Process Control

Operational Excellence
Process Control
Introduction
2/11/2017 Ronald Morgan Shewchuk 1
• Now that you have implemented process mapping, value stream mapping and
5S+Safety within your organization you have effectively leaned-out your
operations.
• You are left with the core processes that add value to your products and/or
services in the eyes of your customer.
• The degree of control that these processes exhibit is directly related to your
company’s profitability.
• The higher the degree of control, the more money your company makes – it’s as
simple as that.
• Process control implies reducing variation.
• Variation, as you will recall, is the enemy of Six Sigma.
• Jack Welch, the CEO of General Electric from 1981 through 2001, was keenly aware
of the effect variation had on his business, and consequently was a leading
proponent of driving Six Sigma throughout General Electric’s operations.
• In this presentation we will review the fundamentals of process control.

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Standard Deviation and Variance
• The DNA of statistics is the standard deviation, which may be visualized as the
average distance of each data point to the mean of all the data points in the
sample set.
• The equation for the standard deviation of a sample extracted from a larger
population is provided in Eqn 6.1.
• This is the same equation used by MS Excel in the STDEV function to calculate the
standard deviation of a data set.
s =
(xi – x)2
 
i = 1
n
n - 1
where s = sample standard deviation
n = number of data points in sample
xi = value of the ith element of x
x = mean of all elements of x sample
Eqn 6.1

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• The units of the standard deviation are the same as the data points.
• Variance is simply the square of the standard deviation, s2.
• Variances are additive whereas standard deviations are not.
• For example, if you wanted to calculate the standard deviation of a machined part
consisting of multiple assemblies you must first add all the variances of each
assembly and then take the square root of the total variance.
• The equation for variance is defined in Eqn 6.2.
s2 =
(xi – x)2
i = 1
n
n - 1
where s2 = sample variance
x = mean of all elements of x sample
Eqn 6.2

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• If your sample size has the same number of elements as your population, that is,
you have performed 100% inspection, then the standard deviation and variance of
the population are denoted by  and 2 respectively and may be calculated by
Eqn 6.3 and 6.4 below.
 =
(xi – )2
 
i = 1
n
n
where  = population standard deviation
n = number of data points in population
 = mean of all elements of x population
Eqn 6.3
2 =
(xi – )2
i = 1
n
n
where 2 = population variance
n = number of data points in population
 = mean of all elements of x population
Eqn 6.4

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Normal Distribution
•  is the Greek letter sigma, thus we can see that six sigma implies six standard
deviations. This implication is best illustrated by considering the normal
distribution.
• If you measured the height of every adult male in the United States and plotted a
graph with measured height on the x-axis and the number of occurrences of the
measured height on the y-axis you would find that the data follows a bell-shaped
curve.
• Scientists have found that many aspects of nature follow a bell-shaped curve and
hence, have applied the name normal distribution to this type of curve, an
example of which is shown in Figure 6.1.
• The probability density function that describes the normal distribution is given in
Eqn 6.5.

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Figure 6.1 Normal Distribution
68.26%
95.46%
99.73%
-2-3 +3+2-1 +1

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Normal Distribution
• An important property of the normal distribution is its relationship to the standard
deviation.
• You will note from Figure 6.1 that 68.26% of the data fall within plus or minus one
standard deviation of the mean.
• Similarly, 95.46% and 99.73% of the data points fall within +/- 2 and 3 standard
deviations of the mean respectively.
• If we extrapolate the graph out to +/- 6 standard deviations of the mean we will
include 99.9997% of the data.
• This is the formal definition of a six sigma capable process, that is, a process that
can consistently manufacture product within specifications 99.9997% of the time
allowing for a maximum of 3.4 defects per million opportunities (DPMO).
f(x) =
1
Eqn 6.5
2
e
x – 

2
-1
2

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Normal Distribution
• Let us consider an industrial example to drive home the importance of the
standard deviation to process control.
• Suppose your company was manufacturing a machined part with target length of
100 mm.
• The measured standard deviation is 3 mm resulting in the part distribution
depicted in Figure 6.2A.
• If you could cut the standard deviation in half through process improvements the
distribution of part sizes would be narrowed to that of Figure 6.2B.
• Clearly, this improvement will not only benefit your manufacturing operations but
also those of your customer since the outbound variation in part length is
dramatically reduced.

