The document outlines statistical process control (SPC) methods, covering concepts such as histograms, measures of location and variability, control charts, and management of variations. It emphasizes the historical development of quality control techniques and the transition from traditional to advanced process control concepts, highlighting the importance of understanding process variations for continuous improvement. Key definitions and steps for creating histograms and identifying data distribution types are also presented.

1. 1 April 9, 2016 – v4.0
Six Sigma Statistical Process Control
by Operational Excellence Consulting LLC

2. 2 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control (SPC) – Table of Contents

3. 3 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

4. 4 April 9, 2016 – v4.0
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

5. 5 April 9, 2016 – v4.0
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

6. 6 April 9, 2016 – v4.0
Process Inspection
Good
Bad
Repair
Scrap
+
Monitor/Adjust
The Traditional Process Control Concept
The Detection Control Scheme

7. 7 April 9, 2016 – v4.0
• The traditional process control concept does not help us
to produce only good products or services.
• Every process outcome, product or service, has to be
inspected.
• Products have to be repaired or even scraped.
• Rendered services result in customer dissatisfaction.
• With respect to productivity and efficiency every activity
after the actual process is a non-value added activity.
The Traditional Process Control Concept

8. 8 April 9, 2016 – v4.0
Prevention Control Scheme
Process Inspection
Good
Bad
Repair
Scrap
+
An Advanced Process Control Concept
Monitor/Adjust
Learn/Improve
Selective measurement
• Product / Service
• Process

9. 9 April 9, 2016 – v4.0
Statistical Thinking - A Definition
All work is a series of
interconnected processes
All processes vary
Understanding and
reducing variation are keys
to success
ASQ

10. 10 April 9, 2016 – v4.0
Customer Satisfaction
or
Customer Dissatisfaction
Process/
System
Material
Machines Methods
Men
Environment
The Variation Management Approach

11. 11 April 9, 2016 – v4.0
A defect is any variation of a required characteristic of the product or
service, which is far enough removed from its nominal value to
prevent the product or service from fulfilling the physical and
functional requirements of the customer.
Variation Management – Defect Definition

12. 12 April 9, 2016 – v4.0
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

13. 13 April 9, 2016 – v4.0
Remarks or Questions ?!?

14. 14 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

15. 15 April 9, 2016 – v4.0
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

16. 16 April 9, 2016 – v4.0
Step 1: Collect at least 25 data points (if possible), but better 50 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

17. 17 April 9, 2016 – v4.0
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.
The Histogram – How to create a Histogram?
Number of Bars:
less than 50
50 - 100
100 - 250
over 250
5 or 7
5, 7, 9 or 11
7 - 15
11 - 19
Number of Data Points:
Minimum = 2.1
Maximum = 3.1
Range = 1.0
Bar Width = 0.2 (5 Bars)

18. 18 April 9, 2016 – v4.0
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.
The Histogram – How to create a Histogram?
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

19. 19 April 9, 2016 – v4.0
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.
The Histogram – How to create a Histogram?
Sorted Measurements
Part Hole Size Bar
5 2.1 1
2 2.3 2
7 2.4 2
6 2.5 3
8 2.5 3
1 2.6 3
10 2.6 3
4 2.7 4
9 2.8 4
3 3.1 5
0
1
2
3
4
5
NumberofDataPoints
2.1 2.3 2.5 2.7 2.9 3.1

20. 20 April 9, 2016 – v4.0
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

21. 21 April 9, 2016 – v4.0
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.
The Histogram – Typical Patterns of Variation

22. 22 April 9, 2016 – v4.0
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.
The Histogram – Typical Patterns of Variation

23. 23 April 9, 2016 – v4.0
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.
The Histogram – Typical Patterns of Variation

24. 24 April 9, 2016 – v4.0
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.
The Histogram – Typical Patterns of Variation

25. 25 April 9, 2016 – v4.0
The Histogram – The Bell-Shaped or Normal Distribution
We will come back to
this one later.

26. 26 April 9, 2016 – v4.0
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

27. 27 April 9, 2016 – v4.0
Remarks or Questions ?!?

