Six Sigma Confidence Interval Analysis (CIA) Training Module

1 April 9, 2016 – v1.0
Six Sigma Confidence Interval Analysis
by Operational Excellence Consulting LLC

2 April 9, 2016 – v1.0
Why should we use Intervals ?
Sometimes these questions tend to be of two types:
“How has something been performing in the past ?”
and
“How will something perform in the future ?”
It is important to recognise that the manner in which we collect and analyse data dictates
the quality of our results.
Unfortunately, it is often assumed that the estimated parameter based on a sample is the
“true” value. This is not accurate !!!
Therefore, the natural question to ask has to be:
“How good is the estimation one obtains from a sample ?”
To answer this question, one must turn to the method of interval estimation. This
technique extend a estimation of a product/process parameter to a so-called confidence
interval of values that contain the “true” parameter with a pre-defined confidence or
probability.

3 April 9, 2016 – v1.0
Data-Based Decision Making
1. Plan:
What do you want to know ?
How do you want to see what it is that you
need to know ?
What type of tool will generate what it is
that you need to see ?
What type of data is required to the
selected tool ?
Where can you get the required type of
data ?
2. Collect Data
3. Analyse Data
4. Interpret Data

4 April 9, 2016 – v1.0
Introduction to Confidence Interval Analysis
This module will help improve decision making and communication
when data are used.
The opportunity for improvement exists because:
• Sometimes we don’t recognise that there is uncertainty in our
data.
• Sometimes we want to report an appropriate interval but the
calculation methods are cumbersome or confusing.
• Sometimes we attempt to make statements about our
uncertainty but do so improperly [like reporting a (± 3 std.dev.)
or ± (max. - min.)/2 interval when some other type of interval
is more appropriate].

5 April 9, 2016 – v1.0
Definition of a Confidence Interval
A 90/95/99% confidence interval for a population parameter (e.g.
mean/average, standard deviation, cp, cpk, defect rate, etc.) is an interval around
the estimated parameter that has a 90/95/99% chance of capturing the true
value of the parameter. The value  is called the risk level of the decision. With
a confidence of 90/95/99% we have a 10/5/1% risk which means
.
5.0%
2.5%
0.5%
5.0%
2.5%
0.5% 99%
95%
90%
xestimation

6 April 9, 2016 – v1.0
Assumptions for Confidence Interval Analysis
There are three fundamental assumptions to all of the following
intervals:
• Control
• The data comes from a single, stable distribution. This assumption should be
verified by plotting the data in time sequence (trend or control chart). Being
in control is important from two standpoints:
• The descriptive statistics lose meaning if the process is significantly out
of control.
• A state of control provides the ability to predict future process
behaviour.

7 April 9, 2016 – v1.0
• Randomness
• The sample is a random sample. The question of a sample being
random should be addressed before the data is collected.
• Normality
• The assumption of fair normality is most important when dealing
with confidence intervals for standard deviations or Process
Capability Indices as cp and cpk. It can be assessed through the use
of histograms (as evidenced by a symmetric, “bell-shaped” curve) or
some statistical methods like the Kolmogorov-Smirnov test or the
Normality Plot.
Assumptions for Confidence Interval Analysis

8 April 9, 2016 – v1.0
We can take a sample in production and estimate the
mean value or average of a product or process, but what is
the “real” mean value ?
Can we say that there is evidence that the estimated mean
value is different from a pre-defined target value T ?
Confidence Interval for a Mean Value

9 April 9, 2016 – v1.0
03.7
02.57
10
1
1
1



estim
estim
s
x
n
Sample 1.
84.9
12.61
50
2
2
2



estim
estim
s
x
n
Sample 2.
Confidence Interval for a Mean Value
Which one is the real mean value ?
What can we conclude about the real mean value based
on the estimated mean value ?
Is the sample size sufficiently large ?

10 April 9, 2016 – v1.0
12.61
50
00.58...33.7300.72
...21





n
xxx
x n
n= 50
84.9
150
)12.6158(...)12.6133.73()12.6172(
1
)(...)()(
222
22
2
2
1







n
xxxxxx
s n
72.00
73.33
65.33
58.00
59.00
88.83
73.67
62.50
51.17
55.67
52.33
55.83
55.00
77.17
53.33
58.00
Confidence Interval for a Mean Value - Example
m
0.0
4.8
9.6
14.4
19.2
24.0
0.00
6.80
13.60
20.40
27.20
34.00C
E
L
L
F
R
E
Q
U
E
N
C
Y
P
E
R
C
E
N
T
CELL BOUNDARY
44.8 53.6 62.4 71.2 80.0 88.8
Fitted curve is a Normal. K-S test: 0.326. Lack of fit is not significant.

