Variance and
Standard Deviation
Variance and standard deviation are measures of the spread
of values around the expected value. As a practical matter for project
practitioners, the larger the spread, the less meaningful is the expected value
per se.
Variance and standard deviation are related functionally:
SD = sqrt(VAR) = √VAR
where VAR (variance) is always a positive number and so,
therefore, is SD (standard deviation). Commonly used notation is σ = SD, σ2 = VAR.
Variance is a measure of distance or spread of a probable outcome
from the expected value of the outcome. Whether the distance is "negative" or
"positive" in some arbitrary coordinate system is not important for judging the
significance of the distance. Thus we first calculate distance from the expected
value as follows:
Distance2 = [Xi - E(X)]2
The meaning of the distance equation is as follows: the
displacement or distance of a specific value of X, say for example a value of "Xi", from the expected value is calculated as
the square of the displacement of Xi from E(X). Figure
2-5 illustrates the idea. Now we must also account for the probability of X taking on the value of "Xi".
Probabilistic distance = p(Xi) * [Xi - E(X)]2
Now, to obtain variance, we simply add up all the probabilistic
distances:
VAR(X) = σ2(X) =
∑ p(X) * [Xi - E(X)]2 for all "i"
which simplifies to:
VAR(X) = σ2(X) =
E(X2) - [E(X)]2
To find the standard deviation, σ, we
take the square root of the variance.
Let's return to the example of task duration used in the expected
value discussion to see about variance. The durations and the probability of
each duration are specified. Plugging those figures into the variance
equation:
σ2(task duration D) = 0.3 * (1.5 - 2.45)2 + 0.5 *
(2 - 2.45)2
+ 0.2 * (5 - 2.45)2
where 2.45 weeks is the expected value of the task duration from
prior calculation, σ2(task duration D) = 1.67 weeks-squared, and σ(task duration D)
= 1.29 weeks. [18]
It is obvious that variance may not have physical meaning,
whereas standard deviation usually does have some physical meaning. [19]
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