Risk
Management
Part 2 – Variability and
Risk: Twin Sons of
Different Mothers
by
Donald E. Shannon, PMP, CFCM, CPCM, DML
The Contract Coach, Albuquerque, NM
Disclaimer
Information in this presentation makes reference to various
software products. This should not be interpreted as a
recommendation or endorsement by any of the sponsors of any
one product. Individuals should conduct appropriate research to
identify a product that best meets their specific needs. Where
appropriate, credit has been given to the software OEM
especially where screen shots of their products have been used.
A listing of the products commonly used to accomplish the
simulation and scheduling functions described herein is included
at the end of this presentation
Where Are We Going?
 Quantitative (numerical or statistical) analysis
 What is variability
 The nature of random variables
 How they are depicted
 Confidence intervals
 Variability (uncertainty) in program / project management
 Why estimates never seem to be right
 Adding risk to uncertainty = real world variability
 Adding risk and uncertainty to our cost/schedule model
Variability
Why things are never
exactly alike….
Trivia questions:
1.
2.
3.
Who is the driver in the above?
What year was the photo taken and how can you tell?
On average how long does a F-1 pit stop take in 2014
Trivia questions:
1.
2.
3.
Who is the driver in the above? – Kimi Raikkonen
What year was the photo taken and how can you tell? – 2008. Because Kimi’s car is #1 and he won
the F1 championship in 2007 giving him that number for the following year.
On average how long does a F1 pit stop take in 2014? – 3 Seconds
A 3-Second Pit Stop?
Let’s Fact Check That Claim
Pit Stop Times
2014 F-1 Grand Prix of Spain
 Is the average pit stop time




actually 3.0 seconds?
Data from 13 pit stops during
the most recent F-1 race
Most consistent team was
Williams with 3 stops each of
which was 3.0 seconds
Fastest team was Red Bull with
an average stop of 2.675 seconds
Further analysis would likely
show that the average time of
2.8 in this sample was within the
expected range and there is not
enough evidence to reject the
supposed average of 3.0 seconds
Driver
Team
VET
RB
3.1
2.3
2.7
RIC
RB
2.9
2.4
2.65
GRO
LOT
2.9
-
2.9
MAS
WIL
3.0
-
3.0
BOT
WIL
3.0
3.0
3.0
HAM
MER
3.8
4.3
4.05
ROS
MER
2.8
3.0
2.9
2.68
2.98
3.04
Average
Stop
1
Stop 2
Average
Variability (Uncertainty)
 The output of any task when repeated will vary to some
degree from its predecessors or successors
 Size, weight, volume etc
 Performance time
 Your morning commute is a good example
 Depending on road conditions and traffic it may vary
significantly from day-to-day
 Even when conditions are perfect the time still varies by a few
seconds or minutes
Random Variables

We call these probabilistic events
because the answer can not be
described by a single value.
 The value of these events (time,
cost, etc.) is variable.
 So how do we describe variability




We establish ranges of possible
values
We define measures of central
tendency (average, mean,
median, mode)
We identify how the values are
spread out (variance)
We quantify how far possible
values are displaced from the
center (standard deviation or zscore)
On Closer Examination
Reviewing some terms you probably know
 The pit stop range (low to





high) was 2 Seconds (2.3 – 4.3)
3.0 was the most common
(mode) time recorded
3.0 was the middle value
(median)
3.04 was the arithmetic
average (also called mean or μ)
of all the times recorded
The standard deviation (σ) was
.54
All data lies within +/- 3 σ
Pit Stop Time
8
7
6
5
4
3
2
1
0
μ
-1 σ
2.3 - 2.5 - 2.8 2.5
2.7
3.0
1σ
3.1 3.3
2σ
3.3 3.5
Time
3.5 3.7
3σ
3.8 - 4.0 4.0
4.3
Random Variables
 Random variables are
either discrete (whole
numbers 1,2,3) or
continuous
 There are separate rules for
displaying the data for
each
 Discrete RVs are
typically shown in
histograms (bar charts)
 Continuous RV are
typically shown in
probability distribution
functions (PDF) or
cumulative distribution
functions (CDF)
Histograms
 Data may be either continuous




or discrete
Discrete data is grouped into
classes or “bins”
Bin values displayed on x-axis
Count or number in class
displayed on y-axis
A smoothed line enclosing the
tips of each column may
provide insight as to the
underlying distribution
(Normal, Lognormal, Beta,
etc.
Probability Distribution Function






