220 ◾ Simple Statistical Methods for Software Engineering
On the same metric, we can pose another inverse problem. If we reject 5% of
events on either side of the Gaussian distribution, what is the range of requirement
volatility within the acceptable region? e upper rejection point has a cumulative
probability of 0.95 and the corresponding x value is as follows:
NORMINV(0.95, 3.3, 4) = 9.879%
e lower rejection point has a cumulative probability of 0.05, and the corre-
sponding x value is as follows:
NORMINV(0.05, 3.3, 4) = −3.279%
erefore, requirement volatility has a range defined by the interval [−3.279,
9.879] when 5% of events are rejected in both tails. Sometimes it helps to discard
extreme ends of the mathematical function and consider the truncated interval
as the practical dispersion. User’s judgment is required to decide on how much to
cut off. Typically, people choose any one of 1%, 5%, and 10% truncations. e
truncated range accordingly shrinks in progression. When it comes to determining
range, it depends on how the problem gets formulated. Whatever be the formula-
tion, the Gaussian can provide a simple and ready answer.
The formulation of the problem is often more essential than
its solution, which may be merely a matter of mathematical or
experimental skill.
Albert Einstein
Box 13.4 is there an average man?
A landmark in the history of the bell curve is the notion of the average man
put forward by Lambert Adolphe Jacques Quetelet (1796–1874), a Belgian
astronomer, mathematician, statistician, and sociologist. He applied the
bell curve to social science, which he called a social physics. He collected the
heights of 100,000 French conscripts and the chest measurements of 5738
Scotch soldiers. e probable error in these measurements was approxi-
mately 2 inches. He found “harmonious” variations around the average.
ere was an astonishing symmetry and also an inevitable mixture. It
looked as if there existed a fictitious average man or ideal man and others
Grand Social Law ◾ 221
Process Capability Indices
A process is said to be capable if two conditions are met: it should show less varia-
tion, and it should be aligned to goal.
Variation must be contained inside the specification window available. e
practical range of the process (truncated) must be less than the gap (USL − LSL).
How much of the specification window is consumed by the process determines
capability. Hence, we use the following two equations in assessing the first index,
known as C
p
, process potential:
Specification window = USL − LSL
C
p
, process potential = specification window/6σ
mere deviations from the ideal. Variations came from “constant causes”
and some extreme perturbations came from “accidental causes.” (We are
reminded of Shewhart’s common causes and special causes, a profound idea
that would appear in 1920, a hundred or so years later, in his statistical
process control.)
Quetelet was convinced that “there is a general law which dominates
our universe.” He presented a most important and extensive role for the
average man. e physician could thus determine the most useful remedies
and the action to be taken, in both usual and unusual cases, by compar-
ing with the fictitious average man. Hence, the artist could predict truth,
the politician could predict public sentiments, the naturalist could predict
racial types, and social scientists could predict laws of birth, growth, and
decay.
Quetelet compared the average man with the center of gravity. Everything
is to be viewed as varying about a normal state in a manner to be accurately
described by beautiful bell-shaped curves of perfect symmetry but of varying
amplitude. us, it is that the individual varies about his normal self and the
members of a group vary about their average. In social physics, the bell curve
represents the true mechanics of human history.
e average man is free from excess and defect. Nature is striving to pro-
duce the average man but fails because of the interference from a multitude
of causes [9].
Lesson learned: By analogy, likewise, the industry strives to achieve ideal
processes but fails because of interferences.
Growth should be judged by averages; variations must be used to
detect problems.
222 ◾ Simple Statistical Methods for Software Engineering
Alignment depends on how close the process mean is to the process target.
Alignment can be measured by the distance between process mean and process
target. Processes tend to drift away from the target. Drift diminishes capability. A
drifted process achieves only a part of its potential. Hence, we have the concept of
achieve process capability index C
pk
, shown as follows:
C
pk
, achieved process capability = C
p
(1 − k)
where k = process drift/half of specification window.
e more the drift, the more will be the value of k and the less will be the value
of C
pk
.
ere is an established tradition that puts C
p
at 1.0 for acceptable quality and
2.0 for excellent quality.
Besides, if C
pk
< C
p
, then the process requires alignment to the target.
An example calculation is shown in Table 13.3. Effort variance metric is used
for this example calculation. e mean is 5%, and the SD is 7%. Specification lim-
its typically are ±10%. In Table 13.3, C
p
is calculated as 0.48, and C
pk
is calculated
as 0.24. Two improvement opportunities emerge from these calculations. e first
opportunity is to improve C
pk
and make it equal to C
p
; this involves process align-
ment, meaning a shift in the central tendency. e next opportunity is to achieve
Table 13.3 Process Capability Indices
Process Metric Effort Variance %
Process Goals
Target 0 %
USL 10 %
LSL –10 %
Process Performance
Mean 5 %
Sigma 7 %
Process Capability Indices
C
p
= (USL – LSL)/6σ
0.48
k = (Drift)/(0.5*(USL – LSL)) 0.50
C
pk
= C
p
*(l – k) 0.24
Grand Social Law ◾ 223
a breakthrough reduction in process variation and hence increase C
p
. Reducing
variation is more challenging than shifting process mean.
Reduce variation. Knowledge about variation is profound
knowledge.
Deming
z Score Calculation
Process z is defined in Equation 13.5. e difference between observed value and
ideal mean is divided by standard deviation to obtain process z:
z
x
=
− µ
σ
(13.5)
e previously mentioned formula is also known as z score. Larger deviations
from mean earn larger values of score. Hence, z score is a metric of deviation from
mean. Because z score is normalized, it is dimensionless.
In practice, what is measured is deviation from the target. Hence, practical z
score has the following formula:
z
x T
=
−
σ
(13.6)
For each project metric, z score can be computed as shown in the example in
Table 13.4. Six development project metrics are considered in the table, and using
Equation 13.6, the z scores have been computed.
Table 13.4 z Score
Metric Target Sigma Performance z Score
Effort variance 0 5 10 2.00
Schedule variance 0 3 5 1.67
Scope creep 0 2 3 1.50
Defect density 0 1 3 3.00
Complexity 30 10 70 4.00
CSAT 8 2 5 –1.50
224 ◾ Simple Statistical Methods for Software Engineering
e choice of metrics strikes a balance between process, product, and business
objectives.
Process Effort variance
Schedule variance
Scope creep
Product Defect density
Complexity
Business CSAT (customer satisfaction)
In this example, the larger the score, the greater the deviation from the target.
e bar length indicates the statistical distance of mean from targets, the magni-
tude of problems. e advantage is that all metrics performance can be shown in
the same chart with a common unit (Figure 13.10).
e picture provides a balanced view of the development project.
e approach of measuring statistical distances using z scores can be used to
compare current year performance from last year performance. Such a comparison
is shown in Table 13.5. e tornedo chart is shown in Figure 13.11.
Negative z scores indicate the statistically significant reduction in problems.
e positive z score of CSAT spells significant improvement. e usual practical
–2 –1
CSAT
Complexity
Defect density
Scope creep
Schedule variance
Effort variance
0 1 2
z score
z score chart
3 4 5
Figure 13.10 z Scores: statistical distance from targets.
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