Grand Social Law ◾ 225
question asked, when improvements are reported, is whether the improvements
are statistically significant. e direction of z scores indicate improvement and the
magnitude indicate statistical significance.
Sigma Level: Safety Margin
As the story goes, in 1986 Bill Smith, the father of Six Sigma, showed a bell curve
to the CEO Bob Galvin and proved that there is a finite probability that processes
cross the limits and explained why defects reach customers despite multiple testing.
Six Sigma began as a reliability model. Jack Welch made Six Sigma a discipline
in GE and later opined, “the mean is fine it is the standard deviation that spells
trouble.”
Table 13.5 Improvement Scores
Metric
Organization
Sigma
Last Year
Mean
Current
Year Mean
Shift in
Mean
z
Score
Effort variance 5 12 10 –2 –0.40
Schedule variance 3 6 5 –1 –0.33
Scope creep 2 6 3 –3 –1.50
Defect density 1 3 2 –1 –1.00
Complexity 10 70 60 –10 –1.00
CSAT 2 5 6 1 0.50
–2 –2
CSAT
Complexity
Defect density
Scope creep
Schedule variance
Effort variance
–1 –1 0
z score
1 1
Figure 13.11 Score improvement.
226 ◾ Simple Statistical Methods for Software Engineering
e entire Six Sigma methodology depends on the normal distribution.
Assuming normal distribution, we can calculate sigma level of safety margin to the
customer using the following definition:
Sigma level USL mean / or
Mean lSL /
Whicheve
= −
= −
( ) ,
( )
σ
σ
rr is smaller
(13.7)
e sigma level can be converted into risk using ideas developed early in this
chapter. Risk is expressed in parts per million or defects per million opportunities
(DPMO). For example, we can consider two-tailed problems and find tail areas
for different distances of tails from the mean: 1, 2, 3, 4, 5, and 6 sigmas as shown
in Table 13.6. e distances are known as sigma levels. e tail areas are known
as defect levels. In Table 13.6, defect levels are presented first in fractions, then in
percentage, and finally in PPM or DPMO in the last column.
Table 13.6 Six Sigma Conversion Table
Sigma Level
Tail Areas
Fraction % PPM (DPMO)
Part A: Pure Scale
1 0.3173105078629 31.731050786 317,310.50786
2 0.0455002638964 4.550026390 45,500.26390
3 0.0026997960633 0.269979606 2699.79606
4 0.0000633424837 0.006334248 63.34248
5 0.0000005733031 0.000057330 0.57330
6 0.0000000019732 0.000000197 0.00197
Part B: Practical Scale (1.5 Sigma Drift Included)
1 0.6976721 69.76721 697,672.13
2 0.3087702 30.87702 308,770.17
3 0.0668106 6.68106 66,810.60
4 0.0062097 0.62097 6209.68
5 0.0002326 0.02326 232.63
6 0.0000034 0.00034 3.40
Grand Social Law ◾ 227
Table 13.6 has two parts. In part A, the tail areas have been computed the usual
way. is suggests a defect level of 1.97 parts per billion for a process to qualify
as a Six Sigma process. is requirement was contested internally in Motorola by
pragmatic managers who wanted an allowance for long-term drifts in processes.
In particular, they wanted the “Shewhart allowance of 1.5 sigma drift in control
charts” to be made available in Six Sigma considerations. Part B of Table 13.6 has
been computed with Shewhart allowance; this table suggests 3.4 PPM defects for
a Six Sigma process, approximately 1726 times more defects than the pure scale.
Finally, the practical scale has prevailed and is widely used as a quality standard.
In Six Sigma culture, we respond to mathematically derived tail probabilities
even if there is no physical event in the tail region.
Statistical Tests
Gaussian properties are extensively used in statistical tests to compare results. If two
processes are represented by adjacently located Gaussian curves and if the tails do not
overlap, they are distinctly different processes. If tails overlap, perhaps they are not so
different. To resolve this problem, we resort to statistical tests, such as z test, t test, and F
test. In all tests, we find a p value, the probability of finding one sample from the other
lot. To calculate p value, in commonly used statistical tests, the Gaussian curve is used.
We have seen how the bell curve can be put to a variety of applications in soft-
ware engineering and management.
