MD 021 - Management and Operations
Statistical Process Control
Outline

Definition of statistical process control (SPC)

Process variation

Characteristics of control charts

Types of control charts

Choosing an SPC method
1
Definition of Statistical Process Control
Statistical process control (SPC) is the application of statistical techniques to
determine whether the output of a process conforms to the product or service
design.
SPC is implemented via control charts that are used to monitor the output of the
process and indicate the presence of problems requiring further action.
2
Process Variation
All processes exhibit variation. For example, a machine that fills cereal boxes will not
put exactly the same amount in each box.
If we were to collect data over time on the amount of cereal in each box, a plot of the
data would be described as a distribution.
A distribution is characterized by its:
 Mean
Cereal Production
 Spread
 Range
 Standard deviation
700
No. of boxes
600
 Shape
 Symmetric
 Skewed
500
400
300
200
100
0
9.7 9.75 9.8 9.85 9.9 9.95 10 10.1 10.1 10.2 10.2 10.3 10.3
Average weight (oz.)
3
Quality Measurements
Control charts can be used to monitor processes where output is measured as
either variables or attributes.
Variables measures - Characteristics of a product or service that can be
measured on a continuous scale. Examples include length, width, and time.
Attributes measures - Characteristics of a product or service that can be
quickly counted for acceptable quality. Examples include the number of defects
in a product or service.
4
Sources of Process Variation
Sources of process variation can be categorized as:
 Common causes - Random, or unavoidable, sources of variation within a
process. A process with only common causes of variation is stable (i.e. the
mean and spread do not change over time). Such a process is said to be “in a
state of statistical control” or “in-control”.
 Assignable causes - Any cause of variation that can be identified and
eliminated, originating from outside the normal process
5
Characteristics of Control Charts
A control chart is a time-ordered diagram to monitor a quality characteristic,
consisting of:
 A nominal value, or center line – The average of several past samples
 Two control limits used to judge whether action is required, an upper
control limit (UCL) and a lower control limit (LCL)
 Data points, each consisting of the average measurement calculated from a
sample taken from the process, ordered over time. By the Central Limit
Theorem, regardless of the distribution of the underlying individual
measurements, the distribution of the sample means will follow a
normal distribution. The control limits are set based on the sampling
distribution of the quality measurement.
6
Purpose of Control Charts
Control charts are intended to reflect only common causes of variation in order to detect
assignable causes of variation.
Question: If, at the time we are constructing a control chart, there are assignable causes
of variation in the process, how can we construct a meaningful control chart?
Answer: By carefully choosing our sample size so that only common causes are found
within a sample.
The control limits are set to:
 Usually detect when the process has gone out of control (narrow control limits
work better)
 Usually not overreact to random variation (wider control limits work better)
The control limits are set to strike a balance between these competing priorities.
7
Control Charts for Variables
Standard Deviation of the Process,  , Known
Control charts for variables (with the standard deviation of the process,  , known ) monitor the
mean, X , of the process distribution.
The control limits are:
UCL  X  z X
LCL  X  z X
where: X = center line of the chart and the average of several past sample means, z is the
standard normal deviate (number of standard deviations from the average),  X   / n and is
the standard deviation of the distribution of sample means, and n is the sample size
8
2. An automatic filling machine is used to fill 1-liter bottles of cola. The machine’s output is
approximately normal with a mean of 1.0 liter and a standard deviation of 0.01 liter. Output is
monitored using means of samples of 25 observations.
a. Determine upper and lower control limits that will include roughly 97 percent of the sample means
when the process is in control.
x
= 1.0 liter,  = .01 liter, n = 25
a. Control limits : x  2

n
,
[z = 2.17 for 97%]
UCL is 1.0  2.17
.01
 1.0043
25
LCL is 1.0  2.17
.01
 .9957
25
b. Given the sample means: 1.005, 1.001, 0.998, 1.002, 0.995 and 0.999, is the process in control?
1.006
b.
1.0043
out
UCL
*
1.002