11 April 9, 2016 – v1.0
Confidence Interval for a Mean or Average
Number of Observations 50
Mean or Average 61.12
Standard Deviation 9.840
Lower limit Average Upper limit ± D t(/2)
Confidence of 90%: 58.79 61.12 63.45 2.33 1.677
Confidence of 95%: 58.32 61.12 63.92 2.80 2.010
Confidence of 99%: 57.39 61.12 64.85 3.73 2.680
OPERATIONAL EXCELLENCE
C O N S U L T I N G
Excel-Calculator for a Mean Value
Input Data

12 April 9, 2016 – v1.0
Can we decide with a confidence of 95(99)% (means a 5(1)% risk)
that the true mean value of the process is better than 58.50% ?
Conclusion: There is a 5 % risk that the true process mean value
is lower than 58.79%. However, there is a 95% confidence that the
true process mean value is greater than 58.79% and so of course
also greater than the target value 58.50%.
57.39
58.32
58.79 63.45
63.92
64.8561.12
Confidence Interval for a Mean Value - Example
85.6439.57:%99
92.6332.58:%95
45.6379.58:%90



true
true
true
x
x
x
58.50
57.39 64.8557.39 64.8557.39

13 April 9, 2016 – v1.0
n
s
txx
n
s
tx estimtrueestim 
22

where is a constant that depends on the risk level  and the
number of observations n in the sample, it can be easily calculated
using the Excel-function “TINV” as =TINV(;n-1)t

t 
Confidence Interval for the Mean Value - Theory
estimxestimx
Small variation higherconfidence Large variation lowerconfidence

14 April 9, 2016 – v1.0
Can we say that two samples are not taken from the
same process ?
Does evidence for a significant difference exist between
two processes (e.g. current/better), two test groups, or
two persons performing the same work/measurements ?
Comparing two Mean Values

15 April 9, 2016 – v1.0
03.7
02.57
10
1
1
1



s
x
n
estim
Sample 1.
84.9
12.61
50
2
2
2



s
x
n
estim
Sample 2.
or
03.7
02.57
10
1
1
1



s
x
n
estim
Sample 1.
84.9
12.61
50
2
2
2



s
x
n
estim
Sample 2.
?
Comparing two Mean Values
or
?
or
?

16 April 9, 2016 – v1.0
(for equal standard deviations)
Sample 1:
Number of Observations 10 Confidence of 90%: 52.94 61.10
Mean or Average 57.02 Confidence of 95%: 51.99 62.05
Standard Deviation 7.030 Confidence of 99%: 49.80 64.24
Sample 2:
Lower limit Difference Upper limit ± D
Confidence of 90%: -1.38 4.10 9.58 5.48
Confidence of 95%: -2.46 4.10 10.66 6.56
Confidence of 99%: -4.63 4.10 12.83 8.73
Pooled Std. Dev.= 9.46
Comparing two MeansOPERATIONAL EXCELLENCE
C O N S U L T I N G
Excel-Calculator for two Mean Values with s1 = s2
Input Data
Input Data

17 April 9, 2016 – v1.0
Process 1
03.7
02.57
10
1
1
1



s
x
n
estim
95% confidence interval:
Process 2
84.9
12.61
50
2
2
2



s
x
n
estim
95% confidence interval:
05.6299.51  truex 92.6332.58  truex
50 55 60 65
Is there evidence for a significant process improvement ?
Comparing two Mean Values - Example

18 April 9, 2016 – v1.0
84.9
12.61
50
2
2
2



s
x
n
estim
03.7
02.57
10
1
1
1



s
x
n
estim
63.483.12:%99
46.266.10:%95
38.158.9:%90
D
D
D
true
true
true
Conclusion: Because zero is inside all confidence
intervals we have not enough evidence to say that
there is a significant difference between the two
processes.
Confidence intervals for the difference of two means:
Comparing two Mean Values with s1 = s2 - Example
Process 2
Process 1

19 April 9, 2016 – v1.0
Comparing two Means
(for unequal standard deviations)
Sample 1:
Sample 2:
Confidence of 90%: -0.48 4.10 8.68 4.58
Confidence of 95%: -1.46 4.10 9.66 5.56
Confidence of 99%: -3.56 4.10 11.76 7.66
Degrees of Freedom: 16.00
C O N S U L T I N G
Excel-Calculator for two Mean Values with s1  s2
Input Data
Input Data

