034_Process Capability (Non Normal)

1
Process Capability
(Non Normal)

4
• However, many processes do not produce normal data
• Non-normal Capability Analysis allows us to analyze
processes that produce non-normal data
Still need to know if the process is meeting customer
expectations!
Non-normal Capability Analysis
How does the process compare to customer expectations?
• Recall that the calculation of Z
scores and other capability
indices requires normal data
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7
Multiple Modes
Common causes of multiple modes
Mixtures of distributions – Operators, shifts,
materials, machines, etc.
6.55.54.53.52.5
9
8
7
6
5
4
3
2
1
0
BiModal
Frequency
1098765
40
30
20
10
0
BiModal2
Frequency
More than one distribution with different means

10
Probability Plot Non-Normal Data
Skewed distributions are parabolic in nature. The left and
right skews look similar but are inverted.
P-Value: 0.000
A-Squared: 2.593
Anderson-Darling Normality Test
N: 100
StDev: 0.437554
Average: 2.54648
321
.999
.99
.95
.80
.50
.20
.05
.01
.001
Probability
Lskew
Normal Probability Plot
P-Value: 0.007
A-Squared: 1.078
N: 50
StDev: 1.47328
Average: 6.20268
1110987654
.999
.99
.95
.80
.50
.20
.05
.01
.001
Probability RSkew

13
Leptokurtic and Platykurtic distributions follow an S
pattern. Leptokurtic has extreme points and the
Platykurtic is concentrated at the center with short tails.
Probability Plot Non-Normal Data
P-Value: 0.000
A-Squared: 12.225
N: 259
StDev: 1.49859
Average: 7.61920
1272
.999
.99
.95
.80
.50
.20
.05
.01
.001
Probability
Lepto
P-Value: 0.001
A-Squared: 1.528
N: 150
StDev: 1.53592
Average: 5.98992
9876543
.999
.99
.95
.80
.50
.20
.05
.01
.001
Probability
Platy

16
Accounting For Non-normality
• There are two methods that we can use to account for
non-normal data when conducting capability studies:
This method will not provide “within” estimates of process
capability
2. Use Minitab’s Non-Normal
Capability Analysis
Feature:
Requires you to select a
distribution that fits
your data type

19
Box-Cox Transformation
The Box-Cox transform is a search procedure in Minitab that
evaluates various transforms and identifies the optimum
transform (powers between -5 and 5 are evaluated)
•Stat>Control Charts> Box-Cox Transformation
1. Enter Raw Data Column
2. Enter the subgroup size
used during data collection
3. Select Options
4. Enter Column
to be used to store
transformed data

22
Skewed Example (Cont.)
Before the transform After the transform
A capability analysis can now be performed. The specification
limits will also need to be transformed
C18
Percent
0.2750.2500.2250.2000.1750.1500.1250.100
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.937
0.1693
StDev 0.03606
N 50
AD 0.165
P-Value
Probability Plot of C18
Normal
Transformed
data is normal
RSkew
Percent
111098765432
99
95
90
80
70
60
50
40
30
20
10
5
1
Mean
0.007
6.203
StDev 1.473
N 50
AD 1.078
P-Value
Probability Plot of RSkew
Normal

25
The Capability in ‘Z’

28
Identify the Distribution

31
Graphical output
RSkew
Percent
12963
99
90
50
10
1
RSkew
Percent
105
99
90
50
10
1
RSkew - T hreshold
Percent
101
99
90
50
10
1
RSkew
Percent
100.010.01.00.1
99.9
90
50
10
1
3-Parameter Lognormal
A D = 0.176
P-V alue = *
Exponential
A D = 14.208
P-V alue < 0.003
Goodness of Fit Test
Normal
A D = 1.078
P-V alue = 0.007
Lognormal
A D = 0.362
P-V alue = 0.430
Probability Plot for RSkew
Normal - 95% C I Lognormal - 95% C I
3-Parameter Lognormal - 95% C I Exponential - 95% C I

34
What about Transformation

37
Multiple Modes

40
Data in the middle of the distribution is sparse, where we
would expect it to be concentrated
Multiple Modes (Cont.)
BiModal
Percent
876543210
99.9
99
95
90
80
70
60
50
40
30
20
10
5
1
0.1
Mean
<0.005
4.199
StDev 1.077
N 60
AD 2.870
P-Value
Probability Plot of BiModal
Normal

43
Multiple Modes (Cont.)
Un-stack data and analyze it separately. Un-stacking data
splits data into two separate distributions:
Data> Unstack Columns
Location of original
data
IDVar generated when
brushing data
Create new worksheet
or add data after last
column of current
worksheet
Check the box to
name the unstacked
columns
Unstack the data into two new columns

46
Capability Analysis Multiple Modes
(Cont.)
Combining the data requires that the proportion of data in
each distribution be accounted for in the overall total. The
general equation for a Bimodal distribution is as follows:
)(PPM
N
n
)(PPM
N
n
PPM 2
2
1
1
Total +=
For our Example:
(29564)
60
29
(252503)
60
31
PPMTotal +=
144,749PPMTotal =
Expected defect rate is 144,749 ppm

49
Accounting for Non-Normality
Granularity
1. Find a better measurement system with better
resolution
2. Take an average of multiple readings (subgroup
samples)
Granularity is typically caused by measurement system
resolution. To combat granularity:

Tools
None
52

1
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034_Process Capability (Non Normal)