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Process Capability and
Statistical Process Control
1.
2.
3.
Explain what statistical quality control is.
Calculate the capability of a process.
Understand how processes are monitored
with control charts for both attribute and
variable data




How many paint defects are there in the
finish of a car?
How long does it take to execute market
orders?
How well are we able to maintain the
dimensional tolerance on our ball bearing
assembly?
How long do customers wait to be served
from our drive-through window?
LO 1

Assignable variation: caused by factors that
can be clearly identified and possibly
managed
◦ Example: a poorly trained employee that creates
variation in finished product output

Common variation: variation that is inherent
in the production process
◦ Example: a molding process that always leaves
“burrs” or flaws on a molded item
LO 1


When variation is reduced, quality is
improved
However, it is impossible to have zero
variation
◦ Engineers assign acceptable limits for variation
◦ The limits are know as the upper and lower
specification limits
 Also know as upper and lower tolerance limits
LO 1


LO 1
Traditional view is that quality within the
range is good and that the cost of quality
outside this range is constant
Taguchi views costs as increasing as
variability increases, so seek to achieve
zero defects and that will truly minimize
quality costs


Taguchi argues that tolerance is not a yes/no
decision, but a continuous function
Other experts argue that the process should
be so good the probability of generating a
defect should be very low
LO 2

Process limits

Specification limits

How do the limits relate to one another?
LO 2
LO 2


Capability index (Cpk) shows how well parts
being produced fit into design limit
specifications
 X  LTL UTL - X 

Cpk = min 
or

3

3



Also useful to calculate probabilities
Z LTL 
LO 2
LTL  X

ZUTL 
UTL  X


Data
◦ Designed for an average of 60 psi
 Lower limit of 55 psi, upper limit of 65 psi
◦ Sample mean of 61 psi, standard deviation of 2
psi

Calculate Cpk
C pk
 x  LSL USL  x 
 min 
,

3 
 3
 61  55 65  61
 min 
,





3
2
3
2


 min 1, 0.6667  0.6667
LO 2
Less than 55 psi
X X
55  61
Z

 3

2
P( Z  3)  0.00135
More than 65 psi
X X
65  61
Z

2

2
P( Z  2)  0.02275
LO 2
P( Z  3 or Z  2)  0.00135  0.02275  0.02410





LO 2
We are the maker of this cereal. Consumer
Reports has just published an article that shows
that we frequently have less than 15 ounces of
cereal in a box.
Let’s assume that the government says that we
must be within ± 5 percent of the weight
advertised on the box.
Upper Tolerance Limit = 16 + .05(16) = 16.8
ounces
Lower Tolerance Limit = 16 – .05(16) = 15.2
ounces
We go out and buy 1,000 boxes of cereal and find
that they weight an average of 15.875 ounces with
a standard deviation of .529 ounces.

Specification or
Tolerance Limits
◦ Upper Spec = 16.8 oz
◦ Lower Spec = 15.2 oz C pk

Observed Weight
◦ Mean = 15.875 oz
◦ Std Dev = .529 oz
 X  LTL UTL  X 
 Min 
;

3 
 3
15.875  15.2 16.8  15.875 
C pk  Min 
;

3(.529) 
 3(.529)
C pk  Min.4253; .5829
C pk  .4253
LO 2
An index that shows how well the
units being produced fit within the
specification limits.
 This is a process that will produce
a relatively high number of
defects.
 Many companies look for a Cpk of
1.3 or better… 6-Sigma company
wants 2.0!

LO 2

Attribute (Go or no-go information)
◦ Defectives refers to the acceptability of product across
a range of characteristics.
◦ Defects refers to the number of defects per unit which
may be higher than the number of defectives.
◦ p-chart application

Variable (Continuous)
◦ Usually measured by the mean and the standard
deviation.
◦ X-bar and R chart applications
LO 3
Statistical
Process Normal Behavior
Control
(SPC) Charts
UCL
LCL
1
2
3
4
5
6
UCL
Samples
over
time
Possible problem, investigate
LCL
1
2
3
4
5
6
UCL
Samples
over
time
Possible problem, investigate
LCL
1
LO 3
6
2
3
4
5
Samples
over
time
x
m
-3
-2
-1
Standard
deviation
units or “z”
units.
LO 3
0
1
2
3
z
We establish the Upper Control Limits (UCL)
and the Lower Control Limits (LCL) with plus
or minus 3 standard deviations from some xbar or mean value. Based on this we can
expect 99.73% of our sample observations to
fall within these limits.
99.73%
LCL
LO 3
UCL
x


Created for good/bad attributes
Use simple statistics to create the
control limits
Total number of defects from all samples
p
Number of samples  Sample size
sp 

p 1 p
n
UCL  p  zs p
LCL  p  zs p
LO 3

LO 3
1 – 2- 5- 7 Rule




1 point above UCL or 1 point below LCL
2 consecutive points near the UCL or 2
consecutive points near the LCL
5 consecutive decreasing points or 5
consecutive increasing points
7 consecutive points above the center line or
7 consecutive points below the center line
LO 3
Total number of defects from all samples
91
p

 0.03033
Number of samples x Sample size
3,000
sp 


p 1 p
0.030331  0.03033

 0.00990
n
300
UCL  p  3s p  0.03033  30.00990  0.06003
LCL  p  3s p  0.03033  30.00990  0.00063
LO 3
LO 3
In variable sampling, we measure actual
values rather than sampling attributes
Generally want small sample size


1. Quicker
2. Cheaper
Samples of 4-5 are typical
Want 25 or so samples to set up chart


LO 3
UCL X  X  zs X
LCLX  X  zs X
where
s  s
X
 Standard deviation of sample means
n
s  Standard deviation of the process distributi on
n  Sample size
X  Average of sample means or a target va lue set for the process
z  Number of standard deviations for a specific confidence level
LO 3
X Chart
UCL X  X  A2 R
LCL X  X  A2 R
R Chart
UCL R  D4 R
LCL R  D3 R
LO 3
LO 3
LO 3
LO 3