Central Limit Theorem

Every bit as important as the Law of Large Numbers is to sampling or diversification, the Central Limit Theorem helps to simplify matters regarding probability distributions to the point of heuristics in many cases. Here is what it is all about. Regardless of the distribution of the random variables in a sum or sample — for instance, (X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + ...) with BETA or Triangular distributions — as long as their distributions are all the same, the distribution of the sum will be Normal with a mean "n times" the mean of the unknown population distribution!

∑ (X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + ...) = S

S will have a Normal distribution regardless of the distribution of X:

E(X₁ + X₂ + X₃ + X₄ + X₅ + X₆ + ...) = E(S) = n * E(X_i) = n * μ

For n = ∑ i

"Distribution of the sum will be Normal" means that the distribution of the sample average, as an example, is Normal with mean = μ, regardless of the distribution of the X_i. We do not have to have any knowledge whatsoever about the population distribution to say that a "large" sample average of the population is Normal. What luck! Now we can add up costs or schedule durations, or a host of other things in the project, and have full confidence that their sum. or their average is Normal regardless of how the cost packages or schedule tasks are distributed.

As a practical matter, even if a few of the distributions in a sum are not all the same, as they might not be in the sum of cost packages in a WBS, the distribution of the sum is so close to Normal that it really does not matter too much that it is not exactly Normal.

Once we are working with the Normal distribution, then all the other rules of thumb and tables of numbers associated with the Normal distribution come into play. We can estimate standard deviation from the observed or simulated pessimistic and optimistic values without calculating sample variance, we can work with confidence limits and intervals conveniently, and we can relate matters to others who have a working knowledge of the "bell curve."

^[21]Chebyshev's Inequality: probability that the absolute distance of sample value X_i from the population mean is greater than some distance, y, varies by 1/y²: p(X_i - μ | ≥ y) ≤ σ²/y². Markov's Inequality applies to positive values of y, so the absolute distance is not a factor. It says that the probability of an observation being greater than y, regardless of the distance to the mean, is proportional to 1/y: p(X ≥ y) ≤ E(X) * (1/y).

^[22]Unlike the sample average, the sample variance, (1/n) ∑ [X_i - α(x)]², is not an unbiased estimator of the population variance because it can be shown that its expected value is not the variance of the population. To unbias the sample variance, it must be multiplied by the factor [n/(n-1)]. As "n" gets large, you can see that this factor approaches 1. Thus, for large "n", the bias in the sample variance vanishes and it becomes a practical estimator of the population variance.