231
Chapter 14
Law of Compliance:
Uniform Distribution
In sharp contrast to the bell curve, the uniform distribution looks plain, flat, and
simple but has some interesting applications. ere are no tails in the uniform
distribution. ere is no peak either. Uniformly distributed processes are rare in
manufacturing, but nearly uniformly distributed processes exist in IT services.
Uniform distribution is a continuous distribution bounded between two limits,
A and B. e probability density function (PDF) may be stated as follows:
F x
A B
A x B( ) =
−
≤ ≤
1
for
(14.1)
e plot of uniform distribution, shown in Figure 14.1, is a rectangle.
Hence, uniform distribution is also known as the rectangular distribution. It
may be recalled that by integrating a PDF, we get a cumulative distribution func-
tion (CDF). It is also known that when we integrate a rectangle, we get a triangle.
In this case, the PDF is rectangle. Integrating the PDF, we get a triangle which is
the CDF. Obviously, the CDF is a triangle.
e PDF has the following statistics:
Mean = (A + B)/2
Median = (A + B)/2
Range = B – A
Variance =
−( )B A
2
12
232 ◾ Simple Statistical Methods for Software Engineering
Standard deviation =
−( )B A
2
12
Skewness = 0
Kurtosis = 9/5
Relative kurtosis = −1.2
e CDF is defined as follows:
F x
x A
B A
A x B( ) =
−
−
≤ ≤for
(14.2)
e CDF is a triangle, as shown in Figure 14.2.
A
Lower
specification
B
Upper
specification SLA
Service delivery time
Probability
Equal probability
of delivering inside
the bounds
Figure 14.1 Probability density function (PDF) of uniform distribution.
A
Lower
specification
B
Upper
specification SLA
Service delivery time
Probability
Cumulative
probability
Figure 14.2 Cumulative distribution function (CDF) of uniform distribution.
Law of Compliance ◾ 233
Bounded Distribution
A principal feature of the uniform distribution is its boundary. It is bounded on
both sides, in sharp contrast with the unbounded bell curve. To qualify as a uni-
form distribution function, three criteria must be met:
1. Bounding limits must be “final minimum,” clear, and not elastic or vague
2. Containment criterion 100%
3. Uniform probability criterion
In practice, there could be challenges in meeting criterion 3; there could be
turbulence between the limits.
ere are many bounded phenomena in software projects. Customer satisfac-
tion data are bounded between 1 and 5. SLA compliance is bounded between
0% and 100%. Time to repair is bounded between 0 and agreed upon maximum
time; hence, criterion 1 is met. e question in criterion 3 is uniformity. e pro-
cess could be any other bounded distributions, beta or truncated Gaussians. Here
comes a need for approximation and simplification. Uniform distribution is simpler
and hence is preferred in most cases.
When the customer does not specify boundary, and it is left to the process QA
to define the boundary based on data, challenges arise. In particular, estimating the
upper bound could be a challenging problem. Data may not show a sharp edge. If
we are using only sample data, then the problem of fixing upper bound is analogous
to the German tank problem relevant to the World War II situation [1].
German tanks were produced according to a uniform distribution [1, N]. e
number of tanks captured was n, a mere sample. Can we estimate N from the
sample data?
N m
m
n
= − +1 (14.3)
where N is the upper bound of the uniform distribution [1, N], n is the number of
tanks captured, and m is the Largest serial number in the sample.
Later, after the war, statistical estimations were found to be much closer to the
truth than intelligence reports, four to one.
Random Number Generators
Random numbers follow uniform distribution. However, it is difficult to gener-
ate random numbers that fit perfectly into a uniform distribution. ere are sev-
eral random number generators (RNGs), but they predict uniform distribution
with varying levels of success. For reliable results in simulation, we need perfect
234 ◾ Simple Statistical Methods for Software Engineering
random numbers, which means their PDF must be super uniform. A few RNGs
aim at smooth uniform PDFs, whereas others accommodate deviations from per-
fect smoothness. To illustrate this point, Figure 14.3 shows a histogram of 1000
random numbers generated by Excel function RAND(). We can see a histogram
with perturbed profile, bounded between 0 and 1. ese are pseudorandom num-
bers, as best as a practical tool can predict.
Shuttle Time
If a shuttle bus has a cycle time of 40 minutes, the waiting time is uniformly distrib-
uted between 0 and 40. A passenger who reaches the bus stand in a random hour and
is unaware of when the last shuttle left is bound to wait up to 40 minutes. From his
viewpoint, the arrival of the shuttle is uniformly distributed from 0 to 50 minutes.
Parkinson’s Law
If a service manager assigns a task to his team and suggests that the team completes
the task within an interval of 4 to 6 days, the team will take anywhere between 4
and 6 days. e team will exploit the goal window and not exercise its own natural
capability to finish the task in a naturally possible time. For all we know, the team
may be able to finish the job in less than 4 days, but human behavior is to stretch
the job according to Parkinson’s law which states,
Work expands to ll the time available.
LSL USL
0.00 0.16 0.32 0.48 0.64 0.80 0.96
Figure 14.3 Distribution of random numbers.
Law of Compliance ◾ 235
If teams exercised their natural capabilities, the PDF of completion time would
have a peak. Because the team negotiates time to meet specified goals, we end up
with a performance that does not have character.
Censored Process
If components out of specifications are removed, the reaming lot shows nearly uni-
form distribution between the specification limits. e censored lot tends to be
more uniform if the original lot shows wide variation and if the specification lim-
its are stringent. An example is when a semiconductor component manufacturer
screens best pieces from the line and sells them at premium prices as close tolerance
devices. He downgrades the rejected components and sells them at a lower price.
e premium components after censoring show uniform distribution.
Perfect Departure
From an auditorium, if people leave in perfect queues, the departure is uniformly
distributed. e probability of people crossing the gate is uniformly distributed
between 0 and a finite time that depends on the number of people and width of the
gate. When people try to break the queue and rush out, the departure is skewed,
the worst case being a stampede.
Estimating Calibration Uncertainty
with Minimal Information
During the calibration of measuring equipment, we need to assess uncertainties.
ere are two types of uncertainties affecting measurement: type A is determined
from data, and type B is guessed. In type B estimates, we might only be able to
estimate the upper and the lower limits of uncertainty. We have to assume the value
is equally likely to fall anywhere in between, that is, a rectangular or uniform dis-
tribution. e standard uncertainty for a rectangular distribution is equal to
a
3
,
where a is the semirange between the upper and the lower limits.
Estimation of uncertainty using the uniform distribution is relevant in the fol-
lowing cases:
◾ Digital resolution uncertainty
◾ RF phase angle
◾ Quantization error
◾ As an expression of ignorance
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