Law for Estimation ◾ 249
Parameter Extraction
Precise parameters make precise models. In the case of the triangular model, we
use expert judgment, Delphi, or wideband Delphi in the absence of data. Group
judgment is more robust, resulting in a less skewed triangle. If data were available,
we use histograms and visually match a triangle first and then, if called for, a least
square error method after optimally binning the histogram. Going for MLE or
other rigorous techniques is not warranted for triangular models. If more precise
judgment of risk is involved, then the beta distribution should be chosen or the tri-
angular model must be refined as van Dorp and Kotz [4] have done: ey extended
the model into a four-parameter version that allows even J-shaped forms. It may be
seen that the TD offers all the facilities available in classical distribution models; it
supports risk estimation and simulation; It is used in 3 point estimation and first
order approximation of processes. All this is accomplished using elementary linear
consideration.
Box 15.2 a crysTal clear world
e triangle offers a tailless view of process. is is a crystal clear world
without the ambiguity. e presence of tails makes comparing two pro-
cesses a complex affair; one often needs hypothesis testing and abstruse
rules. It is quite plain with triangles, and one can make commonsense-
based judgments.
For example, let us consider a productivity model defined by a triangle
(30, 40, and 60) lines of code (LOC) per person day. To answer a question
whether a productivity data point 61 belongs to this process or an outlier is
rather easy. e data point in question is outside the triangle.
Had we used a bell curve, the answer is not so easy; at least it will not be
a straightforward reply. One would say there is chance p that the data point
LSL
Risk
probability
Success
probability
Figure 15.8 Risk estimation using triangular distribution.
250 ◾ Simple Statistical Methods for Software Engineering
Review Questions
1. Why is the triangular distribution preferred in the place of uniform distribu-
tion for estimation models?
2. Can we represent data skew in a triangular model?
3. Compare the way of calculating expected value using the PERT formula with
the method of calculating the expected value using the mean of triangular
distribution.
4. Compare beta distribution with the triangular distribution.
5. Why is the triangular distribution popular in project management?
Exercises
1. In a triangular distribution model for productivity, a = 30, b = 90, and
c = 55. e numbers are LOC/per day, standing for productivity in software
development. e naming conventions are shown in Figure 15.2. Calculate
skew.
2. Calculate mean and median productivity in the above-mentioned situation.
3. If the threshold productivity is 40, what is the risk in productivity perfor-
mance in the above context?
4. What is the standard deviation of the distribution in the above example?
5. Calculate risk using a Gaussian model using the formulas given in Chapter
13 “Bell Curve,” making use of the standard deviation you found in Exercise
4 and the mean you found in Exercise 2.
References
1. S. Kotz and J. R. van Dorp, BEYOND BETA: Other Continuous Families of Distributions
with Bounded Support and Applications, World Scientific Publishing Co Pte Ltd, 2004.
2. A. S. Wahed, e family of curvi-triangular distributions, International Journal of
Statistical Sciences, 6(special issue), 7–18, 2007.
belongs to the process. en we will apply a policy to decide whether the p
value is less than the “critical value.” If the p value is less, then we would say
that the data point is significantly different from the process. It is a round-
about way of saying that p is outside and is certainly confusing to many who
would rather have a simple and transparent answer.
e triangle presents a crystal clear view of process and facilitates straight
decision making.
Law for Estimation ◾ 251
3. M. Brizz, A skewed model combining triangular and exponential features: e two-
faced distribution and its statistical properties, Austrian Journal of Statistics, 35(4),
455–462, 2006.
4. J. R. van Dorp and S. Kotz, A novel extension of the triangular distribution and its
parameter estimation, Journal of the Royal Statistical Society, Series D (e Statistician),
51(1), 63–79, March, 2002.
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