28. 28 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

29. 29 April 9, 2016 – v4.0
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

30. 30 April 9, 2016 – v4.0
Example 1: x1 = 2 x2 = 5 x3 = 4
Construction: 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
median = 4
median = 4.5
?
Measure of Location – The Sample Median

31. 31 April 9, 2016 – v4.0
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

32. 32 April 9, 2016 – v4.0
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

33. 33 April 9, 2016 – v4.0
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:

34. 34 April 9, 2016 – v4.0
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

35. 35 April 9, 2016 – v4.0
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

36. 36 April 9, 2016 – v4.0
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 short-term 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

37. 37 April 9, 2016 – v4.0
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

38. 38 April 9, 2016 – v4.0
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 with respect to process
variation when using appropriate process 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 



39. 39 April 9, 2016 – v4.0
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 %
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(sigma) limits.

40. 40 April 9, 2016 – v4.0
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()”.

41. 41 April 9, 2016 – v4.0
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

42. 42 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

43. 43 April 9, 2016 – v4.0
Attribute Data
(Count or Yes/No Data)
Variable Data
(Measurements)
Variable
subgroup
size
Subgroup
size
of 1
Fixed
subgroup
size
I
MR chart
x-bar
R chart
x-bar
s chart
Count
Incidences or
nonconformities
Fixed
oppor-
tunity
Variable
oppor-
tunity
c - chart u - chart
Yes/No Data
Defectives or
nonconforming units
Fixed
subgroup
size
Variable
subgroup
size
np - chart p - chart
Process Control Charts – Types of Control Charts
Type of Data

44. 44 April 9, 2016 – v4.0
 The I-MR (or Individual – Moving Range) chart is a method of
looking at variation in a variable data or measurement.
 One source is the variation in the individual data points over time
(Individual chart). This represents “long term” variation in the
process.
 The second source of variation is the variation between
successive data points (Moving Range chart). This represents
“short term” variation.
 I-MR charts should be used when there is only one data point to
represent a situation at a given time.
 To use the I-MR chart, the individual sample results should be
“sufficient” normally distributed. If not, the I-MR chart will give
more false signals.
Process Control Charts – The I - MR Chart

45. 45 April 9, 2016 – v4.0
Process Control Charts – Test for Normality
The Normality Test from Anderson & Darling provides a method to determine if
our data comes from a process that creates normally distributed data.
The red line represents
the normal distribution.
If the all the individual data
points fall on the red line, the
sample data itself is perfectly
normally distributed.
As long as the p-value stays
above 0.05, we can assume
that the process creates
normally distributed data.

46. 46 April 9, 2016 – v4.0
Process Control Charts – I-MR Chart Example
Individual chart showing the
individual data points we
collected from our process.
Moving Range chart showing
the difference between two
consecutive individual data
points.

47. 47 April 9, 2016 – v4.0
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
Process Control Charts – The Central Limit Theorem
Carl Friedrich GaussPierre Laplace

48. 48 April 9, 2016 – v4.0
 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 large amount of data collected makes it difficult to analyze
the data using the I-MR chart
 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

49. 49 April 9, 2016 – v4.0
Process Control Charts – (x-bar/R) - Chart Example
Average of the individual data
points in each subgroup. In
this case we had 3 data
points in each subgroup.
Range of the individual data
points in each subgroup.

50. 50 April 9, 2016 – v4.0
 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

51. 51 April 9, 2016 – v4.0
Process Control Charts – (x-bar/R) - Chart Example
Average of the individual data
points in each subgroup. In
this case we had 15 data
points in each subgroup.
Standard Deviation (long-term
formula) of the individual data
points in each subgroup.