Box 13.5 electron charge to
mass ratio measurement
Nobel laureate J. J. omson measured the invisibly small particle electron in
1897 at the well-known Cavendish laboratory in Cambridge, England.
By carefully measuring how the cathode rays were
deflected by electric and magnetic fields, Thomson was
able to determine the ratio between the electric charge
(e) and the mass (m) of the rays. Thomson’s result was
e/m = 1.8 × 10
–11
coulombs/kg
He received the Nobel Prize in 1906 for the discovery
of the electron, the first elementary particle.
Nobelprize.org
228 ◾ Simple Statistical Methods for Software Engineering
Review Questions
1. What are the various names by which the normal distribution is known?
2. What is Gaussian smoothening?
In reality, measurements do not get reported by a single value. We have
either a range of values or a mean with associated standard deviation. We
are discussing e/m measurements of electrons in coulombs per kilogram. e
numbers presented here must be multiplied by 10
11
.
Todays’ accepted mean e/m is 1.758820088 with a standard deviation of
0.000000013.
Earlier trials by J. J. omson gave results between 1.1 and 1.4 [10]. Norton
et al. [11] have shown results with a mean value of 1.60 and a standard devia-
tion value of 0.29. Earlier attempts by Millikan [12] result in a mean value
of 2.82 and a standard deviation of 0.55. ese three results are shown as
reconstructed Gaussian curves in Figure 13.9.
e three bell curves indicate measurement reliabilities available in those
experiments. e broader the curve, the less the measurement reliability. It
may also be noted that broader curves have shorter peaks. In this context, one
can intuitively feel that the height of the peak can also be considered as an
indicator of measurement reliability. Narrower curves indicate “precision” in
measurement (Figure 13.12).
7.00
6.00
5.00
4.00
3.00
2.00
1.00
0.00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
4.1
4.2
4.3
4.4
4.5
4.6
e/m × 10
11
C/kg
Norton et al.
JJT-1
Millikan-1
Figure 13.12 Gaussian error curves.
Grand Social Law ◾ 229
3. Why does the bell curve perfectly fit all the different variance metrics such as
effort variance, schedule variance, and size variance?
4. Name the famous mathematicians and scientists who have contributed to the
development of the bell curve.
5. Mention three important applications of the bell curve.
Exercises
1. In the context of a bell curve with mean = 7 and standard deviation = 6, find
the z score of a data point 25.
2. Dr. Shewhart prescribed three sigma limits to control charts. What are the
tail areas outside these limits?
3. Find the area under the bell curve included inside two sigma limits.
4. Find the percentage of area beyond six sigma limits. Express this in parts per
million.
5. A process peaks at 4. Its specification limits are 2 and 5. If the standard devia-
tion is 1, find the process capability indices C
p
and C
pk
.
References
1. A. de Moivre, e Doctrine of Chances: Or, a Method of Calculating the Probability of
Events in Play, W. Pearson, London, 1718.
2. J. C. Friedrich Gauss. Available at http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss.
3. H. Fischer, A History of the Central Limit eorem: From Classical to Modern Probability
eory, Springer, New York, Dordrecht, Heidelberg, London, 2010.
4. Sir F. Galton, Natural Inheritance, MacMillan, 1889.
5. G. Pólya, Über den zentralenGrenzwertsatz der Wahrscheinlichkeitsrechnung und das
Momentenproblem, MathematischeZeitschrift (in German), 8(3–4), 171–181, 1920.
6. L. Le Cam, e central limit theorem around 1935, Statistical Science, 1(1), 78–91,
1986.
7. D. J. Poirier, Intermediate Statistics and Econometrics: A Comparative Approach, MIT
Press, London, Cambridge, 1995.
8. Elvins, Volume Visualization. Available at http://www.cs.rug.nl/~michael/FANTOM
/FANTOM1a.pdf.
9. F. H. Hankins, Adolphe Quetelet as Statistician, Columbia University, New York, 1908.
10. P. F. Dahl, Flash of the Cathode Rays: A History of JJ omson’s Electron, CRC Press,
1997.
11. M. Norton, C. Bush, B. Atinaja and B. Steven, Electron Charge to Mass Ratio. Available
at http://www.siue.edu/~mnorton/Ratio.pdf.
12. R. A. Millikan, e Electron: Its Isolation and Measurements and the Determination of
Some of Its Properties, e University of Chicago Press, Chicago, 1917.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.