1.000
(liters)
Mean
.998

.9957
.994


LCL
*
9
out
Control Charts for Variables
Standard Deviation of the Process,  , Unknown
Control charts for variables monitor the mean ( X chart) and variability (R chart) of the process
distribution.
R chart: To calculate the range of the data, subtract the smallest from the largest measurement in
the sample.
The control limits are:
UCLR  D4 R and LCLR  D3 R
where: R = average of several past R values and is the central line of the control chart, and
D3 , D4 = constants that provide three standard deviation (three-sigma) limits for a given sample
size
X chart: The control limits are:
UCL X  X  A2 R and LCL X  X  A2 R
where: X = central line of the chart and the average of past sample means, and A2 = constant to
provide three sigma limits for the process mean.
10
Control Chart for Variables Example
X , R Charts
Webster Chemical Company produces mastics and caulking for the
construction industry. The product is blended in large mixers and
then pumped into tubes and capped. Webster is concerned whether
the filling process for tubes of caulking is in statistical control.
The process should be centered on 8 ounces per tube. Several
samples of eight tubes are taken and each tube is weighed in
ounces.
Sample
1
2
3
4
5
6
1
7.98
8.23
7.89
8.24
7.87
8.13
2
8.34
8.12
7.77
8.18
8.13
8.14
3
8.02
7.98
7.91
7.83
7.92
8.11
Tube number
4
5
7.94 8.44
8.41 8.31
8.04 8.00
8.05 7.90
7.99 8.10
8.13 8.14
6
7.68
8.18
7.89
8.16
7.81
8.12
7
7.81
7.99
7.93
7.97
8.14
8.13
8
8.11
8.06
8.09
8.07
7.88
8.14
a) Assuming that taking only 6 samples is sufficient, use the data
in the table to construct three-sigma R-chart and X -chart
control limits. Is the process in statistical control?
b) The process variability for the first and sixth samples appear to
be out of control. Webster looks for assignable causes and
quickly notes that the weighing scale was gummed up with
caulking. Apparently, a tube was not properly capped. The
sticky scale did not correctly read the variation in weights for
the sixth sample.
Delete that data and recalculate R , UCLR , and LCLR . Is the
process in statistical control?
11
Solution to Control Chart for Variables Example
a)
Sample
1
2
3
4
5
6
1
7.98
8.23
7.89
8.24
7.87
8.13
2
8.34
8.12
7.77
8.18
8.13
8.14
3
8.02
7.98
7.91
7.83
7.92
8.11
Tube number
4
5
7.94 8.44
8.41 8.31
8.04 8.00
8.05 7.90
7.99 8.10
8.13 8.14
6
7.68
8.18
7.89
8.16
7.81
8.12
7
7.81
7.99
7.93
7.97
8.14
8.13
8
8.11
8.06
8.09
8.07
7.88
8.14
Avg.
8.040
8.160
7.940
8.050
7.980
8.130
8.050
Range
0.76
0.43
0.32
0.41
0.33
0.03
0.38
X = 8.050, R = 0.38, n = 8
From Table 7.1:
UCLR  D4 R  1.864 (0.38)  0.708
LCLR  D3 R  0.136 (0.38)  0.052
UCLX  X  A2 R  8.050  0.373(0.38)  8.192
LCLX  X  A2 R  8.050  0.373(0.38)  7.908
b) We delete the sixth observation and recalculate the control limits. The
ranges, including the range for the first sample are now all within the
revised control limits, and the process average for the second sample
now falls just inside of the revised control limits.
(0.76  0.43  0.32  0.41  0.33)
 0.45
5
8.040  8.160  7.940  8.050  7.980
X
 8.034
5
UCLR  D4 R  1.864 (0.45)  0.839
R
LCLR  D3 R  0.136 (0.45)  0.061
UCLX  X  A2 R  8.034  0.373(0.45)  8.202
LCLX  X  A2 R  8.034  0.373(0.45)  7.866
12
Control Charts for Attributes
p-Chart
A p-chart is a commonly used control chart for attributes, whereby the quality
characteristic is counted, rather than measured, and the entire item or service can
be declared good or defective.
The standard deviation of the proportion defective, p, is:  p 
p(1  p) / n ,
where n = sample size, and p = average of several past p values and central line
on the chart.
Using the normal approximation to the binomial distribution, which is the actual
distribution of p,
UCL p  p  z p and LCL p  p  z p
where z is the normal deviate (number of standard deviations from the average).
13
Control Chart for Attributes Example
p-Chart
A sticky scale brings Webster’s attention to whether caulking tubes
are being properly capped. If a significant proportion of the tubes
aren’t being sealed, Webster is placing their customers in a messy
situation. Tubes are packaged in large boxes of 144. Several
boxes are inspected and the following number of leaking tubes are
found:
Sample
Tubes
Sample
Tubes
Sample
Tubes
1
3
8
6
15
5
2
5
9
4
16
0
3
3
10
9
17
2
4
4
11
2
18
6
5
2
12
6
19
2
6
4
13
5
20
1
7
2
14
1
Total
72
Calculate p-chart three-sigma control limits to assess whether the
capping process is in statistical control.
Solution: n = 144, p 
72
 0.025
20(144)
p(1  p)
0.025(1  0.025)