The total area under the curve is always
equal to 1 (100%)
The tallest point of the curve is the mode
(most often occurring value)
In symmetric distributions the mean and
median are co-located with the mode
If the distribution is not symmetric we
say it is ‘skewed’ with the direction of
the skew (left or right) being where the
preponderance of the values reside.
The shape of the distribution may give a
clue to the underlying distribution but
beware – sometimes the shape is
misleading
Values on the x-axis may be actual or z
(standard deviations). If z scores those
to the left of the mean are negative and
those to the right are positive.
Cumulative Distribution Function
 Values start at zero
 Values end at 1 (or 100
percent)
 Value on the y axis (vertical) is
at cumulative probability for
the value on the x axis i.e., the
sum of the probabilities from
zero to the selected x value
 The shape of the curve is an
indication of the shape of the
probability distribution
function (PDF).
Confidence Intervals
 What is the likelihood of an
event happening?
 Single point estimate of
continuous variable is
undefined
 Therefore we phrase the
question as:
 Probability that x will exceed
some value
 Probability that x is less than
some value
 Probability that x is between a
and b.
Using a PDF to Answer Questions
 Consider the normal distribution
 Average IQ is 100
 Standard deviation is 10 points
 68% lie between 90 – 110
 95.4% lie between 80 – 120
 99.74 lie between 70 – 130
 PDFs help us answer questions
such as:
 What is the probability of
someone having an IQ between
110 and 120? (13.6%)
 What is the probability of an IQ of
125? (Trick question. If IQ is a
continuous random variable the
probability of one precise value is
undefined)
Uncertainty in Program
Management
Why things always take longer and cost more than you planned ….
Murphy’s Law
 Often stated as: “Anything that can go wrong, will go wrong”
 References to the principle date to at least 1877
 Famous corollaries include:
 If anything just cannot go wrong, it will anyway
 … Usually in the worst possible (or most inconvenient) time in the
worst possible way.
 Things will be lost or damaged in inverse relationship to their value
or need.
 But the real culprit in all of this is nature’s uncertainty.
Uncertainty
 Uncertainty is an admission that the
outcome of any event is a random variable
 Therefore the data describing that event
(cost, performance time etc.) will be variable
 Forecasting the outcome of an uncertain
event can only be stated probabilistically
 Probabilistic outcomes typically involve
 A percent likelihood the event will occur
 Sometimes an indication of underlying
distribution (uniform, binomial, normal,
etc,)
 When we plan for uncertain events we
typically plan for the “most likely” outcome
 “On average” it costs “x” or takes “y” days
 Sometimes we do better
 Sometimes we do worse
Please Fasten Your Seatbelts
Statistics Airline Flight 101 Now Departing
 We can’t talk about uncertainty and quantitative risk
management without delving into statistics.
 Keep in mind that the goal is not to teach you the math
but the underlying concepts and terminology
 We’ll let the computer do the math – all we care about is
the output
 But to understand the output you have to know the
concepts.
Uncertainty in Program
Management
 The time to perform a task (e.g., a pit
stop) is a random variable
 Because of this nature, when we
express a task duration as a specific
value (single point estimate) we
ignore uncertainty
 Single point estimates are typically
“most likely” values
 Better estimates are possible if we
describe 3 points
 Minimum (Best Case)
 Maximum (Worst Case)
 Most Likely
 The three point estimate is especially
useful if dealing with ‘expert opinion”
Task Performance Time
Most Likely (m)
9
6
Minimum (a)
15
Maximum (b)
Triangular Distribution
 Triangle distribution is a continuous
frequency distribution often used to
model random variables (cost or
performance time) in program
management
Most Likely (m)
 Fast
 Easy to use
 Provides reasonable accuracy
9
 Tends to be slightly optimistic i.e.,
values returned tend to be a little less
than what ends up being the case
 Formula
 Mean = a+m+b ÷ 3 = 30 ÷3 = 10
 Mode = m = 9
 STDEV = σ=√(a2+m2+c2-am-ab-mc)
18
15
6
Minimum (a)
μ =10
Maximum (b)
σ=1.58
Note: Don’t worry about doing the number crunching – that’s why we have
software to do this for us!
Triangular Distribution
 Probability of a value of x is given





