Statistical Process Control

The concepts of Statistical Process Control (SPC) were initially
developed by Dr.
Walter Shewhart of
Bell Laboratories in the 1920's, and were expanded upon by Dr.
W. Edwards Deming, who introduced SPC to Japanese industry
after WWII. After early successful adoption by Japanese firms,
Statistical Process Control has now been incorporated by
organizations around the world as a primary tool to improve
product quality by reducing process
variation. 
Dr. Shewhart identified two sources of process variation: Chance variation
that is inherent in process, and stable over time, and Assignable,
or Uncontrolled variation,
which is unstable over time  the result of specific events
outside the system. Dr. Deming relabeled chance variation as Common
Cause variation,
and assignable variation as Special
Cause variation.
Based on experience with many types of process data, and
supported by the laws of statistics and probability, Dr.
Shewhart devised control charts used to plot data over time and
identify both Common Cause variation and Special Cause
variation. 
This tutorial provides a brief conceptual background to the
practice of SPC, as well as the necessary formulas and
techniques to apply it. 
If you have reviewed the discussion of frequency distributions
in the Histogram module, you will recall that many histograms
will approximate a Normal Distribution, as shown below (please
note that control
charts do not require normally distributed data in order to work 
they will work with any process distribution  we use a normal
distribution in this example for ease of representation): 

In order to work with any distribution, it is important to have
a measure of the data dispersion, or spread. This can be
expressed by the range (highest less lowest), but is better
captured by the standard deviation (sigma). The standard
deviation can be easily calculated from a group of numbers using
many calculators, or a spreadsheet or statistics program. 

Often we focus on average values, but understanding dispersion
is critical to the management of industrial processes. Consider
two examples:

If you put one foot in a bucket of ice water (33 degrees
F) and one foot in a bucket of scalding water (127
degrees F), on average you'll feel fine (80 degrees F),
but you won't actually be very comfortable! 

If you are asked to walk through a river and are told
that the average water depth is 3 feet you might want
more information. If you are then told that the range is
from zero to 15 feet, you might want to reevaluate the
trip. 

MoreSteam Hint: Analysis of averages should always be
accompanied by analysis of the variability! 

Statistical tables have been developed for various types of
distributions that quantify the area under the curve for a given
number of standard deviations from the mean (the normal
distribution is shown in this example). These can be used as
probability tables to calculate the odds that a given value
(measurement) is part of the same group of data used to
construct the histogram. Shewhart found that control limits
placed at three standard deviations from the mean in either
direction provide an economical tradeoff between the risk of
reacting to a false signal and the risk of not reacting to a
true signal  regardless the shape of the underlying process
distribution. If the process has a normal distribution, 99.7% of
the population is captured by the curve at three standard
deviations from the mean. Stated another way, there is only a
199.7%, or 0.3% chance of finding a value beyond 3 standard
deviations. Therefore, a measurement value beyond 3 standard
deviations indicates that the process has either shifted or
become unstable (more variability). The illustration below shows
a normal curve for a distribution with a mean of 69, a mean less
3 standard deviations value of 63.4, and a mean plus 3 standard
deviations value of 74.6. Values, or measurements, less than
63.4 or greater than 74.6 are extremely unlikely. These laws of
probability are the foundation of the control chart. 

Now, consider that the distribution is turned sideways, and the
lines denoting the mean and ± 3 standard deviations are
extended. This construction forms the basis of the Control
chart. Time series data plotted on this chart can be compared to
the lines, which now become control limits for the process.
Comparing the plot points to the control limits allows a simple
probability assessment. We know from our previous discussion
that a point plotted above the upper control limit has a very
low probability of coming from the same population that was used
to construct the chart  this indicates that there is a Special
Cause  a source of variation beyond the normal chance variation
of the process. 

Deploying Statistical Process Control is a process in itself,
requiring organizational commitment across functional
boundaries. The flowchart below outlines the major components
of an effective SPC effort. The process steps are numbered for
reference. 

Statistical Process Control is based on the analysis of data, so
the first step is to decide what data to collect. There are two
categories of control chart distinguished by the type of data
used: Variable or Attribute. Variable data comes from
measurements on a continuous scale, such as: temperature, time,
distance, weight. Attribute data is based on upon discrete
distinctions such as good/bad, percentage defective, or number
defective per hundred. 
MoreSteam Hint: Use variable data whenever possible
because it imparts a higher quality of information  it
does not rely on sometimes arbitrary distinctions
between good and bad. 

