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Variance and Standard DeviationVariance and Standard DeviationVariance 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] 2428
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