by the area of interest.
Example. What is the probability
that x is less than 7.25?
f(x) = 2(x-a)
(b-a)(m-a)
f(x) = 2 x (7.25-6)
(15-6)(9-3)
f(x) = 2.5 / 54 = .0462963
p(x) = ½ * .0462963 * 1.25 = .029
or 2.9%
Most Likely (m)
9
6
15
7.25
Minimum (a)
μ =10
Maximum (b)
σ=1.58
Note: Don’t worry about doing the number crunching – that’s why we have
software to do this for us!
Special Use Of the Triangular
Distribution
 Some scheduling applications use a




“weighted” version of the triangle
The PERT Triangular Distribution
distribution called the “PERT”
Most Likely (m)
distribution
PERT (Program Evaluation and
Review Technique) adds (statistical)
9
emphasis to the “Most Likely”
value and weigh it 4 times as likely
as the Minimum or Maximum
Average (mean) is then = a+4m+b /
15
6
6
Criticism is that PERT tends to be
Minimum (a)
μ
Maximum (b)
optimistic
=9.5
σ=1.50
Second criticism is that PERT
ignores likelihood of events outside
3 points (closed interval)
Other Distributions Used
In Program Management
 Beta Distribution is a typically
mound shaped continuous
distribution
 Very flexible and can take on a
number of shapes depending
on the parameters used
 May be either closed form
(upper example) or open form
(lower example)
 Is less optimistic than
triangular
Other Distributions Used
In Program Management
 Lognormal Distribution is a
typically mound shaped
continuous distribution
 Very flexible and can take on a
number of shapes depending
on the parameters used
 Open ended toward the upper
side allowing for extreme
values
 Is the resultant distribution
when two triangular
distributions are combined.
Why Estimates of Cost or
Schedule End Up Being Wrong
Murphy was an optimist ……
Leading Factors
 Optimism bias
 Everything takes longer than planned
 Everything costs more than you thought
 Improper estimates
 Corrupt data
 Failure to properly consider risk
Optimism Bias
 Values predicted are “better” or
“rosier” than real world
experience would show
 Sources of Optimism Bias
 Expert Opinion on average tends
to be as much as 25% optimistic
for commonly performed tasks1
 Use of the triangle distribution
when another (e.g., Beta) is
more appropriate
 The sequence of collecting
estimates for Minimum,
Maximum and Most likely can
introduce bias2
1 Bias in Memory Predicts Bias in Estimation of Future Task Durations, Roy 2007
2 Herding Cats: Why 3 Point Estimates Create False Optimism
Improper Estimates
 Estimates are commonly built
on mathematical models
 Models must be appropriate for
the application selected
 Models must be used within the
range they were designed to
predict
 Another estimating technique is
to compare something new to
something already done
 Effort is often scaled by some
percentage
 Only as good a technique as the
data on which it is based and the
similarity of the two
tasks/projects
Risk
 Risk adds an additional element
to uncertainty
 Broadens the range of possible
values – usually to:
 Increase costs
 Extend performance time
 Risk is an “event” such that it
either takes place or does not
take place
 Risk events have 3 parameters
 Polarity ( +/- )
 Likelihood – their chance of
occurring
 Impact – the cost, delay, or
opportunity associated with the
risk
Risk as a Variable
 Risks are a two-step process
 Step 1 … Does risk occur
 Step 2 … What is the impact
Delay associated with risk
Most Likely (m)
 Risk impacts are often
modeled using the triangular
distribution
3
 Minimum impact (cost, delay,
or both)
 Most likely impact
 Worst case impact
 Data comes from risk register
as was completed in Part 1
1
Minimum (a)
6
Maximum (b)
When Worlds Collide
What happens if we combine a variable (cost or schedule) and risk ….
How Risk Changes an Event
 If the risk event does not