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Figure 6.2 Effect of Standard Deviation on Normal Distribution
88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Machined Part Length (mm)
88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112
Machined Part Length (mm)
A
B
x = 100 mm
s = 3 mm
x = 100 mm
s = 1.5 mm

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Central Limit Theorem
• Many statistical tests assume the data is normally distributed.
• It is common for the parent population from which the data is drawn to not be
normal.
• Consider the case of the sales manager wanting to analyze the sales statistics from
her territory recognizing that her customer base is far from being “normal”.
• Fortunately, this problem may be avoided through effective sampling design and
an understanding of the central limit theorem.
• Population distributions come in all shapes and sizes.
• They may be skewed, uniform, exponential, parabolic, logarithmic, bimodal, etc.
• The central limit theorem states that, regardless of the shape of the parent
population, the distribution of the means of sample subsets extracted from the
parent population will be normal provided that a sufficient number of sample
subsets are extracted.
• This theorem is best visualized in Figure 6.3.

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Figure 6.3 Effect of Sample Size on Mean Sample Distribution

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Sampling Plan Design
• It can be seen that a sample size of 30 or more subgroups will result in an
approximately normal distribution of the means.
• Thus, you do not need to know the type of distribution the parent population
exhibits as long as you extract a minimum of 30 subgroup samples from the parent
population as part of your statistical analysis of the sample set.
• This illustrates the importance of sampling plan design to ensure that your
samples are representative of the true population.
• Today’s factories have a plethora of information due to Supervisory Control and
Data Acquisition (SCADA) systems, Distributed Control Systems (DCS), data
historians, etc.
• On the other hand, some measurements are time-consuming, expensive and
sometimes destructive; resulting in yield losses.
• Consequently, a balance must be struck between extracting enough subgroups to
manage the Producer Risk (also referred to as Type I Error or Alpha Risk - the risk of
falsely rejecting good parts) and the Consumer Risk (also referred to as Type II
Error or Beta Risk – the risk of falsely accepting bad parts).

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• Deciding how many subgroups to sample at what periodicity is the essence of
sampling plan design.
• The American Society for Quality has published national standards for sampling
procedures for attribute data (count or classification) in ANSI/ASQC Z1.4 and
sampling procedures for variable data (measurement) in ANSI/ASQC Z1.9.
• These standards are based upon military standards MIL-STD-105E and MIL-STD-
414 respectively.
• The sampling plan tables and operating characteristic curves of these standards
allow you to maintain the Acceptable Quality Level (AQL) of your process which is
defined as the worst tolerable process defect average that you are willing to accept
when a continuing series of lots is submitted for acceptance sampling.
• Sampling plans are designed to yield a high probability of accepting a lot at the
AQL and a low probability of accepting a lot at the Rejectable Quality Level (RQL)
also known as the Lot Tolerance Percent Defective (LTPD).
• Producer Risk is managed by the selection of AQL and α.
• Consumer Risk is managed by the selection of RQL and β.

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• Thus, our sampling plan will provide us with a 1-α probability of accepting the lot
at the AQL and a β probability of accepting the lot at the RQL.
• A good starting point for α is 0.05 and 0.10 for β.
• The Quality Engineer will typically iterate selections of AQL and RQL to develop a
sampling plan which minimizes the cost of quality while still protecting the
customer from escaped detection.
• This sliding scale of risk management is best depicted as in Figure 6.4.
• The operating window between the AQL and RQL is the 95% to 10% portion of the
Operating Characteristic (OC) Curve, a measure of the discriminating power of the
sampling plan.
• The OC curve plots the probability of accepting the lot versus the lot fraction
defective.

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• Minitab can be used to generate the OC curve for a given AQL, RQL, lot size and
historical standard deviation.
• It is a convenient way for the Quality Professional to compare sampling plans to
manage risk.
• Let’s consider the example where a candy manufacturer is dispensing 20 g of
chocolate into individual serving bags that have a tolerance of ± 0.4 g.
• The historical standard deviation of fill weights is 0.1 g.
• How many bags of candy would have to be sampled out of a lot size of 5,000 bags
to satisfy management’s AQL and RQL agreement of 1% and 3% respectively with
the candy distributor?
Figure 6.4 Acceptance Sampling Plan Risk Management
Ship
Nothing
Ship
Everything
0% Risk 100% Risk
AQL RQL
1-α probability
of acceptinga
lot at AQL
β probability of
acceptinga lot
at RQL

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Figure 6.5 Steps for Generating Acceptance Sampling Plan by Variables
Open a new worksheet. Click on Stat  Quality Tools  Acceptance Sampling by Variables  Create/Compare on the top menu.

Process Control
Select Create a Sampling Plan from the drop down menu in the dialogue box. Select Percent defective for the units for quality levels.
Enter 1 for AQL, 3 for RQL, 0.05 for Alpha, 0.10 for Beta, 19.6 for Lower spec, 20.4 for Upper spec, 0.1 Historical standard deviation, and
5000 for the lot size. Click OK.

