
Lambda
Last IterationInfo
0.693
0.692
0.695
0.393
0.337
0.281
StDevLambda
Up
Est
Low
Box-CoxPlot for Original

33.0
33.0
33.0
originaltransform
originaltransform
originaltransform
USLUSL
LSLLSL
xx



“The Power Ladder“
Which function would transform
the original data to a normal
distribution ?

99 April 9, 2016 – v3.0
2.001.751.501.251.000.750.500.250.00
UpperSpecLowerSpec
s
Mean-3s
Mean+3s
Mean
n
k
LSL
USL
Targ
Cpm
Cpk
CPL
CPU
Cp
Short-TermCapability
0
485
30000
41141
0.00
0.05
3.00
4.11
Obs
PPM<LSL Exp
Obs
PPM>USL Exp
Obs
%<LSL Exp
Obs
%>USL Exp
0.28635
0.08559
1.80369
0.94464
100.000
0.310
0.000
1.442
*
*
0.58
1.10
0.58
0.84
Process CapabilityAnalysis for Transfor
20100
1.25
1.00
0.75
0.50
Xbar and R Chart
Subgr
Means
X=0.9446
3.0SL=1.329
-3.0SL=0.5605
1.5
1.0
0.5
0.0
Ranges
R=0.6660
3.0SL=1.408
-3.0SL=0.000
20100
Last 20 Subgroups
1.7
1.2
0.7
0.2
SubgroupNumber
Values
1.44220.0000
1.803690.08559
Cp: 0.84
CPU: 0.58
CPL: 1.10
Cpk: 0.58
Capability Plot
Process Tolerance
Specifications
StDev: 0.286349
III
III
1.51.00.5
Normal Prob Plot
1.51.00.5
Capability Histogram
Process Capability Sixpack for Transfor
“The Transformed Measurements“
The Capability Study for the transformed
data indicates “only” 500 PPM below the
lower specification limit !

100 April 9, 2016 – v3.0

101 April 9, 2016 – v3.0
The End …
“Perfection is not attainable, but if we chase perfection we can catch
excellence.” - Vince Lombardi

102 April 9, 2016 – v3.0
Terms & Conditions
After you have downloaded the training material to your own
computer, you can change any part of the course material and
remove all logos and references to Operational Excellence
Consulting. You can share the material with your colleagues and
re-use it as you need. The main restriction is that you cannot
distribute, sell, rent or license the material as though it is your
own. These training course materials are for your — and
your organization's — usage only. Thank you.

Six Sigma Process Capability Study (PCS) Training Module

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Six Sigma Process Capability Study (PCS) Training Module