20 April 9, 2016 – v1.0
84.9
12.61
50
2
2
2



s
x
n
estim
03.7
02.57
10
1
1
1



s
x
n
estim
56.376.11:%99
46.166.9:%95
48.068.8:%90
D
D
D
true
true
true
Conclusion: Because zero is inside all confidence
intervals we have not enough evidence to say that
there is a significant difference between the two
processes.
Confidence intervals for the difference of two means:
Comparing two Mean Values with s1  s2 - Example
Process 2
Process 1

21 April 9, 2016 – v1.0
    .if,
2
or
2
11
21
2
2
2
1
21
2
22
2
11
nn
ss
S
nn
snsn
S Pp 





To combine two standard deviations we have to calculate the pooled
standard deviation:
Confidence interval for the difference isDtrue true truex x 1 2
21
2
21
2
1111
nn
St
nn
St pestimtruepestim DDD 
where tcan be calculated as  .2;TINV 212  nnt 
The question is now, do we have evidence to say ? If zero is not
inside the 95% confidence interval we can say with a 95% confidence or a
5% risk that the two process are different.
0Dtrue
Comparing two Mean Values with s1 = s2 - Theory

22 April 9, 2016 – v1.0
 
 
 
  


















11 2
2
2
2
2
1
2
1
2
1
2
2
2
2
1
2
1
n
ns
n
ns
n
s
n
s

To compare two mean values with different standard deviations we have to calculate
a “pooled” degree of freedom:
Confidence interval for the difference is21 truetruetrue xx D
2
2
2
1
2
1
2
2
2
2
1
2
1
2 n
s
n
s
t
n
s
n
s
t estimtrueestim DDD 
where tcan be calculated as  .;TINV2  t
The question is now, do we have evidence to say ? If zero is not inside
the 95% confidence interval we can say with a 95% confidence or a 5% risk that the
two process are different.
0Dtrue
Comparing two Mean Values with s1  s2 - Theory
This value will not be nessarily be an integer. If it is not, we round it down to the
nearest integer.

23 April 9, 2016 – v1.0
Can we say that two data sets are not taken or generated
by the same process or method ?
Does evidence for a significant difference exist between
two methods (e.g. current/better), two test groups, or two
persons performing the same work/measurements ?
Comparing Mean Values for Paired Data

24 April 9, 2016 – v1.0
For example: We can compare measurement results obtained for
example from two different persons measuring the same products.
Data has to be collected pairwise so that one pair contains
measurement results from the same products.
Pair 1 2 3 ... n
Person 1 X1 X2 X3 ... Xn
Person 2 Y1 Y2 Y3 ... Yn
Results D1 D2 D3 ... D n
     
1
...
...
,...3,2,1,
21
21

DDDDDD

DDD
D
D
D
n
s
n
niYX
estimnestimestim
n
estim
iii
(differences within the pairs)
(average of differences within the pairs)
(standard deviation
of differences)
Define:
Comparing Mean Values for Paired Data

25 April 9, 2016 – v1.0
0233.00013.0:%99
0196.00024.0:%95
0180.00040.0:%90
D
D
D
true
true
true
Pair 1 2 3 4 5 6 7 8 9 10
Person 1 13.21 13.15 13.18 13.20 13.19 13.18 13.22 13.20 13.19 13.21
Person 2 13.19 13.14 13.18 13.17 13.18 13.18 13.20 13.21 13.17 13.20
D 0.02 0.01 0.00 0.03 0.01 0.00 0.02 -0.01 0.02 0.01
0120.0011.0 D Dsestim
Confidence intervals for the Differences:
Conclusion: Zero is not included in the 95% confidence interval. Therefore, we
can say, with a 5% risk, that the two persons are measuring in significantly
different ways. However, zero is included in the 99% confidence interval, thus
with a confidence of 99% we can not say that the persons are doing the
measurement in different way.
Comparing Mean Values for Paired Data - Example

26 April 9, 2016 – v1.0
Comparing Paired Data
Number of Pairs 10
Average of Differences 0.011
Standard Deviation of Differences 0.012
Lower limit Average Upper limit ± D t(α/2)
Confidence of 90%: 0.004 0.011 0.018 0.007 1.833
Confidence of 95%: 0.002 0.011 0.020 0.009 2.262
Confidence of 99%: -0.001 0.011 0.023 0.012 3.250
C O N S U L T I N G
Excel-Calculator for Paired Data
Input Data

27 April 9, 2016 – v1.0
n
s
t
n
s
t estimtrueestim
DD
DDD
22

where tcan be calculated as  1;TINV2  nt 
If zero is not inside the 95% confidence interval we can say (with a
5% risk) that the two persons are performing the measurement
process in significantly different ways or as exist at least some reason
why they are getting significantly different measurement results.
Comparing Mean Values for Paired Data - Theory

28 April 9, 2016 – v1.0
Confidence Interval for a Standard Deviation
We can take a sample in production and estimate the
standard deviation, but what is the real standard
deviation of the process ?
Can we say that the process standard deviation is
small enough compared to a given tolerance interval ?