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A graph is generated for the Operating Characteristic Curve, the Average Outgoing Quality Curve and the Average Total Inspection Curve.
The required sample size is 44. If one of the bags of candy audited for fill weight is outside of the 20.0 ± 0.4 g of chocolate, the entire lot
must be rejected and 100% inspection performed. Click Window  Session on the top menu.

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The session window displays the descriptive statistics of the sampling plan. Let’s say the Quality Engineer is concerned about escaped
detection and wants to introduce some buffer into the RQL agreed upon with the candy distributor. Press CTRL-E to return to the last
dialogue box.

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Reduce the RQL level to 2% defective. Click OK.

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The sample size is increased from 44 to 116 by reducing the RQL from 3% to 2%. The Quality Engineer decides to leave well enough alone
and to remain with the sampling plan that represents the agreed upon terms with the distributor.

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• It is a common occurrence for manufacturers to struggle with the definition of lot
size and lot number.
• If you conduct a supplier audit and the answer to your question “how do you
assign lot numbers to your finished product?” results in the response “well … we
use your P.O. number” or “we use the date of manufacture” you know that it will
be a lengthy audit.
• ANSI Z1.4 provides the following guidance for the formation of lots or batches:
“Each lot or batch shall, as far as is practicable, consist of units of product of a
single type, grade, class, size, and composition, manufactured under essentially
the same conditions, and at essentially the same time.”
• This means that a part selected from the chronological front of the lot, the end of
the lot or anywhere in between will have the same quality characteristics.
• A lot number should be a unique code assigned to a lot, used once, and then
retired for life.
• The lot number must have traceability to raw material lot numbers, manufacturing
location, manufacturing line and process conditions.

Process Control
• Typical reasons for a lot number change include a change in raw material lots, a
change to a different production line, downtime exceeding x hours, a process
upset, a process change to correct a defectives situation, or any other change that
alters the quality characteristic of the process output.
• It is common for continuous process manufacturers to assign a maximum part
count or maximum production time to a lot number to limit their exposure in the
event of a quality nonconformance discovered after production.
• Samples should be collected in rational subgroups.
• For example, if your plant is running three shifts per day you will want to collect
your samples to identify shift-to-shift differences.
• Thus, it is logical to collect five subgroup samples from each shift per day and track
the trends of the means via Statistical Process Control.
• But before we venture into statistical process control we need to test the data set
to confirm that it follows a normal distribution.

Process Control
Normality Testing
• Normality testing compares the values of the data set to the probability density
function of Eqn 6.5.
• Let’s look at the measurement data compiled in Figure 6.6 representing a
continuous process where assay wt % is a key process output variable.
• We will use Minitab to quickly assess if the data follow a normal distribution, the
steps for which are captured in Figure 6.7.

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Figure 6.6 Assay Weight % Measurements – Continuous Process
Shift Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Avg Assay wt %
A 92.13 92.67 92.02 92.83 92.50 92.43
B 93.38 93.45 92.76 92.78 92.89 93.05
C 92.66 92.42 92.46 92.20 92.35 92.42
A 92.89 92.50 92.33 92.51 92.74 92.59
B 93.40 92.55 93.44 92.64 93.25 93.06
C 92.30 92.46 92.99 92.23 92.25 92.45
A 92.25 92.73 92.46 92.07 92.83 92.47
B 92.64 93.15 93.33 93.17 93.44 93.15
C 92.30 92.02 92.61 92.81 92.77 92.50
A 92.94 92.65 92.00 92.99 92.87 92.69
B 92.58 93.17 92.54 93.29 92.71 92.86
C 92.27 92.20 92.78 92.30 92.55 92.42
A 92.05 92.11 92.18 93.00 92.30 92.33
B 92.67 92.63 93.01 92.81 93.24 92.87
C 92.70 92.25 92.91 92.53 92.79 92.64
A 92.66 92.97 92.65 92.73 92.05 92.61
B 93.00 93.44 92.58 92.58 93.20 92.96
C 92.16 92.63 92.44 92.17 92.80 92.44
A 92.27 92.65 92.87 91.93 92.18 92.38
B 93.07 93.18 92.52 92.68 93.38 92.97
C 92.09 92.57 92.71 92.87 92.64 92.58
A 92.97 92.06 92.21 92.08 92.04 92.27
B 92.68 92.58 92.64 93.05 93.24 92.84
C 92.05 92.62 92.93 92.00 92.43 92.41
A 92.08 92.17 92.90 92.57 92.54 92.45
B 93.43 93.97 93.28 93.17 93.57 93.48
C 92.67 92.63 92.87 92.67 92.36 92.64
A 92.09 92.34 92.54 92.87 92.69 92.51
B 92.92 92.86 93.27 92.72 92.51 92.86
C 92.70 92.66 92.72 92.69 92.34 92.62

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Figure 6.7 Normality Testing Steps
Copy and paste the assay weight % measurements into a Minitab worksheet.

