Expected Value
and Average
The best-known statistic familiar to everyone is "average"
(more properly, arithmetic average), which is arithmetically equal to a specific
case of expected value. Expected value, E, is the most
important statistic for project managers. The idea of expected value is as
follows: In the face of uncertainty about a random variable that has values over
a range, "expected value" [14] is the "best" single number to represent a range of estimated value of
that random variable. "Best" means expected value is an unbiased maximum
likelihood estimator for the population mean. [15] We will discuss unbiased maximum likelihood estimators
more in subsequent paragraphs. Reducing a range of value to a single number
comes into play constantly. When presenting a budget estimate to project sponsors, more often than not, only one
number will be acceptable, not a range. The same is true for schedule, resource
hours, and a host of other estimated variables.
Mathematically, to obtain expected value we add up all the values
that a random variable can take on, weighting or multiplying each by the
probability that that value will occur. Sound familiar? Except for "weighting
each value by the probability," the process is identical to calculating the
arithmetic average. But wait: in the case of the arithmetic average, there
actually are weights on each value, but every weight is 1/n. Calculating
expected value, E:
where pi is the probability of specific value Xi occurring. If pi =
1/n, where "n" is the number of values in the summation, then E(X) is mathematically equal to the calculation of arithmetic average.
Consider this example: a work package manager estimates a task
might take 2 weeks to complete with a 0.5 probability, optimistically 1.5 weeks
with a 0.3 probability, but pessimistically it could take 5 weeks with a 0.2
probability. What is the expected value of the task duration?
Check yourself on the "p"s:
p1 + p2 + p3 = 0.3 + 0.5
+ 0.2 = 1
There are a couple of key things of which to take note. E(X) is not a random variable. As such, E(X) will have a deterministic value. E(X) does not have a distribution. E(X) can be manipulated mathematically like
any other deterministic variable; E(X)
can be added, subtracted, multiplied, and divided.[16]
Transforming a space of random variables into a deterministic
number is the source of power and influence of the expected value. This concept
is very important to the project manager.
Very often, the expected value is the number that the project manager carries to
the project balance sheet as the estimate for a particular item. The range of
values of the distributions that go into the expected value calculation
constitutes both an opportunity (optimistic end of the range) and a threat
(pessimistic end of the range) to the success of the project. The project
manager carries the risk element to the risk portion of the project balance
sheet.
If X is a continuous
random variable, then the sum of all values of X morphs into integration as we saw before. We know that
pi is the shorthand for the probability function f(X | Xi), so
the expected value equation morphs to:
E(X) = ∫ X * f(X | Xi) dX, integrated over all values of "X"
Fortunately, as a practical matter, project managers do not
really need to deal with integrals and integral calculus. The equations are
shown for their contribution to background. Most of the integrals have
approximate formulas amenable to solution with arithmetic or tables of values
that can be used instead.
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