Statistical Process Control
Processes that are not in a state of statistical control
show excessive variations or exhibit variations that
change with time
 Control charts are used to detect whether a process
is statistically stable.
 Control charts differentiates between variations that
is normally expected of the process
due chance or common causes that change over time
due to assignable or special causes

Common cause variation
Variations due to common causes
 are inherent to the process because of:
◦ the nature of the system
◦ the way the system is managed
◦ the way the process is organized and operated

can only be removed by
◦ making modifications to the process
◦ changing the process

are the responsibility of higher management
Special Cause Variation
Variations due to special causes are
 localized in nature
 exceptions to the system
 considered abnormalities
 often specific to a
◦ certain operator
◦ certain machine
◦ certain batch of material, etc.
Investigation and removal of variations due to special
causes are key to process improvement
Causes of Variation

Two basic categories of variation in output include common
causes and assignable causes.

Common causes are the purely random, unidentifiable
sources of variation that are unavoidable with the current
process.
◦ If process variability results solely from common causes of variation, a
typical assumption is that the distribution is symmetric, with most
observations near the center.

Assignable causes of variation are any variation-causing
factors that can be identified and eliminated, such as a machine
needing repair.
Statistical Process Control (SPC)
Charts

Statistical process control (SPC) charts are used to help us
distinguish between common and assignable causes of variation.
We will cover 2 types:

Variables Control Charts: Service or product characteristics
that is continuous and can be measured, such as weight, length,
volume, or time.

Attributes Control Charts: Service or product
characteristics that can be counted...pass/fail, good/bad, rating
scale. Color, inspection, test results (pass/fail), number of defects,
types of defects
SPC charts Explained
Control Charts

Control Charts are run charts with
superimposed normal distributions
Purpose of Control Charts

Control charts provide a graphical means for testing
hypotheses about the data being monitored.
Control and warning limits
The probability of a sample having a particular value is given by
its location on the chart. Assuming that the plotted statistic is
normally distributed, the probability of a value lying beyond
the:
 warning limits is approximately 0.025 or 2.5% chance (plus or
minus 2-sigma from the mean)
 control limits is approximately 0.001 or 0.1% chance (plus or
minus 3 sigma from the mean), this is rare and indicates that

◦ the variation is due to an assignable cause
◦ the process is out-of-statistical control
Out of Control Processes
Run rules are rules that are used to indicate out-of-statistical control
situations. Typical run rules for Shewhart X-charts with control
and warning limits are:
 a point lying beyond the control limits
 2 consecutive points lying beyond the warning limits
(0.025x0.025x100 = 0.06% chance of occurring)
 7 or more consecutive points lying on one side of the mean (
0.57x100 = 0.8% chance of occurring and indicates a shift in the
mean of the process)
 5 or 6 consecutive points going in the same direction (indicates
a trend)
 Other run rules can be formulated using similar principles
Control Charts
Variations
UCL
Nominal
LCL
Sample number
Control Charts
Variations
UCL
Nominal
LCL
Sample number
Control Charts
Variations
UCL
Nominal
LCL
Sample number
Control Charts
Variations
UCL
Nominal
LCL
Sample number
Sampling

Sampling plan: A plan that specifies a sample
size, the time between successive samples, and
decision rules that determine when action
should be taken.
Sample size: A quantity of randomly selected
observations of process outputs.
Why do I need to sample? The case of potato
chips

Means and Ranges
The mean of the sample is the sum of all observations in
a sample divided by the number of observations in the
sample.
The range of the sample is the difference between the
largest observation in a sample and the smallest.
The mean of the process is the sum of all sample means
divided by the number of samples
The mean range for the process is the sum of all ranges
divided by the number of samples
Statistical Process
Control Methods

Control Charts for variables are used to monitor the
mean and variability of the process distribution.

