Some remarks on measuring sigma coefficient
in six sigma multidimensional processes
Grzegorz Kończak1
Department of Statistics, University of Economics, ul. Bogucicka 14,
40-522 Katowice,
Poland
koncz@ae.katowice.pl
Summary. In the paper a proposition of the method of calculating Sigma level in
multidimensional assessing quality of products or processes is presented. This coefficient is a generalization of the classical Sigma level coefficient which is widely used in
the one dimensional case. The proposed coefficient can be used for determining the
Sigma level when the product is described by many characteristics. This coefficient
is a natural extension of the classical Sigma measure which is most widely used in
Six Sigma procedures.
Key words: Six Sigma, sigma coefficient, quality control, multidimensional process
1 Introduction
The managers of corporations should assure high quality of products in their companies. This is especially important nowadays when products have to meet even more
demanding quality standards. In modern management various statistical methods
play a vital role in quality improvement. Such methods as correlation analysis, scatter plots, hypothesis testing, control charts (see [Bru04]) are widely used in quality
improvement. Montgomery D.C. [Mon96] defines quality as ”fitness for use” or in
another way as ”inversely proportional to variability”. According to this definition
one of the main goal of quality improvement is reduction of variability.
One of the most known and widely used statistical methods of improving quality
are the Shewhart control charts. They are usually used for monitoring process stability. The construction of control charts is based on the well known statistical 3-sigma
rule. Another very popular method of quality improvement, especially nowadays, is
Six Sigma. The name of this method refers to the mentioned 3-sigma rule. The main
goal of Six Sigma is to improve quality of entire process. There are many statistical
procedures which are used in Six Sigma (see [Bru04], but the first step in Six Sigma
procedures is usually calculating the Sigma level which corresponds to fraction defectives parts of product or process. Determining sigma levels of processes allows
process performance to be compared throughout an entire organization, because it
is independent of the process.
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Grzegorz Kończak
2 Sigma coefficient
2.1 The construction of the Sigma coefficient
Walter A. Shewhart has leaned the construction of control charts on the 3-sigma rule.
The name ’Six Sigma’ refers to the same rule. In Six Sigma activities it is important
to distinguish between the ’Sigma’ measure used by the pioneers of this method
and now used in Six Sigma activities on the one hand and the ’sigma’ symbol for
standard deviation. Under the assumption of normality and stability of the process
the probability of signal occurrence on the standard control chart is c.a. 0.27%. In
Six Sigma we often say that there are average 2,700 defects per million opportunities
(dpmo) and we usually say that the company works on 3 Sigma level (see fig. 1). In
company working on 2 Sigma level there are 45,500 defects per million opportunities
expected. The coefficient dpmo for the company working on 4 Sigma level is equal
to 63 and for 6 Sigma level it is equal to 0.002. In the Six Sigma philosophy, this
is considered to be an idealized situation realized only in short term. In Six Sigma
we analyze processes considering 1.5 sigma drift which can be expected in a single
direction (see fig. 2). Over time, other long term influences come and go which move
the population and change some of its characteristics. This is called shift and drift.
This shift and drift impacts the position of the mean and shifts it 1.5 sigma from
its original position. For the processes operating on 6 Sigma level conventionally we
say that the process is in 4.5 standard deviations. The expected values of defectives
per million opportunities for the processes with 1.5 sigma shift and without shift
and corresponding to them Sigma’s level are presented in table 1.
Fig. 1. The probabilities of no failure occurrence for various Sigma level
The characteristics of the product made by factory working on 6 Sigma level
with and without 1.5 sigma shift are shown in fig. 3. In the first case the average
number of failures per million opportunities is 3.4 (it refers to the number of defects
in company which works on the 4.5 sigma level with no shift assumptions), and in
the second case it refers to 0.002 dpmo (see table 1).