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Click Stat  Basic Statistics  Normality Test on the top menu.

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Select C7 Avg Assay wt% for the Variable field in the dialogue box. Click OK.

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A normal probability plot is created in the Minitab project file.

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Data Transformation for Normality
• The measured data do not closely follow the blue line representing a normal
distribution.
• The P-value is less than 0.05 which allows us to conclude at the 95% confidence
level that the data set does not follow a normal distribution (we will learn more
about P-values and confidence levels under the section entitled hypothesis testing).
• Although we have practiced good sampling design, averaged rational subgroups,
and taken thirty samples from the parent population the data set is still not normal.
• What can we do? We can transform the data set.
• In 1964 George Box and David Cox derived a power transformation to convert non-
normal data to normal data. In 1993 Norman Johnson developed an alternative
transformation approach.
• In practice, it does not matter which transformation you use. Typically, analysts will
select the approach which provides the best normalization of the data set. Figure
6.8 captures the steps required to transform data to conform to a normal
distribution.

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Figure 6.8 Data Transformation Steps for Normalization
Return to the active worksheet.

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Click Stat  Quality Tools  Johnson Transformation on the top menu.

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Select C7 Avg Assay wt% for the Single column field in the dialogue box. Enter C8 to store the transformed data in column 8. Click OK.

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The Johnson Transformation probability plot is created in the Minitab project file. The transformation was successful to normalize the data set.
The transformed data has a P-value which is well above 0.05. The derived transformation function is given as
1.35408 + 1.07249 * ln[(X - 92.2172)/(93.9892 - X)] where X is the original average assay wt% variable.

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Return to the active worksheet. Notice that the transformed data has been entered in column 8. Label the column as Johnson Transf Avg
Assay.

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Distribution Identification
• The transformed data set would subsequently be used as the source data to
perform analyses such as Statistical Process Control and process capability.
• Periodically, data sets will prove to be particularly resilient to normal distribution
transformation.
• You can do everything including standing on your head, but fail to normalize the
data.
• In this case, it will be necessary to identify the distribution type which most closely
matches the data in order to select the appropriate statistical analysis tool.
• Let’s say we have the active ingredient concentration measurements as captured in
the Minitab worksheet of Figure 6.9.
• The raw data does not follow a normal distribution (P-value less than 0.05) as
shown in Figure 6.10.
• Johnson transformation and Box-Cox transformation fail to normalize the data as
shown in Figures 6.11 and 6.12 respectively. Our next option is to identify the
distribution type which most closely models the data set – the screen shots for
which are captured in Figure 6.13.

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Figure 6.9 Active Ingredient Concentration Worksheet

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Figure 6.10 Active Ingredient Normality Test – Raw Data

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Figure 6.11 Active Ingredient Normality Test – Johnson Transformation

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Figure 6.12 Active Ingredient Normality Test – Box-Cox Transformation

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Figure 6.13 Steps for Distribution Identification
Return to the active worksheet. Click Stat  Quality Tools  Individual Distribution Identification on the top menu.

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Select C1 Active Ingredient (ppm) for the Single column field in the dialogue box. Enter 1 for the subgroup size. Ensure that the radio toggle
button to use all distributions and transformations is checked. Click OK. Fifteen probability plots are generated for the distributions and
transformations.

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Distribution Identification
• Notice that all the distribution and transformation P-values are less than 0.05.
• None of the distributions are a good match for the data.
• The best of the worst is the Weibull distribution with the lowest Anderson Darling
test statistic, AD = 4.435.
• The Anderson Darling test statistic measures the goodness of fit of the data set to
each distribution probability density function.
• The lower the AD test statistic, the more closely the data set follows the distribution
in question.
• We will utilize this information when we explore process capability analysis of non-
normal distributions.

Process Control
Statistical Process Control
• In the mid 1920’s Walter A. Shewhart of Bell Laboratories developed the first control
charting procedures.
• Dr. Shewhart recognized that control charts are a powerful tool to determine if a
process is operating in a state of statistical control or if there are special causes of
variation present which require root cause investigation.
• Statistical Process Control (SPC) charts are useful to establish a benchmark for the
current process variation, detect special causes of variation, ensure process stability,
enable predictability, and to confirm the impact of process improvement activities.
• The data must be plotted in time-series order and it is recommended to plot a
minimum of thirty (30) data points before establishing control limits.
• Control limits are calculated from the data set according to the formulas shown in
Figure 6.14 and utilize the coefficients of Figure 6.15 which are a function of the
subgroup size within the data set.