29 April 9, 2016 – v1.0
03.7
02.57
10
1
1
1



estim
estim
s
x
n
Sample 1.
84.9
12.61
50
2
2
2



estim
estim
s
x
n
Sample 2.
Which one is the real standard deviation ?
What can we conclude about the real standard deviation
based on the estimated standard deviation ?
Is the sample size sufficiently large ?

30 April 9, 2016 – v1.0
Number of Observations 50
Standard Deviation 9.840
Lower limit Std. Dev. Upper limit
Confidence of 90%: 8.457 9.840 11.825
Confidence of 95%: 8.220 9.840 12.262
Confidence of 99%: 7.788 9.840 13.195
C O N S U L T I N G
Excel-Calculator for a Standard Deviation
Input Data

31 April 9, 2016 – v1.0
Confidence Interval for a Standard Deviation - Example
Can we say that our sample gives us enough evidence to say that the
process standard deviation is smaller than the target standard
deviation equal to s = 8 ?
84.9
50


estims
n
20.1379.7:%99
26.1222.8:%95
82.1146.8:%90



true
true
true
s
s
s
Conclusion: The target value of s = 8 is only inside the 99%
confidence interval. Therefore, we can say with 97.5% confidence
(means 2.5% risk) that the true standard deviation is higher than the
target value. However, we can’t say so with a 99.5% confidence,
because the lower bound of the 99% confidence interval is smaller
than 8.
Sample: Confidence intervals:

32 April 9, 2016 – v1.0
Confidence Interval for a Standard Deviation - Theory
2
2
1
2
2
11
  




n
ss
n
s estimtrueestim
where and are constants depending on the risk level  and the
number of observations n in the sample. Those can easily be calculated using
the Excel-function “CHIINV” as
and
If the target standard deviation is higher than the upper limit of the 90%
confidence interval we have only a 5% risk that the real process standard
deviation is not small enough.
 1;2CHIINV2
2  n
 1;21CHIINV2
21  n 
2
2 2
21  where and are constants depending on the risk level  and the
and
2
22
2where and are constants depending on the risk level  and the
and

33 April 9, 2016 – v1.0
Comparing two Standard Deviations
Is there evidence for a significant process improvement
with respect to a decreased standard deviation, i.e. is
there a significant difference between the standard
deviations of the two processes ?

34 April 9, 2016 – v1.0
03.7
10
1
1


s
n
Sample 1.
84.9
50
2
2


s
n
Sample 2.
or
03.7
10
1
1


s
n
Sample 1.
84.9
50
2
2


s
n
Sample 2.
?
or
??

35 April 9, 2016 – v1.0
Sample 1: Confidence of 90%: 5.13 11.57
Sample 2: Confidence of 90%: 8.46 11.82
Lower limit Ratio Percentage Upper limit
Confidence of 90%: 0.50 0.71 40.0 1.20
Confidence of 95%: 0.46 0.71 40.0 1.33
Confidence of 99%: 0.41 0.71 40.0 1.67
C O N S U L T I N G
Excel-Calculator for two Standard Deviations
Input Data
Input Data

36 April 9, 2016 – v1.0
Comparing two Standard Deviations - Example
Is there evidence for a significant process improvement ?
03.7
10
1
1


s
n
95% confidence interval
for the standard deviation:
Process 2
84.9
50
2
2


s
n
for the standard deviation:
83.1284.4  trues 26.1222.8  trues
4 86 1210
Process 1

37 April 9, 2016 – v1.0
84.9
50
2
2


s
n
03.7
10
1
1


s
n
714.0
2
1

estim
estim
estim
s
s

Confidence intervals for the ratio :true
67.141.0:%99
33.146.0:%95
20.150.0:%90



true
true
true



Conclusion: Because is inside all confidence
intervals we have not enough evidence to conclude that
there is a significant difference between the standard
deviations of the two processes.
Process 2
1true
Comparing two Standard Deviations - Example
Process 1 Confidence intervals for the ratio :
Conclusion: Because is inside all confidence
intervals we have not enough evidence to conclude that
there is a significant difference between the standard
deviations of the two processes.