52. 52 April 9, 2016 – v4.0
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

53. 53 April 9, 2016 – v4.0
The number of defect phones produced per hour were
1. hour: 100 phones and 10 defect phones.
2. hour: 110 phones and 12 defect phones.
3. hour: 90 phones and 9 defect phones.
4. hour: 95 phones and 10 defect phones.
5. hour: 115 phones and 13 defect phones.
6. hour: 120 phones and 15 defect phones.
7. hour: 80 phones and 7 defect phones.
8. hour: 85 phones and 5 defect phones.
9. hour: 100 phones and 8 defect phones.
10. hour: 110 phones and 11 defect phones.
11. hour: 75 phones and 5 defective phones. Something wrong ???
Process Control Charts – Attribute “Yes/No” Data

54. 54 April 9, 2016 – v4.0
Number of defective Items
10 2 3 4 5 6 7 8 9
Average
Process Control Charts – The Binomial Distribution

55. 55 April 9, 2016 – v4.0
 The p - chart is used to look at variation in the yes/no attribute data.
It can for example be used to determine the percentage p of
defective items in a group of items.
 The number n of items in each group has not to be constant, but
should not vary more than 25 %.
 Operational definitions must be used to determine what constitutes
a defective item.
Process Control Charts – The p - Chart
n
np
p 
itemsofnro.
itemsdefectiveofnro.
The percentage of defective items is given by

56. 56 April 9, 2016 – v4.0
Process Control Charts – The p – Chart Example
Proportion of defects in each
subgroup. In this case we had
always 100 “items” coming out of
our process (subgroup size = 100)
and in average 14.44% were
defective.

57. 57 April 9, 2016 – v4.0
 The np - chart, like the p - chart, is used to look at variation in
yes/no type attributes data.
 np - charts are used to determine the number np of defective
items in a group of items, while p - chart looked at the
percentage of defective items in a group of items. Because the
np - chart uses the number of defects, it is easier to use.
 However, the major difference between the np - chart and the p
- chart is that the subgroup size has to be constant for the np -
chart. This is not necessary for the p - chart.
Process Control Charts – The np - Chart

58. 58 April 9, 2016 – v4.0
Process Control Charts – The p – Chart Example
Number of defects in each
subgroup. In this case we had
always 100 “items” coming out of
our process (subgroup size = 100)
and in average we had 14.44
defects per subgroup.

59. 59 April 9, 2016 – v4.0
The number of wrong assembled components in SMD made on 20 PCBs were
1 - 20: 10 wrong assembled components
21 - 40: 8 wrong assembled components
41 - 60: 7 wrong assembled components
61 - 80: 5 wrong assembled components
81 - 100: 6 wrong assembled components
101 - 120: 9 wrong assembled components
121 - 140: 7 wrong assembled components
141 - 160: 5 wrong assembled components
161 - 180: 2 wrong assembled components. Something wrong ???
Process Control Charts – The Attribute “Count” Data

60. 60 April 9, 2016 – v4.0
Number of Incidences
10 2 3 4 5 6 7 8 9
Average
Process Control Charts – The Poisson Distribution

61. 61 April 9, 2016 – v4.0
 The c - chart is used to look at variation in counting-type
attributes data. It is used to determine the variation in the number
of defects in a constant subgroup size.
 For example, a c - chart can be used to monitor the number on
injuries in a plant. In this case, the plant is the subgroup.
 To use the c - chart, the opportunities for incidences to occur in
the subgroup must be very large, but the number that actually
occur must be small.
Process Control Charts – The c - Chart

62. 62 April 9, 2016 – v4.0
Process Control Charts – The c – Chart Example
Number of incidences over time. In
this case the average number of
incidences is 14.72%.

63. 63 April 9, 2016 – v4.0
 A u - chart is used to examine the variation in counting-type
attributes data.
 For example, a u - chart can be used to monitor the number on
infections in a hospital during a specific time period.
 The u - chart is very similar to the c - chart. The only difference is
that the subgroup size for the c - chart must be constant. This is
not necessary for the subgroup size of a u - chart.
 To use the u - chart, the opportunities for incidences to occur in the
subgroup must be very large, but the number that actually occur
must be small.
Process Control Charts – The u - Chart

64. 64 April 9, 2016 – v4.0
Process Control Charts – The u – Chart Example
Proportion of incidences over time
for a specific subgroup size. In this
case we used a subgroup size of
200. The average percentage of
incidences over time was 7.36%.