Process Control
Figure 6.14 Control Limit Formulas - Continuous Data
Centerline X mR
Average of data points Average of the moving ranges
UCL X + 2.66mR D4R
LCL X - 2.66mR D3R
Centerline X R
Average of subgroup averages Average of subgroup ranges
UCL X + A2R D4R
LCL X - A2R D3R
Centerline X S
Average of subgroup averages Average of subgroup std deviations
UCL X + A3R B4S
LCL X - A3R B3S
Individuals - Moving Range Chart (IMR Chart)
Subgroup Averages - Range Chart (Xbar-R Chart)
Subgroup Averages - Standard Deviation Chart (Xbar-S Chart)

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Figure 6.15 Coefficients for Control Limit Formulas - Continuous Data
Subgroup
Size (n)
A2 A3 B3 B4 D3 D4
2 1.88 2.66 0.00 3.27 0.00 3.27
3 1.02 1.95 0.00 2.57 0.00 2.57
4 0.73 1.63 0.00 2.27 0.00 2.28
5 0.58 1.43 0.00 2.09 0.00 2.11
6 0.48 1.29 0.03 1.97 0.00 2.00
7 0.42 1.18 0.12 1.88 0.08 1.92
8 0.37 1.10 0.19 1.82 0.14 1.86
9 0.34 1.03 0.24 1.76 0.18 1.82
10 0.31 0.98 0.28 1.72 0.22 1.78
11 0.29 0.93 0.32 1.68 0.26 1.74
12 0.27 0.89 0.35 1.65 0.28 1.72
13 0.25 0.85 0.38 1.62 0.31 1.69
14 0.24 0.82 0.41 1.59 0.33 1.67
15 0.22 0.79 0.43 1.57 0.35 1.65
16 0.21 0.76 0.45 1.55 0.36 1.64
17 0.20 0.74 0.47 1.53 0.38 1.62
18 0.19 0.72 0.48 1.52 0.39 1.61
19 0.19 0.70 0.50 1.50 0.40 1.60
20 0.18 0.68 0.51 1.49 0.42 1.59

Process Control
• The tests developed by Dr. Shewhart to check for special causes of variation assume
the data are normally distributed and independent (i.e. a measured value is not
influenced by its past values).
• There are eight generally accepted rules to check for special causes of variation.
• These rules are summarized in Figures 6.16 through 6.23 for the example process of
milk ultra pasteurization to prolong its shelf life.
• Notice that the control limits are separated from the overall mean by three zones
labeled C, B and A respectively.
• These zones correspond to  three standard deviations from the centerline where
the standard deviation is not derived from equation 6.1 but from the control limit
definitions of Figure 6.14.
• For example, the standard deviation for an individuals chart would be calculated as
in Eqn 6.6.

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s =
|xi+1 – xi|
i = 1
n - 1
n - 1
where s = short term process standard deviation
mR = moving range average
Eqn 6.62.66
3
2.66
3
mR=
Equation 6.6 Individuals Chart - Short Term Process Standard Deviation

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Figure 6.16 SPC Rule 1: One or More Points are Outside the Control Limits
28252219161310741
292
290
288
286
284
282
280
Observation
IndividualValue
_
X=284.93
UCL=290.16
LCL=279.71
1
Ultra Pasteurization Process Temp (°F)
A
B
C
C
B
A

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Figure 6.17 SPC Rule 2: Seven Consecutive Points are on the Same Side of the Centerline
28252219161310741
295
290
285
280
275
Observation
IndividualValue
_
X=284.67
UCL=293.20
LCL=276.14
2
2
2
A
B
C
C
B
A

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Figure 6.18 SPC Rule 3: Seven Consecutive Intervals are Entirely Increasing or Entirely Decreasing
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295
290
285
280
275
Observation
IndividualValue
_
X=284.6
UCL=293.50
LCL=275.70
3
3
A
B
C
A
B
C

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Figure 6.19 SPC Rule 4: Fourteen Consecutive Points Alternate Up and Down Repeatedly
28252219161310741
295
290
285
280
275
Observation
IndividualValue
_
X=285.33
UCL=294.60
LCL=276.07
4
4
4
4
4
A
B
C
A
B
C

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Figure 6.20 SPC Rule 5: Two out of Three Consecutive Points are in the Same Zone A or Beyond
28252219161310741
292
290
288
286
284
282
280
Observation
IndividualValue
_
X=285.6
UCL=291.74
LCL=279.46
5
A
A
B
B
C
C

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Figure 6.21 SPC Rule 6: Four out of Five Consecutive Points are in the Same Zone B or Beyond
28252219161310741
292
290
288
286
284
282
280
278
276
Observation
IndividualValue
_
X=284.67
UCL=292.00
LCL=277.33
6
A
A
B
B
C
C