38 April 9, 2016 – v1.0
We simply compare with by looking at their ratio
Confidence interval for the ratio :
1estims 2estims
Comparing two Standard Deviations - Theory
2
1
estim
estim
estin
s
s

2
1
true
true
true
s
s









 estimtrueestim F
F
 2
1
1
Where F1 and F2 are constants which depend on the risk level  and the
number n of observations in the samples. Those can be easily calculated using
the Excel-function “FINV” as F1 = FINV(/2;n1–1;n2–1) and F2 = FINV(/2;n2–
1;n1–1).
Do we have evidence to say ? If is not inside the 95% confidence
interval, then we can say that with a 95% confidence or a 5% risk the standard
deviations of the two process are significantly different.
1true 1true

39 April 9, 2016 – v1.0
Confidence Interval for the and
The estimated and capability indices are for
example based on 25 subgroups with 5 observations in
each subgroup, i.e. 125 measurements were taken.
What are the real capability indices ?
Can we say that our process is not capable in other
words, can we say that is smaller than 1.33 ?
pkcˆ
pkcˆ
pcˆ
pcˆ
pkcˆ

40 April 9, 2016 – v1.0
A Measure of Process Capability
 For any critical characteristic the Capability Index (a
“hat” (^) will denote an estimated quantity), also known as
the design margin, is quantified by :
ST
p
s
LSLUSL
c



6
-
)variationprocess(or totalCapabilityProcess
erance)design tol(orion WidthSpecificat
ˆ
pcˆ

41 April 9, 2016 – v1.0
The Capability Index pcˆ
average
= nominal value or target
LSL USLLCL UCL
1.00 < cp < 1.33
average
+ 3*sigma
average
- 3*sigma

42 April 9, 2016 – v1.0
A Measure of Process Performance
 is the Capability Index adjusted for k, which
considers actual operation of the process and takes
into account any difference between the design
nominal and the actual process mean value x-bar.
)
3
-
,
3
-
(minˆ
STST
pk
s
LSLx
s
xUSL
c


pkcˆ

43 April 9, 2016 – v1.0
The Capability Index pkcˆ
nominal
value
LSL USLLCL UCL
average
1.33 < cp < 2.00, but 1.00 < cpk < 1.33
average
- 3*sigma
average
+ 3*sigma

44 April 9, 2016 – v1.0
Confidence Interval for the and
Mean value
= Nominal value or Target
LSL USL
= ?
Nominal
value
LSL USL
Mean
value
?
pcˆ
pcˆ
pkcˆ
pkcˆpkcˆ
Uncertainty in standard deviation Uncertainty in mean value

45 April 9, 2016 – v1.0
Number of Observations: 90
Cp: 5.00
Cpk: 2.30
PPM: 0
Lower limit Cp Upper limit ± D
Confidence of 90%: 4.38 5.00 5.62 0.62
Confidence of 95%: 4.27 5.00 5.73 0.73
Confidence of 99%: 4.03 5.00 5.97 0.97
Lower limit Cpk Upper limit ± D
Confidence of 90%: 2.01 2.30 2.59 0.29
Confidence of 95%: 1.96 2.30 2.64 0.34
Confidence of 99%: 1.85 2.30 2.75 0.45
Confidence Interval for Capability Indices
C O N S U L T I N G
Excel-Calculator for Capability Indices
Input Data

46 April 9, 2016 – v1.0
Sample
30.2ˆ
00.5ˆ
90518



pk
p
c
c
m
97.503.4:%99
73.527.4:%95
62.538.4:%90



p
p
p
c
c
c
75.285.1:%99
64.296.1:%95
59.201.2:%90



pk
pk
pk
c
c
c
Conclusion: With a risk of less than 0.5% we can say that the process is
capable. The lower limit of every cp -confidence interval is greater than 1.33 and
the lower limit of every cpk - confidence interval is greater than 1.33.
nominal
value
LSL USLLCL UCL
averageAverage - 3*s Average + 3*s
Confidence Interval for the and - Examplepcˆ pkcˆ

47 April 9, 2016 – v1.0
   12
ˆ
ˆ
12
ˆ
ˆ 22




m
c
zcc
m
c
zc
p
pp
p
p 
Where m = n · N is the number of measurements (n is the number of products/
measurements in each subgroup and N is the number of subgroups) and is a
constant that depends on the risk level . This constant can be easily calculated
using the Excel-function “NORMSINV” as
.
If for example the 90% confidence interval includes the value 1.33 we can not
make any decision about the process capability. If the upper limit of the 90%
confidence interval is < 1.33 we can say, with a 5% risk, that the process is not
capable. If the lower limit of the 90% confidence interval is  1.33 we can say,
with a 5% risk, that the process is capable.
z 2
)2-NORMSINV(12  z
pcˆConfidence Interval for the - Theory