Some remarks on measuring sigma coefficient
Fig. 2. Process with 1.5 sigma shift in long term
Table 1. Measuring performance on the Sigma scale
Sigma Percent of Defects per million opportunities
level the good parts
Without
With
1.5 sigma shift 1.5 sigma shift
1.0
2.0
3.0
4.0
5.0
6.0
30.85
69.15
93.32
99.38
99.977
99.99966
317 311
45 500
2 700
63
0.57
0.002
691 462
308 538
66 807
6 210
233
3.4
Fig. 3. Representation of 6 Sigma process (with and without 1.5 sigma shift)
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Grzegorz Kończak
2.2 Measuring Sigma level for one characteristic
Let n be the number of controlled elements and x be the number of nonconformities.
For calculating the Sigma level first we can calculate number of defects per million
opportunities from the following formula:
dpmo =
x
· 1, 000, 000
y
(1)
The Sigma level for this product can be obtained as follows (assuming 1.5 sigma
shift) (see [HS00])
Sigma = F −1 1 −
dpmo
1, 000, 000
+ 1.5
(2)
it can also be written
Sigma = F −1 1 −
x
y
+ 1.5 = F −1 (1 − p) + 1.5
(3)
where F −1 is the inverse function of the standard normal cumulative distribution
function. The formula (3) takes into account 1.5 sigma process shift. Let Sigma* be
the coefficient without 1.5 sigma shift then the appropriate formula for evaluating
Sigma*is following:
Sigma∗ = F −1 1 −
dpmo
1, 000, 000
(4)
it can also be written
Sigma = F −1 1 −
x
y
= F −1 (1 − p)
(5)
The classical formulas mentioned above can be used when only one characteristic
of the product is observed. These classical coefficients related to Six Sigma will be
spread out to the case of the joint describing many characteristic of the product.
3 Sigma level for joint monitoring many characteristics
3.1 Multidimensional Sigma level
The method of calculating Sigma level described above can be used only for one
observed variable, when each element is classified as either good or bad. Harry M.
and Schroeder R. (see [HS00]) emphasize that in one company various products can
be characterized by even very different Sigma levels. In this case it is very difficult
to establish joint Sigma level for the whole company. The same problem is when
Some remarks on measuring sigma coefficient
1529
the product is described by a set of characteristics (height, width, color, etc.) with
according various Sigma levels.
In this part the multidimensional Sigma level coefficient is presented. This
method can be used either for determining Sigma level for a set characteristics
in one product or for determining Sigma level for a whole company where are many
processes functioning on various Sigma levels.
3.2 Using the classical Sigma coefficient to determining the
multidimensional Sigma
Let Sigma(i) denotes the Sigma coefficient related to the i − th characteristic
(i = 1, 2, . . . , k). Basing on these values Harry M.and Schroeder R. (see [HS00])
calculate sigma level for multivariate processes. First the joint failure is calculated
under observed variables independence assumption and then the multivariate Sigma
level is obtained in the same way as in one dimensional case. The formulas for this
multivariate case can be written as follows:
- with the 1.5 sigma shift
Sigma = F
−1
k Y
i=1
xi
1−
yi
!
+ 1.5 = F
−1
k
Y
i=1
!
(1 − pi )
+ 1.5
(6)
- without the shift
∗
Sigma = F
−1
k Y
i=1
xi
1−
yi
!
=F
−1
k
Y
i=1
!
(1 − pi )
(7)
The characteristics independence assumption isn’t always proper. We need a
formula which takes into account the possibility dependence between the product
characteristics. The generalized coefficient for the measure multidimensional Sigma
level should fulfill the following properties:
- for k = 1 proposed coefficient should be equal to the classical Sigma given by
(3),
- for independent characteristics it should be equal to classical Sigma given by
(6),
- should takes into account dependencies between product characteristics.
3.3 Generalization of the Sigma coefficient for joint k
characteristics of the product
Basing on the number of defective elements and all produced elements the fraction
of defective parts is calculated for each attribute. Under defective independence for
each attribute assumption the joint fraction of defective is calculated and then the
Sigma level is derived. In many situations the assumption mentioned above is not
right. The multidimensional Sigmak coefficient presented below allows us to describe
the level of analyzed process in Sigma scale. This coefficient can be evaluated in the
case of dependence in percent of defective for the attributes.