65. 65 April 9, 2016 – v4.0
Yes/No
Defective Items
Count
Incidences
Constant
Subgroup Size
Variable
Subgroup Size
np - chart c - chart
u - chartp - chart
Process Control Charts – Charts for Attribute Data

66. 66 April 9, 2016 – v4.0
Process Control Charts – Exercise 4
Black Beads Process:
Step 1: Take 15 beads out of the bag and record the number of black beads in the
sample.
Step 2: Repeat Step 1 20 times until you have 20 data points.
Step 3: Draw the histogram of the 20 data points in the left spreadsheets below.
Step 3: Select the correct process control chart and draw it in the right spreadsheet.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Histogram
Process Control Chart
Data
Data # of Black Beads
1 4
2 5
3 2
4 0
5 3
6 1
7 5
8 3
9 6
10 2
11 4
12 3
13 1
14 5
15 3
16 4
17 2
18 1
19 0
20 2

67. 67 April 9, 2016 – v4.0
Remarks or Questions ?!?

68. 68 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

69. 69 April 9, 2016 – v4.0
Process Control Limit – 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.
Lower
Control Limit
Upper
Control Limit

70. 70 April 9, 2016 – v4.0
Upper Control Limit (UCL) = average + 3*s(igma)
Lower Control Limit (LCL) = average - 3*s(igma)
•
•
•
•
• •
•
•
average
average + 3*sigma
average + 2*sigma
average + 1*sigma
average - 1*sigma
average - 3*sigma
average - 2*sigma
Process Control Limit – Upper & Lower Control Limit

71. 71 April 9, 2016 – v4.0
Because the variation of a process is not known beforehand, one
cannot calculate or define the control limits in advance.
The calculation of the control limits should be based on at least 20 to
25 data points from a process that was in statistical control (stable).
Control limits are characteristics of a stable process. They bound the
variation of the process that is due to common causes.
The limits should not be recalculated and modified unless there is a
reason to do so (e.g. a process change or improvements).
Process Control Limit – Upper & Lower Control Limit

72. 72 April 9, 2016 – v4.0
where the constant d2 depends on the number of items in each
subgroup used to calculate the range.
The LT and ST subscripts represent long-term and short-term
variability.
The difference between sLT and sST gives an indication of how much
less variability you can have in your process when using SPC.
     
)1(
...
22
2
2
1



N
xxxxxx
s N
LT
2dRsST 
STLT ss 
Process Control Limit – Two Types of Process Variability

73. 73 April 9, 2016 – v4.0
Upper control limit =
Lower control limit =
Upper control limit =
Lower control limit =
The I - chart
The MR- chart
,
where x1, x2, ..., xN are the measurements, N the number of measurements,
, and .
Process Control Limit – The I-MR 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

74. 74 April 9, 2016 – v4.0
Upper control limit =
Lower control limit =
The R- chart
Upper control limit =
Lower control limit =
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
  RAxndRx  223
RD 4
RD 3
N
xxx
x
N

...21
N
RRR
R N

...21minmax
iii xxR 

75. 75 April 9, 2016 – v4.0
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

76. 76 April 9, 2016 – v4.0
Upper control limit =
Lower control limit =
Upper control limit =
Lower control limit =
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

77. 77 April 9, 2016 – v4.0
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

78. 78 April 9, 2016 – v4.0
Lower control limit =
Upper control limit =
with
and
where np1, np2, ..., npN are the number of defective items in each subgroup
of constant size n, and N the number of subgroups.
13)
2
1
( 






n
pn
pnpn
13)
2
1
( 






n
pn
pnpn
np
np np np
N
N

  
1 2
3
...
n p
n
np
n
n
np
n
n
np
n
N
N
  
        
( )
( ) ( ) ... ( )
1
1 1 1
3
1 2
Process Control Limit – The np - Chart

79. 79 April 9, 2016 – v4.0
Lower control limit =
Upper control limit =
for i = 1, 2, 3,..., N, where (np)1, (np)2, ..., (np)N are the number of defective items in
the subgroups and n1, n2, ..., nN are the number of items in the N subgroups.
Note: The sample sizes should not vary more than 25% around the average sample
size when using control limits for the whole chart.
  npp
n
p  13)
2
1
(
,
)...( 21
N
nnn
n N
with and,
..
)(...)()(
21
21
N
N
nnn
npnpnp
p