A critical but often overlooked step in the process is to
qualify the measurement system. No measurement system is without
measurement error. If that error exceeds an acceptable level,
the data cannot be acted upon reliably. For example: a Midwest
building products manufacturer found that many important
measurements of its most critical processes had error in excess
of 200% of the process tolerance. Using this erroneous data, the
process was often adjusted in the wrong direction  adding to
instability rather than reducing variability. See theMeasurement
Systems Analysis section
of the Toolbox for additional help with this subject. 
Develop a sampling plan to collect data (subgroups) in a random
fashion at a determined frequency. Be sure to train the data
collectors in proper measurement and charting techniques.
Establish subgroups following a rational subgrouping strategy so
that process variation is captured BETWEEN subgroups
rather than WITHINsubgroups.
If process variation (e.g. from two different shifts) is
captured within one subgroup, the resulting control limits will
be wider, and the chart will be insensitive to process shifts.
The type of chart used will be dependent upon the type of
data collected as well as the subgroup size, as shown by the
table below. A bar, or line, above a letter denotes the average
value for that subgroup. Likewise, a double bar denotes an
average of averages. 
Consider the example of two subgroups, each with 5 observations.
The first subgroup's values are: 3,4,5,4,4  yielding a subgroup
average of 4 (X̅_{1}). The second subgroup has the
following values: 5,4,5,6,5  yielding an average of 5 (X̅_{2}).
The average of the two subgroup averages is (4 + 5)/2 = 4.5,
which is called X doublebar (),
because it is the average of the averages. 
You can see examples of charts in Section 9 on Control Limits. 

Each process charted should have a defined reaction plan to
guide the actions to those using the chart in the event of an
outofcontrol or outofspecification condition. Read Section 9
below to understand how to detect outofcontrol conditions. One
simple way to express the reaction plan is to create a flow
chart with a reference number, and reference the flow chart on
the SPC chart. Many reaction plans will be similar, or even
identical for various processes. Following is an example of a
reaction plan flow chart: 

More Steam Note: Specifications should NEVER be
expressed as lines on control charts because the plot
point is an average, not an individual. The only
exception is the moving range chart, which is based on a
subgroup size of one. Consider the case of a subgroup of
three data points: 13, 15, 17. Suppose the upper
specification limit is 16. The average of the subgroup
is only 15, so the plot point looks like it is within
the specification, even though one of the measurements
was out of spec.! However, specifications should be
printed on the side, top, or bottom of the chart for
comparing individual readings. 

A control plan should be maintained that contains all pertinent
information on each chart that is maintained, including: 

Chart Type 

Chart Champion  Person(s) responsible to collect and
chart the data 

Chart Location 

Measurement Method 

Measurement System Analysis (Acceptable Error?) 

Reaction Plan 

Gauge Number  Tied in with calibration program 

Sampling Plan 

Process Stability Status 

Cp & Cpk 

The control plan can be modified to fit local needs. A template
can be accessed through the Control
Plansection of the Toolbox. 
Terms used in the various control chart formulas are summarized
by the table below: 

Formulas are shown below for Attribute and Variable data.


(Here n = subgroup or sample size and k = number of subgroups or
samples) 
Values for formula constants are provided by the following
charts:


Subgroup Size (n) 
A_{2} 
D_{3} 
D_{4} 
d_{2} 
2 
1.880 
0 
3.267 
1.128 
3 
1.023 
0 
2.574 
1.693 
4 
0.729 
0 
2.282 
2.059 
5 
0.577 
0 
2.114 
2.326 
6 
0.483 
0 
2.004 
2.534 
7 
0.419 
0.076 
1.924 
2.704 
8 
0.373 
0.136 
1.864 
2.847 
9 
0.337 
0.184 
1.816 
2.970 
10 
0.308 
0.223 
1.777 
3.078 


Subgroup Size (n) 
A_{2} 
B_{3}

B_{4} 
11 
0.927 
0.322 
1.678 
12 
0.886 
0.354 
1.646 
13 
0.850 
0.382 
1.619 
14 
0.817 
0.407 
1.593 
15 
0.789 
0.428 
1.572 

Chart examples: 




The area circled denotes an outofcontrol condition, which is
discussed below. 
For more specific help in constructing SPC charts, see the MoreSteam
Online SPC Course offering. 
After establishing control limits, the next step is to assess
whether or not the process is in control (statistically stable
over time). This determination is made by observing the plot
point patterns and applying six simple rules to identify an
outofcontrol condition. 
Out of Control Conditions: 