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Figure 6.22 SPC Rule 7: Fourteen Consecutive Points are in Either Zone C
28252219161310741
292
290
288
286
284
282
280
Observation
IndividualValue
_
X=285.7
UCL=291.94
LCL=279.46
7
A
A
B
B
C
C

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Figure 6.23 SPC Rule 8: Eight Consecutive Points are Outside Either Zone C
28252219161310741
292
290
288
286
284
282
280
Observation
IndividualValue
_
X=285.37
UCL=290.87
LCL=279.86
8
A
A
B
B
C
C

Process Control
• Rule 1 detects a shift in the mean, an increase in the standard deviation or a single
anomaly in the process. Check the associated range chart to see if increases in
variation are the source of the special cause.
• Rule 2 detects a shift in the process mean.
• Rule 3 detects an increasing or decreasing trend in the process mean.
• Rule 4 detects systematic effects such as two alternately used machines, vendors or
operators.
• Rule 5 detects a shift in the process mean or increase in the standard deviation.
• Rule 6 detects a shift in the process mean.
• Rule 7 illustrates the symptom of “hugging the centerline”. If the special cause of this
out-of-control symptom was a process change designed to reduce variation the result is
understandable. If this phenomenon occurred on its own the measurement system
may have lost resolution.
• Rule 8 violations can occur when the measurement system develops a blind spot at the
process centerline or when operator interventions result in over-steering the process.

Process Control
• The type of control chart selected will depend upon your data type and subgrouping.
• Attribute data where you can count the number of occurrences but not the number of
non-occurrences (eg the number of defects in a plate of glass) follow the poisson
distribution and are analyzed by c-charts or u-charts depending on whether the sample
size is fixed or not respectively.
• Attribute data of the pass/fail type (eg the number of dropped calls per day at a call
center) follow the binomial distribution and are analyzed by np-charts or p-charts
depending on whether the sample size is fixed or not respectively.
• Variables data which are measured are sampled such that the sample means follow the
normal distribution.
• A subgroup sample size of one derived from continuous data such as the pasteurization
process temperature example we reviewed earlier is analyzed by individuals, moving
range charts
• A subgroup sample size of two to nine is best analyzed by an Xbar-R chart which plots
averages of subgroups and the range within subgroups.

Process Control
• A subgroup sample size equal to or larger than ten is best analyzed by an Xbar-S chart
which plots averages of subgroups and the standard deviation within subgroups.
• Figure 6.24 summarizes the logical decision process for selecting a control chart type.
Variables
Attributes
Attributes or
Variables
Data?
No
Subgroup
Size = 1?
No
Subgroup
Size < 10?
Assess Data Type
IMR Chart
Yes
Yes
Xbar-R Chart
Yes
Xbar-S Chart
Count
ClassificationCount or
Classification
Data?
Yes
NoConstant
Sample Size?
c-Chart
u-Chart
Constant
Sample Size?
Yes
p-Chart
np-Chart
No
Figure 6.24 Control Chart Type – Decision Tree

Process Control
• In general, it is best to convert attribute data into continuous data and conduct SPC
analysis via Xbar-R or IMR charts since the zone rules of Figures 6.16 through 6.23 do
not apply to attribute data.
• For example, the number of defects in a plate of glass could be divided by the weight of
the plate glass.
• The resulting measure of defects/lb of glass could be analyzed via an IMR chart.
• For data that occurs infrequently (such as the occurrence of a safety incident) consider
monitoring the time between incidents rather than the binomial attribute data of
yes/no a safety incident has occurred.
• Also, consider to track leading continuous indicators such as days between near misses.
• Minitab makes creating SPC charts easy. We will consider a sample data set each from
manufacturing and the service sector to illustrate the process of generating SPC charts
and the implications that can be derived from the analysis of these charts.
• Let us first consider a data set from our previous example process of the ultra
pasteurization of milk. Figure 6.25 captures the screen shots of the SPC chart
generation steps.

Process Control
Figure 6.25 SPC Chart Generation Steps – Manufacturing Example
Make sure that your data is entered into the worksheet in the correct format and in chronological order. Click on Stat  Control Charts
 Variables Charts for Individuals  I-MR on the top menu.

Process Control
Highlight C2 Temp_F in the dialogue box and click Select. Click Scale.

Process Control
Click the radio toggle button for Stamp under the X Scale in the dialogue box. Place your cursor in the Stamp columns box, highlight C1
Time and click Select. Click OK.

Process Control
Click I-MR Options.

Process Control
Click on the tab for Tests. Check the boxes to perform special cause analysis for test numbers 1, 2, 5 and 6. Click OK.