 ,3pni3)1(  pni
  npp
n
p  13
2
1
  npp
n
p  13
2
1
or
or
Control limits
for whole process
Control limits
for each subgroup
Process Control Limit – The p - Chart

80. 80 April 9, 2016 – v4.0
Lower control limit =
Upper control limit =
with
where c1, c2, ..., cN are the number of defects in each subgroup of constant
size and N the number of subgroups.
Process Control Limit – The c - Chart
cc  3
 0,3max cc 
2
...21



N
ccc
c N

81. 81 April 9, 2016 – v4.0
u u n 3
Lower control limit =
Upper control limit =
with andu
c c c
n n n
N
N

  
  
1 2
1 2
...
..
, n
n n n
N
N

  ( ... )1 2
where c1, c2, ..., cN are the number of defects in the subgroups and n1, n2, ...,
nN are the number of items in each of the N subgroups.
Note: The sample sizes should not vary more than 25% around the average
sample size.
 0,3max nuu 
c
c c c
N
N

  
1 2
2
...
Process Control Limit – The u - Chart

82. 82 April 9, 2016 – v4.0
Process Control Charts – Exercise 5
Task #1: Calculate the average and the upper and lower control limit for exercise #2 and create a Process
Control Chart.
Task #2: Calculate the average and the upper and lower control limit for exercise #4 and create a Process
Control Chart.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

83. 83 April 9, 2016 – v4.0
Process Control Charts – Exercise 5 Results
Task #1
Subgroup Range
1 59 66 63 62 7
2 60 66 69 65 9
3 65 62 71 72 10
4 68 65 67 69 4
5 65 66 70 68 5
6 64 64 73 73 9
7 63 67 71 68 8
8 63 68 65 68 5
Average Range 7.1
Overall Average: 66.4
Short-Term Standard Deviation: 3.46
Lower Control Limit (LCL): 61.2
Upper Control Limit (UCL): 71.6
Lower Control Limit (LCL): 0
Upper Control Limit (UCL): 16.3
Measurements
x-bar Chart
x-bar / R Chart
R Chart
Task #2 c - Chart
Data # of Black Beads
1 4 Overall Average: 2.8
2 5
3 2 Lower Control Limit (LCL): 0.0
4 0 Upper Control Limit (UCL): 7.8
5 3
6 1
7 5
8 3
9 6
10 2
11 4
12 3
13 1
14 5
15 3
16 4
17 2
18 1
19 0
20 2

84. 84 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

85. 85 April 9, 2016 – v4.0
Upper Specification Limit (USL)
large variation problem exist
root cause analysis
process improvement
trend problem occurs
root cause analysis
corrective action
Defect
nominal value
Out-of-Control & Process Improvement
Lower Specification Limit (LSL)
Defect
nominal value
Upper Specification Limit (USL)
Lower Specification Limit (LSL)

86. 86 April 9, 2016 – v4.0
50 55 = USLLSL= 45
Sample

Process Tempering & Overcontrol
LSL USL
50 5545
Sample 
LSL USL
50 5545

LSL USL
50 5545

LSL USL

87. 87 April 9, 2016 – v4.0
50 5545
LSL USL
Process tampering may substantially increase the product variability
since the process average is shifted each time an adjustment to the
process is made as a reaction to a product or service defect.
Process Tempering & Overcontrol
Original Process
Tempered Process

88. 88 April 9, 2016 – v4.0
Common Causes: Causes that are implemented in the process due
to the design of the process and the random variation in the process
inputs, and affect all outcomes of the process. Identifying these types
of causes often requires Six Sigma methods and tools, including
Multi-vari Study and Design of Experiment (DOE).
Walter A. Shewhart (1931)
Out-of-Control Criteria – Two Causes of Variation
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. These
causes could be the result of a new process input
entering the process, or an existing process input
behaving very differently than normal. Special
causes can be identified using SPC.

