Process Variability and
Capability
Operations Management
Dr. Ron Lembke
Designed Size
10
11
12
13
14
15
16
17
18
19
20
Natural Variation
14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4
Theoretical Basis of
Control Charts

1
N
N
 x   
i 1
2
i
95.5 % of values
within 2σ
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-6
-4
-2
X 
0
2
4
6
Theoretical Basis of Control Charts
99.7 % of values
within 3σ
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-6
-4
-2
X 
0
2
4
6
16
14
12
10
8
6
4
2
0
Skewness


Lack of symmetry
Pearson’s coefficient of
skewness: 3( x  Median )
Positive Skew > 0
s
16
14
12
10
8
6
4
2
0
Skewness = 0
Negative Skew < 0
16
14
12
10
8
6
4
2
0
0.45
Kurtosis
0.4
0.35
0.3
0.25

0.2
Amount of peakedness
or flatness

 (x  x)
ns
Kurtosis < 0
0.15
0.1
0.05
0
-6
4
-4
-2
0
2
Kurtosis = 0
4
Kurtosis > 0
4
6
Heteroskedasticity
Sub-groups
with different
variances
Design Tolerances

Design tolerance:





Determined by users’ needs
UTL -- Upper Tolerance Limit
LTL -- Lower Tolerance Limit
Eg: specified size +/- 0.005 inches
No connection between tolerance and 

completely unrelated to natural variation.
Process Capability and 6
LTL
UTL
3


LTL
UTL
6
A “capable” process has UTL and LTL 3 or more
standard deviations away from the mean, or 3σ.
99.7% (or more) of product is acceptable to
customers
Process Capability
Capable
Not Capable
LTL
UTL
LTL
UTL
LTL
UTL
LTL
UTL
Process Capability



Specs: 1.5 +/- 0.01
Mean: 1.505 Std. Dev. = 0.002
Are we in trouble?
Process Capability

Specs: 1.5 +/- 0.01



LTL = 1.5 – 0.01 = 1.49
UTL = 1.5 + 0.01 = 1.51
Mean: 1.505 Std. Dev. = 0.002


LCL = 1.505 - 3*0.002 = 1.499
UCL = 1.505 + 0.006 = 1.511
Process
Specs
1.49
1.499
1.50
1.511
1.51
Capability Index



Capability Index (Cpk) will tell the position of
the control limits relative to the design
specifications.
Cpk ≥ 1.0, process is capable
Cpk < 1.0, process is not capable
Process Capability, Cpk

Tells how well parts
produced fit into specs
C pk
 X  LTL UTL  X 
 min 
or

3 
 3
Process
Specs
LTL
3
X
3
UTL
Process Capability

Tells how well parts produced fit into specs
C pk



 X  LTL UTL  X 
 min 
or

3 
 3
For our example:
1.505  1.49 1.51  1.505 
C pk  min 
or
0.006 
 0.006
Cpk= min[ 0.015/0.006, 0.005/0.006]
Cpk= min[2.5,0.833] = 0.833 < 1 Process not capable
Process Capability: Re-centered


If process were properly centered
Specs: 1.5 +/- 0.01



LTL = 1.5 – 0.01 = 1.49
UTL = 1.5 + 0.01 = 1.51
Mean: 1.5 Std. Dev. = 0.002


LCL = 1.5 - 3*0.002 = 1.494
UCL = 1.5 + 0.006 = 1.506
Process
Specs
1.49
1.494
1.506 1.51
If re-centered, it would be Capable
C pk
C pk
1.5  1.49 1.51  1.5 
 min 
,

0.006 
 0.006
 0.01 0.01 
 min 
,
  1.67
 0.006 0.006 
Since 1.67 > 1, process would now be Capable
Process
Specs
1.49
1.494
1.506 1.51
What if Not Capable?

We can’t sell everything we
make to these people


LTL
UTL

Shifting the Mean




UTL
Not trivial, may be easy
Adjust a setting?
Reducing variability – Hard!

LTL
Find less choosy customers?
Measure & sort all for them?

Process varies over time
Find and get rid of variability
Training workers, new equip?
Summary

All processes exhibit variability



Reviewed some basic statistical concepts
Defined process capability measurement, Cpk
Difficulties of making a process capable