R-chart (Range Chart) is used to monitor process
variability.
- x-chart is used to see whether the process is
generating output, on average, consistent with a target
value set by management for the process or whether its
current performance, with respect to the average of the
performance measure, is consistent with past
performance.
◦ If the standard deviation of the process is known, we can place
UCL and LCL at “z” standard deviations from the mean at the
desired confidence level.
Another way to develop control
charts so for small sample sizes
The control limits for the x-chart are:
=
UCL–x = x + A2R and LCLx– = x= - A2R
Where
=
X = central line of the chart, which can be either the average of past
sample means or a target value set for the process.
A2 = constant to provide three-sigma limits for the sample mean.
The control limits for the R-chart are UCLR = D4R and LCLR = D3R
where
R = average of several past R values and the central line of the
chart.
D3,D4 = constants that provide 3 standard deviations (threesigma) limits for a given sample size.
Control Chart Factors
TABLE 5.1 |
|
Size of
Sample (n)
FACTORS FOR CALCULATING THREE-SIGMA LIMITS FOR
THE x-CHART AND R-CHART
Factor for UCL and
LCL for x-Chart (A2)
Factor for LCL for
R-Chart (D3)
Factor for UCL for
R-Chart (D4)
2
1.880
0
3.267
3
1.023
0
2.575
4
0.729
0
2.282
5
0.577
0
2.115
6
0.483
0
2.004
7
0.419
0.076
1.924
8
0.373
0.136
1.864
9
0.337
0.184
1.816
10
0.308
0.223
1.777
Over the past 2 years, Professor Matta has been asked to
teach one section of Process Analytics in each quarter (4
Quarters/ Year). Each time he taught, he would give 3
exams. The class average grade on these exams over the
last 8 quarters have been as follows:
Year
Quarter
Exam 1
Exam 2
Exam 3
2009
Quarter 1
85
82
90
2009
Quarter 2
73
67
77
2009
Quarter 3
85
85
85
2009
Quarter 4
90
73
83
2010
Quarter 1
70
83
98
2010
Quarter 2
89
81
83
2010
Quarter 3
95
93
91
2010
Quarter 4
72
83
95
Exam Grades in Prof Matta’s PA class
during the past 2 years with Averages:
Year
Quarter
Exam 1
Exam 2
Exam 3
Average
2009
Quarter 1
85
82
90
85.67
2009
Quarter 2
73
67
77
72.33
2009
Quarter 3
85
85
85
85.00
2009
Quarter 4
90
73
83
82.00
2010
Quarter 1
70
83
98
83.67
2010
Quarter 2
89
81
83
84.33
2010
Quarter 3
95
93
91
93.00
2010
Quarter 4
72
83
95
83.33
Overall Average
83.67
Exam Grades in Prof Matta’s PA class
during the past 2 years with Averages and
Ranges:
Year
Quarter
Exam 1
Exam 2
Exam 3
Average
Range
2009
Quarter 1
85
82
90
85.67
8.00
2009
Quarter 2
73
67
77
72.33
10.00
2009
Quarter 3
85
85
85
85.00
0.00
2009
Quarter 4
90
73
83
82.00
17.00
2010
Quarter 1
70
83
98
83.67
28.00
2010
Quarter 2
89
81
83
84.33
8.00
2010
Quarter 3
95
93
91
93.00
4.00
2010
Quarter 4
72
83
95
83.33
23.00
83.67
12.25
Overall Average
For Prof. Matta’s Case:
The sample size = 3 (3 exams/quarter)
A2
1.02
D3
0
D4
2.57
X-Bar
83.67
R-Bar
12.25
UCLx = X-Bar + A2 * R-Bar
96.16
LCLx = X-Bar - A2 * R-Bar
71.17
UCLr = D4* R-Bar
31.48
LCLr = D3 * R-Bar
0.00
Control Charts for Prof Matta’s Exams:
120.00
100.00
80.00
Avergae
Mean x
60.00
UCLx
40.00
LCLx
20.00
0.00
1
2
3
4
5
6
7
8
9
35.00
30.00
25.00
Range
20.00
Mean r
15.00
UCLr
LCLr
10.00
5.00
0.00
1
2
3
4
5
6
7
8
9
Control Charts Example:

At Quikie Car Wash, the wash process is
advertised to take less than 7 minutes.
Consequently, management has set a
target average of 390 seconds for the
wash process. Suppose that the average
range for a sample of 9 cars is 10 seconds.
Establish the means and ranges control
limits using this data.
Solved Example
390 sec, n = 9, R= 10 sec
From Table 5.1 in your book,
 A2 = 0.337, D3 = 0.184, D4 = 1.816
 UCLR= D4 R= 1.816(10 sec) = 18.16 sec
 LCLR= D3 R= 0.184(10 sec) = 1.84 sec
 UCLx = x + A2 R= 390 sec + 0.337(10 sec) = 393.37 sec
 LCLx = x - A2 R = 390 sec – 0.337(10 sec) = 386.63 sec

X=
Marlin Bottling Company Problem 5
in the book chapter 2
Marlin Company
Sample
1
2
3
4
X-BAR
R
1
0.60
0.61
0.59
0.60
0.60
0.02
2
0.60
0.60
0.61
0.60
0.60
0.01
3
0.58
0.57
0.59
0.59
0.58
0.02
4
0.62
0.61
0.60
0.59
0.60
0.03
5
0.59
0.61
0.61
0.60
0.60
0.02
6
0.59
0.58
0.62
0.58
0.59
0.04
Marlin Co. Bottling – In class
Calculations
Size of
Sample
(n)
Factor for UCL
and LCL for
x-Charts
(A2)
Factor for
LCL for
R-Charts
(D3)
2
3
4
5
6
1.880
1.023
0.729
0.577
0.483
0
0
0
0
0
Factor
UCL for
R-Charts
(D4)
3.267
2.575
2.282
2.115
2.00
Control Charts
for Attributes

p-chart: A chart used for controlling the
proportion of defective services or
products generated by the process.
p =
p(1 – p)/n
Where
n = sample size
p = central line on the chart, which can be either the historical
average population proportion defective or a target value.
–  and LCL = p−z
– 
Control limits are: UCLp = p+z
p
p
p
z = normal deviate (number of standard deviations from the average)
Hometown Bank
Example
The operations manager of the booking services department of Hometown Bank
is concerned about the number of wrong customer account numbers recorded by
Hometown personnel.
Each week a random sample of 2,500 deposits is taken, and the number of
incorrect account numbers is recorded. The results for the past 12 weeks are
shown in the following table.
Is the booking process out of statistical control? Use three-sigma control limits.
Hometown Bank
Using a p-Chart to monitor a process
n = 2500
p=
p =
p =
147
= 0.0049
12(2500)
p(1 – p)/n
0.0049(1 – 0.0049)/2500
p = 0.0014
UCLp = 0.0049 + 3(0.0014)
= 0.0091
LCLp = 0.0049 – 3(0.0014)
= 0.0007
Sample
Number
Wrong
Account #
Proportion
Defective
1
2
3
4
5
6
7
8
9
10
11
12
15
12
19
2
19
4
24
7
10
17
15
3
0.006
0.0048
0.0076
0.0008
0.0076
0.0016
0.0096
0.0028
0.004
0.0068
0.006
0.0012
Total
147
Hometown Bank
Using a p-Chart to monitor a process
Example
In class Problem
In Class Problem
Control Charts
Two types of error are possible with
control charts
 A type I error occurs when a process is
thought to be out of control when in fact
it is not
 A type II error occurs when a process is
thought to be in control when it is
actually out of statistical control
These errors can be controlled by the
choice of control limits