 0.013
n
144
UCL p  p  z p  0.025  3(0.013)  0.064
p 
LCL p  p  z p  0.025  3(0.013)  0.014 (adjusted to zero)
The highest proportion of defectives occurs in sample #10, but is
still within control limits: p10  9 / 144  0.0625 . Therefore, the
process is in statistical control.
14
Control Charts for Attributes
c-Chart
A c-chart is another type of control chart for attributes, whereby the quality
characteristic is counted as the number of defects/ unit.
Using the normal approximation to the Poisson distribution, which is the actual
distribution of c,
UCLc  c  z c and LCLc  c  z c
where c is the average number of defects/unit and the center line of the c-chart.
15
Control Chart for Attributes Example
c-Chart
At Webster Chemical, lumps in the caulking compound could
cause difficulties in dispensing a smooth bead from the tube.
Testing for the presence of lumps destroys the product, so Webster
takes random samples. The following are the results of the study:
Tube #
1
2
3
4
Lumps
6
5
0
4
Tube #
5
6
7
8
Lumps
6
4
1
6
Tube #
9
10
11
12
Lumps
5
0
9
2
Determine the c-chart two-sigma upper and lower control limits for
this process.
Solution:
c
( 6  5  0  4  6  4  1  6  5  0  9  2)
4
12
c  c  4  2
UCLc  c  z c  4  (2)(2)  8
LCLc  c  z c  4  (2)(2)  0
The eleventh tube has too many lumps (9), so the process is
probably out of control.
16
Process Capability Exercise
Webster Chemical’s nominal weight for filling tubes of caulk is
8.00 ounces  0.60 ounces. The target process capability ratio is
1.33. The current distribution of the filling process is centered on
8.054 ounces with a standard deviation of 0.192 ounces. Compute
the process capability index to assess whether the filling process is
capable and set properly.
Solution:
Process capability ratio:
Cp 
Upper specification - Lower specification 8.6  7.4

 10417
.
6
6(0192
. )
Process capability index:
C pk  miminum of
X - Lower specification Upper specification - X
,
3
3
8.054  7.400
8.600 - 8.054
 1135
. ,
 0.948
3(0192
. )
3(0192
. )
C pk  0.948
The process is not capable of consistently meeting specifications
according to the minimum capability level set by Webster.
17
Supplementary Material:
1. Calculating P
xi
pi 
n
pi = proportion of defective products
x i = number of the defected products
n = sample size
m = number of samples
m
x
x1 x2
xi
  ... m

( p1  p 2  ...  p m ) n n
(
x

x

...

x
)
n  1 2
m
p

 i 1
m
m
mn
mn
total number of defected products
p
total number of products
18