89. 89 April 9, 2016 – v4.0
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

90. 90 April 9, 2016 – v4.0
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

91. 91 April 9, 2016 – v4.0
Everything is possible, but is it likely?
What is the “chance” to
loose the coin flip 11
times in a row?
1 =
2 =
…
…
…
11 =
What is the “chance” to
loose the coin flip 11
times in a row?
1 = 50% or 0.50
2 = 25% or 0.50*0.50
…
…
…
11 = 0.049% or 0.5011

92. 92 April 9, 2016 – v4.0
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

93. 93 April 9, 2016 – v4.0
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.

94. 94 April 9, 2016 – v4.0
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.

95. 95 April 9, 2016 – v4.0
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.

96. 96 April 9, 2016 – v4.0
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.

97. 97 April 9, 2016 – v4.0
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.

98. 98 April 9, 2016 – v4.0
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.

99. 99 April 9, 2016 – v4.0
Special Causes showing in the MR, R, or s Chart
Note: Special causes in the process performance that (also) show up in
the MR, R or s chart, i.e. special causes resulting in a “more than usual”
increase or decrease in process performance from one data point to the
next, will increase the short-term standard deviation of the process.
As a result, that will increase the upper and decrease the lower control
limit of the I or x-bar chart. That will then result in a reduced ability to
actually identify special causes in the I or x-bar chart.
Solution: Special causes that (also) show up in the MR, R or s chart,
need to be eliminated from the data set and the control chart needs to
be re-calculated to make sure that all special causes in the I or x-bar
chart are identified.
However, that does not mean that these special causes do not need to
be investigated and eliminated.

100. 100 April 9, 2016 – v4.0
Special Causes showing in the MR, R, or s Chart
The process performance data indicates two special causes in process.
Both special causes show only in the MR chart, increasing the average MR and
therefore the short-term standard deviation used to calculate the control limits
for the I chart.

101. 101 April 9, 2016 – v4.0
Special Causes showing in the MR, R, or s Chart
After we exclude the individual data points
causing the two special causes shown in the
MR chart, we can now see an additional
special cause in the I chart (2 out of 3 points
below the 2 Sigma line).

102. 102 April 9, 2016 – v4.0
x x / R x / s cunp p
Criteria 1
Criteria 2
Criteria 3
Criteria 4
Criteria 5
Criteria 6
Criteria 7
Criteria 8
Criteria 9
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Chart
--
SPC Out-of-Control Criteria – Summary

103. 103 April 9, 2016 – v4.0
SPC Out-of-Control Criteria – Exercise 6
Efficiency Out-of-Control Conditions:
Determine why the process control chart below indicates that the efficiency of production line H300 is out-
of-control.
1 0 .0 4
C HA R
No kia
F rank
5 - 2
o f
AVERAGES.0
.0
.0
.0
.0
.0
2
1
1
2
AA*
A V
L C
UC
RANGES
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
L C
UC
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 u
A uto
C L O
C urve
K -S :
A V E R
P R O
UC L
L C L
re e n
Notes:

104. 104 April 9, 2016 – v4.0
Remarks or Questions ?!?

105. 105 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Statistical Process Control – Table of Contents

106. 106 April 9, 2016 – v4.0
 Applying the Central Limit Theorem to make the average of the
subgroups normally distributed.
 Dividing the sources of variation in the process outcomes into
two different subgroups (short-term and long-term variation).
 Optimizing the probability of identifying a shift in the process
average with the next observation.
Sample Size and Frequency – Rational Subgrouping

107. 107 April 9, 2016 – v4.0
avg
Sample Size and Frequency – Subgroup Size and Sensitivity
USL
avg + STs3
UCLLCL
avg - STs3 avg + E
E
Defects