48 April 9, 2016 – v1.0
pcˆConfidence Interval for the
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.4000
1.6000
0 100 200 300 400 500 600
m=N*n
delta
cp=0.5
cp=1.0
cp=1.5
cp=2.0

49 April 9, 2016 – v1.0
    mm
c
zcc
mm
c
zc
pk
pkpk
pk
pk








9
1
12
ˆ
ˆ
9
1
12
ˆ
ˆ
2
2
2
2 
Where m = n · N is the number of measurements (n is the number of products/
measurements in each subgroup and N is the number of subgroups) and is
a constant that depends on the risk level . This constant can easily be
calculated using the Ecxel-function “NORMSINV” as
.
If the upper limit of the 90% confidence interval is < 1.33 we can say, with a 5%
risk, that the process is not capable (location of distribution is wrong). If the
lower limit of the 90% confidence interval is 1.33 we can say, with a 5% risk,
that the process is capable.
z 2
)2-NORMSINV(12  z
pkcˆConfidence Interval for the - Theory

50 April 9, 2016 – v1.0
Confidence Interval for the
0.0000
0.2000
0.4000
0.6000
0.8000
1.0000
1.2000
1.4000
1.6000
0 100 200 300 400 500 600
m=N*n
delta
cpk=0.5
cpk=1.0
cpk=1.5
cpk=2.0
pkcˆ

51 April 9, 2016 – v1.0
We can take a sample in production or in pre-
production and estimate the defect rate ( ) of a
process, but what is the real defect rate (p) ?
Do we have enough evidence to say that a defined
target is achieved ?
pˆ
Confidence Interval for Defect Rates

52 April 9, 2016 – v1.0
A new IC (160 solder joints per IC) shall be used for a new product.
The product development team decided to purchase 100 ICs and test
they solderability. An experiment was organised in pre-production
and during inspection two (2) poor solder joints were identified. This
results in a estimated defect rate for one solder joint of
%01.0%
100160
2
100%
partsof#ies/partoppurtunitdefectof#
defectsof#
100ˆ 



p
Which is equal to a PPM level of
12510
%100
ratedefect 6
PPM
Confidence Interval for Defect Rates - Example
Is this evidence enough to conclude that with a confidence of 95%
the defect rate for this part will be less than 250 PPM ?

53 April 9, 2016 – v1.0
Number of Tested Defect Opportunities: 16000
Number of Defects Found: 2
Lower limit p (%) Upper limit
Confidence of 90%: 0.00% 0.01% 0.04%
Confidence of 95%: 0.00% 0.01% 0.05%
Confidence of 99%: 0.00% 0.01% 0.06%
Lower limit p (DPM) Upper limit
Confidence of 90%: 22 125 393
Confidence Interval for a Defect Rate
C O N S U L T I N G
Excel-Calculator for Defect Rates
Input Data

54 April 9, 2016 – v1.0
Sample:
PPM
p
d
n
125=ˆ
%01.0ˆ
2
00016



Note, that the number of defect opportunities is given by
jointssolder00016
component
jointssolder
160components100 
Conclusion: With a 5% risk the true defect rate (in PPM) will be above 393
PPM. However, with a confidence of 90% the true defect rate will be in the
interval [22, 393] PPM. For a target value of for example less than 250 PPM
we can’t make any conclusion. If the range of possible defect rates is too
larger we just have to place, solder and inspect more ICs components.
%06.0%00.0:%99
%05.0%00.0:%95
%04.0%00.0:%90



p
p
p
PPMpPPM
PPMpPPM
PPMpPPM
5796:%99
45115:%95
39322:%90




55 April 9, 2016 – v1.0
Sample:
PPMp
d
n
0=%00.0ˆ
0
00016



Conclusion: With a 5% risk the true defect rate (in PPM) will be greater than
187 PPM. However, with a confidence of 95% the true defect rate will be
between 0 PPM and 187 PPM. For a target value less than 250 PPM we do
have a confidence of about 95% that the solderability of the IC’s is good
enough. However, the upper limit of the 99% confidence interval is greater
than the target value and so we have a risk higher than 1% but lower than 5%
that the solderability will not be good enough.
%03.0%0:%99
%02.0%0:%95
%01.0%0:%90



p
p
p
PPMpPPM
PPMpPPM
PPMpPPM
2880:%99
1870:%95
1440:%90




56 April 9, 2016 – v1.0
In cases where the estimated defect rate is equal to zero , the
natural lower limit of the confidence interval is zero. Thus, there
exists no risk that the true value of the defect rate p is smaller than
zero. Therefore, the upper limit of the confidence interval has to be
calculated in such way that with a risk of 100·% the true value of
the parameter p is greater than the upper limit.
 0ˆ p
%100%0
2
2
Fn
F
p


