Grand Social Law ◾ 215
Left tail = 0.0117
Right tail = 0.3446
Total risk = 0.3563
In previously mentioned risk analysis, cost escalation seems to be the dominat-
ing risk.
Table 13.1 Risk Calculation for Tails
Distance from Mean Tail Area % Tail Area
0.2 0.420740 42.074
0.4 0.344578 34.458
0.6 0.274253 27.425
0.8 0.211855 21.186
1.0 0.158655 15.866
1.2 0.115070 11.507
1.4 0.080757 8.076
1.6 0.054799 5.480
1.8 0.035930 3.593
2.0 0.022750 2.275
2.2 0.013903 1.390
2.4 0.008198 0.820
2.6 0.004661 0.466
2.8 0.002555 0.256
3.0 0.001350 0.135
Data 13.1 Two-Sided Risk Estimation
Metric Effort Variance (%)
Historic Data
Mean 14
Sigma 15
Specification Limits
USL 20
LSL −20
Left tail 0.011705
Right tail 0.344578
Total risk 0.356284
216 ◾ Simple Statistical Methods for Software Engineering
Combining Normal Probability Density
Functions (PDF): The Law of Quadrature
A very useful property of the normal distribution is that we can easily combine
several normal PDFs using a simple rule set:
Add the means to obtain the overall mean.
Add the variances to obtain the overall variance.
For example, the schedule performance of milestones can be combined using
this property. e overall schedule for the project is the sum of schedules of mile-
stones. e overall variance in the project schedule is the sum of individual mile-
stone schedule variances. An example is available in Table 13.2.
e root cause for risk is variance, and Table 13.2 provides variance data across
the project at every declared milestone. ese milestones constitute on the critical
path. eir variances are added by using the law of quadrature to obtain the overall
Box 13.3 gaussian smoothening
In image reconstruction Gaussian distribution is used.
In the domain of electromagnetic radiation, antenna beam widths are
Gaussian reconstructed from the half power beam widths, which are easier to
measure. e empirical construction of the beam with multiple data points is
time consuming and looks less attractive when Gaussian smoothening is an
accepted scientific practice. Gaussian smoothening saves time and money, and
yet succeeds in constructing truth. In image processing, Gaussian smoothening
is widely used.
An example of a common algorithm used to perform image smoothening
is Gaussian. Each pixel is convolved with a Gaussian kernel and summed up;
the result is suppression of noise, better signal-to-noise ratio, and better qual-
ity image. e bell curve is used to beat noise.
In digital signal processing, the Gaussian filter retrieves truer signals. In
spatial smoothening MRI images, Gaussian smoothening is used to enrich the
picture. Gaussian smoothening blurs the noise. e degree of smoothening is
determined by the standard deviation of the Gaussian. Larger standard devia-
tion Gaussians, of course, require larger convolution kernels to be accurately
represented. “e Gaussian outputs a ‘weighted average’ of each pixel’s neighbor-
hood, with the average weighted more towards the value of the central pixels.”
During the reconstruction of scanned images, Gaussian smoothening is
like a low-pass filter. e Gaussian window is an attractive option for volume
visualization in CT scans [8].
Grand Social Law ◾ 217
variance in the project. e perception of risks pervades the project during its life
cycle. Figure 13.8a shows the bell curves for milestone deliveries.
e overall variance is filled in black. If the milestones were nonnormally dis-
tributed, the overall variance in a project is typically obtained by a procedure called
Monte Carlo simulation. e overall variance is obtained in an elegant and simple
method because we assume normal distribution in our case.
e overall project delivery variance and mean are shown separately in Figure 13.8b
for further analysis. e delivery day is marked as 226, and the tail beyond is known as
Table 13.2 Milestones Schedule Estimates
Milestone
No. Milestone
Estimated Schedule Days
Mean Cumulative Sigma Variance
1 Start 0 0 3 9
2 Requirement
gathering
3 3 1 1
3 Requirement
documentation
4 7 1 1
4 High level design 12 19 2 4
5 Detailed design 20 39 2 4
6 Code for selected
10modules
40 79 3 9
7 Code for next
10modules
60 139 3 9
8 Code for remaining
modules
30 169 2 4
9 System test 15 184 3 9
10 Integration test 12 196 2 4
11 User acceptance test 12 208 1 1
12 Finish 3 211 3 9
Overall mean 211
Overall variance 64
Overall sigma
(sqrtofvar)
8
Note: The milestones above are on the critical path.
218 ◾ Simple Statistical Methods for Software Engineering
project schedule risk. From the Gaussian formula, risk can be computed as 3%. From
the graph, we can also see the following three PERT values used in project estimation:
Optimistic value = 186
Pessimistic value = 236
Most likely value (mean) = 211
Using the golden rule, we can also estimate the delivery date, as follows:
Delivery date
opt pess most likely
=
+ +t t kt
6
where k is a constant, normally taken as 4 but can be changed depending on the
nature of the project.
Delivery date
Risk 3 %
0.06
0.40
MS 1 MS 2 MS 3 MS 4 MS 5 MS 6 MS 7 MS 8 MS 9 MS 10 MS 11 MS 12 Overall
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0
0
8
16
24
32
40
48
56
64
72
80
88
96
104
112
120
128
136
144
152
160
168
176
184
192
200
208
216
224
232
240
248
0.05
0.04
0.03
0.02
0.01
0
186
196
206
216
Project day
Project day
Milestone delivery bell curves Milestone delivery bell curves
226
236
246
(a)
(b)
Figure 13.8 (a) Milestone delivery bell curves and (b) project delivery bell curve.
Grand Social Law ◾ 219
We feel that the above example illustrates the most admirable capability of
Gaussian distribution. It has made simulation a transparent job, which otherwise
stays as a black box technique using sophisticated but less understood tools.
An Inverse Problem
Let us revisit Equation 13.2, which defines the Gaussian PDF. e integration of
this PDF leads to the cumulative distribution function (CDF), F(x), defined as
follows:
F x e dx
x
x
( ) =
−
−
−∞
∫
1
2
2
2
π
µ
σ
(13.4)
is CDF can be plotted using Excel NORMDIST(x, mean, SD, 1). As an
example, we plot the CDF for requirement volatility with mean = 3.3% and stan-
dard deviation = 4% in Figure 13.9. e inverse problem is given that the Gaussian
F(x) = 0.78; what is x? is question and its answer are shown in Figure 13.9. e
answer is marked as 6.39. is solution has been obtained graphically.
We can use the Excel function NORMINV to find x, as follows:
NORMINV(0.78, 3.3, 4) = 6.3888
F(x) = 0.78
x = 6.39
–10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
–8 –6 –4 –2 0 2 4
Requirements volatility %
Cumulative probability
6 8 10 12 14 16
Figure 13.9 Gaussian cumulative distribution function of requirements volatility.
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