108. 108 April 9, 2016 – v4.0
If a shift in the process average of “E” units will harm the customer or
one of the next process stages, the necessary subgroup sample size
(n) can be calculated as:
2
)28.4( Esn ST
The next plotted point will with 90% confidence identify a process shift
larger than “E” units, that means the next point will be above or below 3
sigma control limits.
Sample Size and Frequency – Subgroup Size and Precision

109. 109 April 9, 2016 – v4.0
 The frequency of sampling of two consecutive subgroups can be
determined by dividing the average time period between two out-
of-control situations by at least 3 but not more than 6.
Example: If experience shows that your process produces defects
or goes out-of-control once every 12-hour shift, the you should
collect measurements from your process every 2 to 4 hours.
 However, no general rule can be defined about which time interval
works best. You have to start with a good (conservative) guess
and refine the time interval if necessary.
Sample Size and Frequency – Sample Frequency

110. 110 April 9, 2016 – v4.0
Remarks or Questions ?!?

111. 111 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

112. 112 April 9, 2016 – v4.0
Activators (out-of-control decision rules)
Checkpoints (list of possible assignable causes)
Terminators (corrective actions)
Out-of-Control-Action-Plans (OCAP)
An OCAP is a flowchart that guides the operator through a defined
and repeatable response to “any” out-of-control situation.

113. 113 April 9, 2016 – v4.0
Start
Checkpoints
Activators
Corrective ActionsNo
No
No
Yes
Yes
Yes
Yes
Yes
Yes
End
No
No
Out-of-Control-Action-Plans (OCAP)

114. 114 April 9, 2016 – v4.0
1. One point outside the 3-sigma control limits.
2. A run of at least seven or eight consecutive points, where the type of
run could be either a run up or down, a run above or below the
center line.
3. Two out of three consecutive points plot beyond from the 2-sigma
warning level.
4. Four out of five consecutive points at a distance of 1-sigma or
beyond.
5. One or more consecutive points near a 2-sigma warning or 3-sigma
control level.
6. …
Out-of-Control-Action-Plans – Activators

115. 115 April 9, 2016 – v4.0
The checkpoints instruct the operator to investigate specific items as
possible assignable causes for the out-of-control situation.
Once a checkpoint has identified a probable assignable cause for the
out-of-control situation, the OCAP will flow into a terminator or
corrective action.
Out-of-Control-Action-Plans – Checkpoints

116. 116 April 9, 2016 – v4.0
The terminator contains a detailed description of the corrective action
that the operator has to take to resolve the out-of-control situation.
Out-of-Control-Action-Plans – Terminators

117. 117 April 9, 2016 – v4.0
... typically generate one or more of the following actions:
 Eliminate the most common assignable causes
 Analyze the activators
 Revise the order of the checkpoints and terminators
 Train the operators to perform more of the corrective actions included
into the OCAP to resolve out-of-control situations quickly
An Analysis of Out-of-Control-Action-Plans ...

118. 118 April 9, 2016 – v4.0
 The OCAP is a systematic and ideal problem-solving tool for
process problems because it reacts to out-of-control situations in
real time.
 OCAPs standardize the best problem-solving approaches from the
most skilled and successful problem solvers (experts/operators).
 The OCAP also allows (and requires) off-line analysis of the
terminators to continually improve OCAP efficiency.
Some Benefits of Out-of-Control-Action-Plans

119. 119 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Section 9: Process Control Plan
Statistical Process Control – Table of Contents

120. 120 April 9, 2016 – v4.0
Process Control Plan → Objective
A Control Plan is a written statement of an organization’s quality planning
actions for a specific process, product, or service.
The Objective of an effective Process Control Plan is to
 operate processes consistently on target with minimum variation,
which results in minimum waste and rework
 assure that product and process improvements that have been
identified and implemented become institutionalized
 provide for adequate training in all standard operating procedures,
work instructions and tools
Customer
Requirements
Product & Part
Characteristics
Process
Input & Output
Characteristics
Process
Controls
Process
Control
Plan