Where n is the number of tested defect opportunities and
. nF  2;2;FINV2 
Confidence Interval for Defect Rates - Theory

57 April 9, 2016 – v1.0
In cases where the estimated defect rate is greater than zero,
means , a two sided confidence interval is calculated by 0ˆ p
%
)1(
)1(
100%
)1(
100
2
2
1 Fddn
Fd
p
Fdnd
d





where n is the number of tested defect opportunities (sample
size) and d the number of defects found. The values and can
be easily calculated using the Excel-function “FINV” as
and
21 FF
)2);1(2;2/(1 ddnFINVF   ))(2);1(2;2/(2 dndFINVF  
If the lower bound of 90% confidence interval is greater than the
target value for the defect rate we can say, with a 5% risk, that the
defect rate is too high compared with the target value.
Confidence Interval for Defect Rates - Theory

58 April 9, 2016 – v1.0
Comparing two Defect Rates
If we compare the defect rates of two processes (e.g.
two production lines) can we say that one process is
producing more defects than the other ?
Can we say that there is evidence for a significant
process improvement with respect to the defect rates ?

59 April 9, 2016 – v1.0
Comparing two Defect Rates - Example
Process 2
PPM0005ˆ%5.0ˆ005.0ˆ
50
00010
1
1
1
1
1



n
d
p
d
n
PPM0001ˆ%1.0ˆ001.0ˆ
10
00010
1
1
2
2
2



n
d
p
d
n
for the defect rate:
for the defect rate:
PPM5876PPM7133
%66.0%37.0
1
1


p
p
PPM8381PPM480
%18.0%05.0
2
2


p
p
PPM
Process 1

60 April 9, 2016 – v1.0
Comparing Two Defect Rates
Sample 1:
Number of Defects Found: 50 Defect Rate: 0.50%
Sample 2:
Number of Defects Found: 10 Defect Rate: 0.10%
Confidence of 90%: 0.26% 0.40% 0.54% 0.14%
Confidence of 95%: 0.24% 0.40% 0.56% 0.16%
Confidence of 99%: 0.19% 0.40% 0.61% 0.21%
C O N S U L T I N G
Excel-Calculator for two Defect Rates
Input Data
Input Data

61 April 9, 2016 – v1.0
005.0ˆ
50
10000
1
1
1



p
d
n
001.0ˆ
10
10000
2
2
2



p
d
n
    
  3109990,10,9950,50min
ˆ1,ˆ,ˆ1,ˆmin
%4.0ˆ004.0ˆˆ
22221111
21



pnpnpnpn
pp
Confidence intervals for the difference:
%6.0%2.0:%99
%6.0%2.0:%95
%5.0%3.0:%90
21
21
21



pp
pp
pp
Conclusion: A defect rate equal to zero is not inside any of the confidence
intervals. Therefore, we can say that with a 99.5% confidence (means 0.5%
risk) process 1 is producing more defects than process 2.
Comparing two Defect Rates - Example
Process 1
Process 2

62 April 9, 2016 – v1.0
Comparing defect rates can be done by looking at the difference .)ˆˆ( 21 pp 
      DD 212121
ˆˆˆˆ pppppp
Where
and .
   
212
22
1
11
2 2
1
2
1ˆ1ˆˆ1ˆ
nnn
pp
n
pp
z 



D 
 2
1NORMSINV
2
 Z
Note: This formula is based on the assumption that
     3ˆ1,ˆ,ˆ1,ˆmin 22221111  pnpnpnpn
Do we have evidence to say ? If is not inside the
95% confidence interval we can say that, with a 5% risk, and
so that one process is producing more defects than the other.
  021  pp
021  pp
  021  pp
Comparing two Defect Rates - Theory

63 April 9, 2016 – v1.0
Confidence Interval for the Counts
Assume a operator has inspect 100 lenses and has found
5 scratches. How many scratches one can expect on 100
lenses of the same quality ? What about in 1, 1000 or
10000 lenses ?
Do we have enough evidence to say that the agreed
quality level (e.g. 2 scratches in average per 100 lenses)
has not been achieved by the supplier ?