A. 
If one or more points falls outside of the upper control
limit (UCL), or lower control limit (LCL). The UCL and
LCL are three standard deviations on either side of the
mean  see section A of the illustration below. 
B. 
If two out of three successive points fall in the area
that is beyond two standard deviations from the mean,
either above or below  see section B of the
illustration below. 
C. 
If four out of five successive points fall in the area
that is beyond one standard deviation from the mean,
either above or below  see section C of the
illustration below. 
D. 
If there is a run of six or more points that are all
either successively higher or successively lower  see
section D of the illustration below. 
E. 
If eight or more points fall on either side of the mean
(some organization use 7 points, some 9)  see section E
of the illustration below. 
F. 
If 15 points in a row fall within the area on either
side of the mean that is one standard deviation from the
mean  see section F of the illustration below. 


When an outofcontrol condition occurs, the points should be
circled on the chart, and the reaction plan should be followed. 

When corrective action is successful, make a note on the chart
to explain what happened. 
MoreSteam Hint: Control charts offer a powerful medium
for communication. Process shifts, outofcontrol
conditions, and corrective actions should be noted on
the chart to help connect cause and effect in the minds
of all who use the chart. The best charts are often the
most cluttered with notes! 

If an outofcontrol condition is noted, the next step is to
collect and analyze data to identify the root cause. Several
tools are available through the MoreSteam.com Toolbox function
to assist this effort  see the Toolbox
Home Page. You can use MoreSteam.com's Traction^{®} to
manage projects using the Six Sigma DMAIC and DFSS processes. 
Remember to review old control charts for the process if they
exist  there may be notes from earlier incidents that will
illuminate the current condition. 
After identifying the root cause, you will want to design and
implement actions to eliminate special causes and improve the
stability of the process. You can use the Corrective
Action Matrix to
help organize and track the actions by identifying
responsibilities and target dates. 
The ability of a process to meet specifications (customer
expectations) is defined as Process Capability, which is
measured by indexes that compare the spread (variability) and
centering of the process to the upper and lower specifications.
The difference between the upper and lower specification is know
as the tolerance. 
After establishing stability  a process in control  the
process can be compared to the tolerance to see how much of the
process falls inside or outside of the specifications. Note:
this analysis requires that the process be normally distributed.
Distributions with other shapes are beyond the scope of this
material. 
MoreSteam Reminder: Specifications are not related to
control limits  they are completely separate.
Specifications reflect "what the customer wants", while
control limits tell us "what the process can deliver". 

The first step is to compare the natural sixsigma spread of the
process to the tolerance. This index is known as Cp. 
Here is the information you will need to calculate the Cp and
Cpk: 

Process average, or 

Upper Specification Limit (USL) and Lower Specification
Limit (LSL). 

The Process Standard Deviation ().
This can be calculated directly from the individual
data, or can be estimated by:


Cp is calculated as follows: 
Following is an illustration of the Cp concept: 

Cp is often referred to as "Process Potential" because it
describes how capable the process could be if it were centered
precisely between the specifications. A process can have a Cp in
excess of one but still fail to consistently meet customer
expectations, as shown by the illustration below: 

The measurement that assesses process centering in addition to
spread, or variability, is Cpk. Think of Cpk as a Cp calculation
that is handicapped by considering only the half of the
distribution that is closest to the specification. Cpk is
calculated as follows: 

The illustrations below provide graphic examples of Cp and Cpk
calculations using hypothetical data: 

The Lower Specification Limit is 48
The Nominal, or Target Specification is 55 The Upper
Specification Limit is 60 Therefore, the Tolerance is 60 
48, or 12 As seen in the illustration, the 6Sigma process
spread is 9.
Therefore, the Cp is 12/9 or 1.33.

The next step is to calculate the Cpk index: 

Cpk is the minimum of: 5748/4.5 = 2, and 6057/4.5 = 0.67 
So Cpk is 0.67, indicating that a small percentage of the
process output is defective (about 2.3%). 
Without reducing variability, the Cpk could be improved to a
maximum1.33, the Cp value, by centering the process. Further
improvements beyond that level will require actions to reduce
process variability. 
The last step in the process is to continue to monitor the
process and move on to the next highest priority. 
More Steam Hint: Statistical Process Control requires
support from the top, like any program. The process will
be most effective if senior managers make it part of
their daily routine to review charts and make comments.
Some practitioners initial charts when they review them
to provide visual support. Charts that are posted on the
floor make the best working tools  they are visible to
operators, and are accessible to problemsolving teams. 