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Grzegorz Kończak
Let us assume that the product is assessed with regard to its k attributes. Let
(y1 , y2 , . . . , yk ) and (x1 , x2 , . . . , xk ) are respectively vectors of numbers of assessed
elements and numbers of defectives elements. With these assumptions
P = (p1 , p2 , ..., pk )
is the vector of the fractions of bad elements for k attributes, where
pi =
xi
yi
.
Let Fk be the cumulative density function of the k dimensional normal distributed random variable vector with mean Θ = (0, 0, ..., 0) and covariance matrix
2
σ11
6σ21
6
Σ=4
...
σk1
σ12
σ22
...
σk2
...
...
...
...
3
σ1k
σ2k 7
7
...5
σkk
where
σpi pj = cov(pi , pj )
These variances and covariances can be estimated on the basis of the empirical
data. The density function for the vector P can be written as follows
g(p1 , p2 , . . . , pk ) = √
1
2π |Σ|
1
1
2
e− 2
P
k
2
i=1 pi
(8)
An extension of the coefficient Sigma for the measuring sigma in multidimensional case can be expressed by following formula
Sigmak = F −1 (1 − pi ) + 1.5
under assumption pi > 0 for i = 1, 2, . . . , k is a classical Sigma coefficient
obtained for one dimensional case.
The procedure of determining the Sigmak coefficient for monitoring k characteristics can be described as follows
1. For each characteristic the Sigma coefficient is determined based on formula
(3).
2. On the base of k values of the Sigma coefficients the value of standard normal
cumulative density function is determined.
3. The value of Sigmak is determined as an inversion function of standard normal
cumulative density function.
The three steps for determining the Sigmak value are schematically presented
in the Fig. 4.
The defined coefficient Sigmak has following properties:
- It is a function of nonconformities p1 , p2 , . . . , pk ,
- Sigmak ≥ 0
- Sigmak ≤ mini Sigma(i),
- If Sigma(1) = Sigma(2) = . . . = Sigma(k) = c then Sigmak ≤ c,
- If the nonconformities of the characteristic increases then Sigmak decreases,
Some remarks on measuring sigma coefficient
1531
Fig. 4. The scheme of determining the Sigmak coefficient for k dimensional estimation Sigma
- If nonconformities for the characteristics are independent then Sigmak is equal
to Sigma evaluated with (6)
In table 2 there are presented values of the proposed Sigmak coefficient for
k = 2 where p1 = p2 . The case of independence of characteristics and cases of ρ
being equal to 0.3, 0.6 and 0.9 are considered. In the case of two variables sensible
differences in sigma coefficients for various values of Pearson coefficients can be seen,
especially for the defectiveness greater than 0.001. We can expect greater differences
for larger number of analyzed characteristics. These facts argue the use in practice
the proposed Sigmak coefficient.
Table 2. The values of Sigmak coefficient for various levels of p1 = p2 and ρ
p1 = p2 Sigma(1)=Sigma(2)
0.01
0.005
0.001
0.0005
0.0001
0.00005
0.00001
0.0000034
3.83
4.08
4.59
4.79
5.22
5.39
5.77
6.0
ρ
0.0 0.3 0.6 0.9
3.56
3.83
4.37
4.59
5.04
5.22
5.61
5.85
3.57
3.83
4.38
4.59
5.04
5.22
5.61
5.85
3.59
3.86
4.39
4.60
5.05
5.22
5.61
5.85
3.68
3.93
4.46
4.66
5.09
5.27
5.65
5.88
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Grzegorz Kończak
References
[Bru04]
[HS00]
[Mon96]
Brussee W.: Statistics for Six Sigma Made Easy, McGraw-Hill, New York
(2004)
Harry M., Schroeder R.: Six Sigma, Doubleday, New York-LondonToronto-Sydney-Auckland (2000)
Montgomery D.C.: Introduction to Statistical Quality Control, John Willey and Sons, Inc., New York (1996)