Process Control
Click OK.

Process Control
A graph is created in the Minitab project file with the stacked Individuals – Moving Range SPC Charts. Out of control data points are
highlighted in red and include a superscript number indicating which special cause test has been violated. In this case, test number 6
(4 out of 5 points greater than 1 standard deviation from the center line) is indicative of special causes which have resulted in a mean shift.
20:3019:0017:3016:0014:3013:0011:3010:0008:3007:00
295
290
285
280
Time
IndividualValue
_
X=285.54
UCL=294.33
LCL=276.74
20:3019:0017:3016:0014:3013:0011:3010:0008:3007:00
12
8
4
0
Time
MovingRange
__
MR=3.31
UCL=10.80
LCL=0
666
I-MR Chart of Temp_F

Process Control
• In this example the data points are all within the temperature specification of 285  5F
but the process has shifted high beginning at point number 18 (15:30 hrs) indicating
the occurrence of a special cause which requires investigation.
• It could be that a heater control module has failed or a heat exchanger valve has stuck
open.
• This might result in the product developing a “cooked” aftertaste, which is
objectionable to customers.
• Without seeing this shift in a control chart format, this special cause could be
overlooked.
• In our second example, we will analyze the process control condition of dropped calls
at a 24/7 customer service call center.
• Please refer to Figure 6.26.

Process Control
Figure 6.26 SPC Chart Generation Steps – Service Example
Make sure that your data is entered into the worksheet in the correct format and in chronological order. Click on Stat  Control Charts
 Variables Charts for Individuals  I-MR on the top menu.

Process Control
Highlight C2 % Dropped Calls in the dialogue box and click Select. Click Scale.

Process Control
Click the radio toggle button for Stamp under the X Scale in the dialogue box. Place your cursor in the Stamp columns box, highlight C1
Time and click Select. Click OK.

Process Control
Click I-MR Options.

Process Control
Click on the tab for Tests. Select the drop down menu to perform all tests for special causes. Click OK.

Process Control
Click OK.

Process Control
A graph is created in the Minitab project file with the stacked Individuals – Moving Range SPC Charts. Out of control data points are
highlighted in red and include a superscript number indicating which special cause test has been violated. In this case, test number 1
(one point greater than three standard deviations from the center line) is indicative of special cause variation.
05:0003:0001:0023:0021:0019:0017:0015:0013:0011:0009:0007:00
6.0%
4.0%
2.0%
0.0%
Time
IndividualValue
_
X=1.3%
UCL=3.9%
LB=0.0%
05:0003:0001:0023:0021:0019:0017:0015:0013:0011:0009:0007:00
6.0%
4.0%
2.0%
0.0%
Time
MovingRange
__
MR=1.0%
UCL=3.3%
LCL=0.0%
1
1
1
I-MR Chart of % Dropped Calls

Process Control
• In this case a spike in dropped calls has occurred at 13:00 hrs indicating a special
cause.
• As it turns out, this data point records the percentage of dropped calls which
occurred between the hours of 12:00 pm and 1:00 pm.
• This corresponds to the lunch hour where the number of customer calls increase
while the number of call center associates decrease resulting in a spike of dropped
calls because customers get tired of waiting in the incoming call queue.
• This indicates the need for a staffing schedule change to split the call center
associates’ lunch hour and to provide additional support staffing during the lunch
hour rush.

Process Control
Process Capability
• Analysis of your data may indicate that your process is in control but is it capable of
meeting your customer requirements?
• Capability implies comparison of your process mean and standard deviation to the
specification limits, the upper and lower bounds for which you and your customer have
mutually agreed upon.
• This could be an external customer, in the case of a measured quality characteristic that
is reported on your Certificate of Analysis (C of A) or it could be an internal customer,
the next downstream process.
• A commonly used measure of process capability is the short term process capability
index Cpk as defined by Eqn 6.7 and the long term process capability index Ppk as
defined by Eqn 6.8.
• The difference between these two indices lies in the calculation of the standard
deviation.
• Short term process capability utilizes the standard deviation as derived from control
limits (remember X-bar ± 3s) whereas long term process capability utilizes the standard
deviation as calculated from the overall data set (eg the STDEV function of Excel).