121. 121 April 9, 2016 – v4.0
Process Control Plan → Template
Supplier: Product:
Key Contact: Process:
E-Mail / Phone:
Product
Characteristic
Process
Characteristic
Characteristic
Process Step
Specification
(LSL, USL &Target)
Date (Orig):
Date (Rev):
Control Method Reaction Plan
Control Limits
(LCL & UCL)
Measurement
System
Sample Size Sample Frequency
Operational Excellence
Process Control Plan

122. 122 April 9, 2016 – v4.0
 Process: Name of the process to be controlled
 Process Step: The process steps of the process to be controlled
 Characteristic (Product/Process): Name of the characteristic of a process
step or a product, which will actually be controlled.
 Specification: Actual specification, which has been set for the characteristic
to be controlled. This may be verified e.g. in standards, drawings,
requirements or product requirement documents.
 Control Limits: Control limits are specified for characteristics that are
quantifiable and selected for trend analysis (x-bar/R, x/mR, p charts). When
the process exceeds these limits, corrective actions are required.
 Measurement System: Method used to evaluate or measure the
characteristic. This may include e.g. gages, tools, jigs and test equipment or
work methods. An analysis of the repeatability and the reproducibility of the
measurement system must first be carried out (e.g. Gage R&R Study).
Process Control Plan → Template

123. 123 April 9, 2016 – v4.0
 Sample Size: Sample size specifies how many parts are evaluated at any
given time. The sample size will be “100 %” and the frequency
“continuous” in case of 100% inspection.
 Sample Frequency: Sample frequency specifies the how often a sample
will be taken, e.g. once per shift or every hour.
 Control Method: Brief description of how the information/data will be
collected, analysed/controlled and reported. More detailed information
may be included in a named work instruction.
 Reaction Plan: Necessary corrective actions to avoid producing non-
conforming products or operating out-of-control. Corrective actions should
normally be in the responsibility of the person closest to the process, e.g.
the machine operator. This is to secure, that immediate corrective actions
will take place and the risk of non-conforming products will be minimized.
More detailed information may be included in a named work instruction.
Process Control Plan → Template

124. 124 April 9, 2016 – v4.0
 Process maps detail manufacturing
steps, material flow and important
variables
 Key product variables identified with
importance to customer, desired target
value and specification range defined
 Key and critical process input variables
identified with targets, statistically
determined control limits & control
strategies defined
 Measurement systems are capable
with calibration requirements specified
 Sampling, inspection and testing plans
include how often, where and to whom
results are reported
 Reaction plan in place for out-of-spec
conditions and material
 Operating procedures identify actions,
responsibilities, maintenance schedule
and product segregation requirements
 Training materials describe all aspects
of process operation and responsibili-
ties
 Process improvement efforts fully
documented and available for refe-
rence
 Control plan is reviewed and updated
quarterly and resides in the operating
area
Process Control Plan → Check List

125. 125 April 9, 2016 – v4.0
Remarks or Questions ?!?

126. 126 April 9, 2016 – v4.0
Section 1: Introduction
Section 2: The Histogram
Section 3: Measure of Location and Variability
Section 4: Process Control Charts
Section 5: Process Control Limits
Section 6: Out-of-Control Criteria
Section 7: Sample Size and Frequency
Section 8: Out-of-Control Action Plan
Statistical Process Control – Table of Contents

127. 127 April 9, 2016 – v4.0
 People are trained without regard for the need to know or
implementation timing.
 Once the necessary charts are created, they are rarely reviewed.
 Charts have characteristics or parameters that do not really
represent the process.
 Control limits are not reviewed or adjusted, or conversely, they
are adjusted too often.
 Someone other than the process operator maintains the chart.
(This is not always bad, however)
 The process is not capable or set up well off target.
 Corrective actions and significant events are not recorded on the
chart.
When SPC fails, look in the mirror ...

128. 128 April 9, 2016 – v4.0
The End …
“Perfection is not attainable, but if we chase perfection we can catch
excellence.” - Vince Lombardi

129. 129 April 9, 2016 – v4.0
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.