64 April 9, 2016 – v1.0
Confidence Interval for a Counts
Number of Events: 5
Lower limit Events Upper limit
Confidence of 90%: 2.0 5 10.5
C O N S U L T I N G
Excel-Calculator for Counts
Input Data

65 April 9, 2016 – v1.0
Confidence Interval for the Counts - Example
100 lenses were inspected and 5 events were found. The confidence
intervals for the events per 100 lenses is given by:
1.141.1:%99
7.116.1:%95
5.100.2:%90






The 95% confidence interval for one single lens is then determined by
dividing the lower and upper limit of the 95% confidence interval for
100 lenses by 100.
 117.0,016.0
100
7.11
,
100
6.1






If we are interested in the 95% confidence interval for events per 1000
lenses we have to multiply the limits for a single lens by 1000:
   117,16117.01000,016.01000 

66 April 9, 2016 – v1.0
Let us note as the number of events (e.g. scratches or suggestions).
The confidence interval for the true value of events () is given by
ˆ
2
)1ˆ(2;
2
2
ˆ2;
2
1
2
1
2
1

  
Where and are constants which depend on the risk
level  and the number of events. Those can be easily calculated
using the Excel-function “CHIINV” as
and
  1 2 2
2
 ;   2 2 1)
2
; (  
  
ˆ2;
2
1CHIINV2
ˆ2;
2
1

 )1ˆ(2;
2
CHIINV2
)1ˆ(2;
2

 
If the lower limit of the 95% confidence interval is greater than the
target value of events for a certain number of inspected parts, we can
say that with a 97.5% confidence the number of events is too high.
Confidence Interval for the Counts - Theory

67 April 9, 2016 – v1.0
Comparing two Counts
Is there evidence for a significant difference
between two numbers of events ?
Can we say that one process is producing 2 or 3
(or even more) times more events than another
process ?

68 April 9, 2016 – v1.0
Comparing two Counts - Example
Yesterday, the morning shift has inspected 100 lenses and 5 scratches were
found ( ). Today, again 100 lenses were inspected by the morning shift
and now 15 scratches were found ( ). Is there evidence that the quality
of the batch used yesterday was significant better than the batch used today
(means 
5ˆ
1 
15ˆ
2 
Confidence intervals for the ratio  are:
492.12783.0:%99
424.9024.1:%95
095.8182.1:%90



true
true
true



Conclusion: The 90% and 95% confidence intervals lower bound is greater
than 1. Therefore, we can with a 97.5% confidence say that the batch used
yesterday was significantly better than the batch used today. However, we
can’t say this with a 99.5% confidence because the lower bound of the 99%
confidence interval is smaller than 1.

69 April 9, 2016 – v1.0
Sample 1: Sample 2:
Number of Events: 5 Number of Events: 15
Lower limit True Ratio Upper limit
Confidence of 90%: 1.2 3.0 8.1
Confidence of 95%: 1.0 3.0 9.4
Confidence of 99%: 0.8 3.0 12.5
Note: Sample 2 must be greater or equal than Sample 1
Comparing two Counts
C O N S U L T I N G
Excel-Calculator for two Counts
Input Data
Input Data

70 April 9, 2016 – v1.0
We define as a target ratio of two numbers of events and
1
2


 
    
  .
ˆˆ
2/1ˆˆ
ˆ
21
2
212






We have evidence for a significant difference between the true ratio and
the target ratio with a confidence of
635.6îf:%99
841.3îf:%95
706.2îf:%90
2
2
2






Frequently we are interested in the ratio = 1, means that the number of
events of the two processes are or better could be equal (). In this
case one simply obtains
  .
ˆˆ
1ˆˆ
ˆ
21
2
212






Comparing two Counts - Theory

71 April 9, 2016 – v1.0
The End …
“Perfection is not attainable, but if we chase perfection we can catch
excellence.” - Vince Lombardi

72 April 9, 2016 – v1.0
Terms & Conditions
After you have downloaded the training material to your own
computer, you can change any part of the course material and
remove all logos and references to Operational Excellence
Consulting. You can share the material with your colleagues and
re-use it as you need. The main restriction is that you cannot
distribute, sell, rent or license the material as though it is your
own. These training course materials are for your — and
your organization's — usage only. Thank you.

Six Sigma Confidence Interval Analysis (CIA) Training Module

Recommended

Recommended

More Related Content

What's hot

What's hot (19)

Viewers also liked

Viewers also liked (20)

Similar to Six Sigma Confidence Interval Analysis (CIA) Training Module

Similar to Six Sigma Confidence Interval Analysis (CIA) Training Module (20)

More from Frank-G. Adler

More from Frank-G. Adler (13)

Recently uploaded

Recently uploaded (20)

Six Sigma Confidence Interval Analysis (CIA) Training Module