Process Control
Process Capability
Cpk =
USL – x
3s
where Cpk = short term process capability index
USL= Upper Spec Limit
LSL= Lower Spec Limit
s = short term process standard deviation
x = overall process mean
Eqn 6.7
x – LSL
3s
or
min
Ppk =
USL – x
3σ
where Ppk = long term process capability index
USL= Upper Spec Limit
LSL= Lower Spec Limit
σ = overall process standard deviation
x = overall process mean
Eqn 6.8
x – LSL
3σ
or
min
Equation 6.7 Short Term Process Capability Index
Equation 6.8 Long Term Process Capability Index

Process Control
Process Capability
• Let us consider the example SPC chart of Figure 6.27 to understand the implications
of process capability.
• The process appears to be in control with individual data points randomly
distributed about the mean.
• If we add upper and lower spec limits to the control chart we notice that the
process is operating in the upper half of the spec range as shown in Fig 6.28.
• The mean for this process is 287.58 F with an overall standard deviation of 1.212 F.
• Thus, we may calculate the Ppk as follows.
Ppk =
290 – 287.58
3(1.212)
or
min
3(1.212)
287.58 – 280
Ppk = 0.665 or
min
2.08
Ppk = 0.665

Process Control
Figure 6.27 Process Capability Implications – SPC Chart
8:30
PM
7:00
PM
5:30
PM
4:00
PM
2:30
PM
1:00
PM
11:30
AM
10:00
AM
8:30
AM
7:00
AM
292
290
288
286
284
282
280
IndividualValue
_
X=287.58
UCL=291.64
LCL=283.53

Process Control
Figure 6.28 Process Capability Implications – SPC Chart with USL and LSL

Process Control
Process Capability
• A Ppk value below one indicates poor process capability.
• The typical goal for long term process capability is 1.33 or above.
• This corresponds to a sigma level of 4.
• Avoid the temptation to widen the spec limits to improve the Ppk.
• Centering the process mean over the process target while simultaneously reducing
the standard deviation maximizes process capability.
• Minitab can be used to perform process capability analysis on your data and
generate tiled charts which provide information on process control, data set
normality and process capability.
• Figure 6.29 captures the screen shots of the process capability analysis steps.

Process Control
Figure 6.29 Process Capability Analysis Steps – Manufacturing Example
Make sure that your data is entered into the worksheet in the correct format and in chronological order. Click on Stat  Quality Tools 
Capability Analysis  Normal on the top menu.

Process Control
Click on Single column in the dialogue box and highlight C2 Temp_F. Click Select. Enter a Subgroup size of 1. Enter the lower spec and
upper spec in the appropriate fields. Click OK.

Process Control
A graph is created in the Minitab project file with the process capability analysis results. The graph indicates that the process is biased toward
the upper specification limit. The short term process capability index, Cpk is 0.60 and the long term process capability index, Ppk is 0.66.

Process Control
Return to the active worksheet. Click on Stat  Quality Tools  Capability Sixpack  Normal on the top menu.

Process Control
Click on Single column in the dialogue box and highlight C2 Temp_F. Click Select. Enter a Subgroup size of 1. Enter the lower spec and
upper spec in the appropriate fields. Click OK.

Process Control
A new graph is created in the Minitab project file with the process capability sixpack analysis results.

Process Control
Process Capability
• The Individuals Chart and Moving Range Chart indicate no special causes of variation.
• The Last 25 Observations chart indicates randomly distributed points about the mean
(a desirable result).
• The Capability Histogram indicates a bias of the process toward the upper spec limit.
• The Normal Probability Plot indicates that the source data is normally distributed since
the P-value is greater than 0.05.
• The Capability Plot stacks the short term and long term process capability over the
spec range.
• This is a powerful collection of charts enabling the analyst to understand the current
state of process control and capability.
• But what should we do if the data set is not normally distributed as we experienced
with the active ingredient concentration data of Figure 6.9?
• We can conduct a non-normal process capability analysis as shown in Figure 6.30
provided that we have identified the distribution type which most closely matches the
data set as shown in Figure 6.13.

Process Control
Figure 6.30 Process Capability Analysis Steps – Non-normal Distribution
Open the worksheet with the non-normal data you want to conduct process capability analysis on. Click on Stat  Quality Tools 
Capability Analysis  Nonnormal on the top menu.

Process Control
Click on Single column in the dialogue box and highlight C1 Active Ingredient (ppm). Click Select. Enter a Subgroup size of 1. Select
Weibull from the Fit distribution drop down menu. Enter the lower spec and upper spec in the appropriate fields. In this case we have no
lower spec, so the process capability analysis will be one-tailed. Click OK.

Process Control
A new graph is created in the Minitab project file with the process capability sixpack analysis results.

Process Control
Summary
• We now have tools for evaluating the degree of control a process exhibits and the
capability of that process to meet customer requirements.
 Standard Deviation
 Variance
 Normality Testing
 Distribution Identification
 Sampling Plan Design
 Data Transformation for Normality
 Statistical Process Control Charts
 Process Capability Analysis
• Remember that variation has many sources.
• Before we pass judgment on the health of the process we must first understand
the variation contribution caused by the measurement system.

Process Control

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