happen, no change to event
 If the risk event happens then
the task performance time will
be the sum of two probabilities
 Task Performance time (no
risk)
 Risk Impact
 Summing probability
distributions is a bit tricky
Task Performance Time
Most Likely (m)
9
15
6
Minimum (a)
Maximum (b)
+
Delay associated with risk
Most Likely (m)
3
1
Minimum (a)
6
Maximum (b)
Combining Probability
Distributions
Using Method of Moments
to Combine PDF’s
 Method of Moments Technique
 Analytical technique
 Used to calculate the “moments” of
the combined distribution
 The resultant distribution from
adding two triangular distributions is
a lognormal distribution.1
 The Moments of that are:
 Mean = μ = μ1 + μ2 … μn
 Variance = σ2 = it depends2
 Skewedness1 = ϑ =
 Kurtosis = κ = 12/5 = 2.387
 The math needed to calculate these
is outside the scope of this
presentation
1. Analytic Method for Cost and Schedule Risk Analysis, Raymond P.
Covert, NASA, 5 April, 2013, pp 34 - 37
2. Calculating variance for the sum of two distributions is complicated
when the two distributions are correlated. Formula shown is for
correlated data
Using Simulation to
Combine PDF’s
 Monte Carlo Technique
 Generate random number
between 0 and 1
 Convert random number to
duration based on triangular
distribution 6,9,15
 Generate second random number
between 0 and 1
 Convert second random number
to duration based on triangular
distribution 1,3,6
 Add duration 1 to duration 2
 Record value
 Repeat numerous times
 Compute statistics such as mean
etc. from collected data
Why Do I Favor Simulation?
 Mostly because it provides
equivalent results without having
to be a math major.
 Praised for
 Ability to provide statistics of a
simulated CDF or PDF formed by
complex modeling of random
variables
 Ease of use
 Criticized for:
 Non-uniform sampling
 Unable to correlate two
distributions using Pearson
product – moment correlation
coefficients
 Does not provide reasonable
results with small number of trials
Simulation output of 3,460 iterations
© Intavar Institute Risky Project 6.0
Going Forward
Adding risk and variability to our cost and schedule estimates ….
Time to Build Our Model
 Our model of the effort
(contract, project, etc) must
include:
 Tasks to be performed
 Resources to be used in
accomplishing those tasks
 Labor
 Materials
 Other (Travel, subcontracts,
etc.)
 Cost data for the each of the
resources and tasks
 Schedule data
 Risk Data
Tools to be Used
 Excel or Numbers spreadsheet
 Project, Primavera, FastTrack,
or other scheduling application
 Simulation program such as:
 @Risk
 Risky Project
 Primavera
 Full Monty
What We Will Do
 Create Work Breakdown






Structure
Expand (decompose) WBS into
various tasks
Identify resources for each
task
Create risk register
Map risks to each task
Determine cost baseline with
and without risk
Create probabilistic estimates
for cost at completion and
completion date.
Summary
 The language of quantitative




analysis is statistics – learn the
language
Schedule (task) duration and
cost are random variables
Nothing is certain until it
occurs
Risk is another random
variable that modifies cost,
schedule, or both
You don’t need a PhD to do
the math … let the software do
that while you interpret the
results
Produced by: The
Contract Coach
The Contract Coach
5338 La Colonia Dr NW
Albuquerque, NM 87120
(505) 259-8485
http://www.contract-coach.com
Monte Carlo Simulation Software
 MS Excel Add-in
 @Risk
 Oracle Crystal Ball
 Risk Solver
 General Purpose Products
 Analytica
 GoldSim
 Reno
 Oracle Crystal Ball
 SPSS
 MS Project Add-in
 Full Monty
 Risky Project (also
supports Primavera P6)
 Enterprise Systems
 Risk+ (Deltek)
 Primavera EPPM &
Primavera Risk (Oracle)