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ThreePoint Estimate Approximations
Quite useful results for project statistics are obtainable
by developing threepoint estimates that can be used in equations to calculate
expected value, variance, and standard deviation. The three points commonly used
are:

Most pessimistic value that yet has some
small probability of happening.

Most optimistic value that also has some
small probability of happening.

Most likely value for any single instance
of the project. The most likely value is the mode of the
distribution.
It is not uncommon that the optimistic and most likely values are
much closer to each other than is the pessimistic value. Many things can go
wrong that are drivers on the pessimistic estimate; usually, there are fewer
things that could go right. Table
25 provides the equations for the calculation of approximate values of
statistics for the most common distributions.
Table 25: Statistics for Common
Distributions
Statistic 
Normal ^{[*]} 
BETA^{[**]} 
Triangular 
Uniform^{[***]} 
Expected value or mean 
O + [(P  O)/2] 
(P + 4 * ML + O)/6 
(P + ML + O)/3 
O + [(P  O)/2] 
Variance, ϭ^{2} 
(P  O)^{2}/36 
(P  O)^{2}/36 
[(O  P)^{2} + (ML  O) * (ML  P)]/18 
(P^{3}  O^{3})/ [3 * (P  O)]  (P 
O)^{2}/4 
Standard deviation, ϭ 
(P  O)/6 
(O  P)/6 
Sqrt(VAR) 
Sqrt(VAR) 
Mode or most likely 
O + [(P  0)/2] 
By observation or estimation, the peak of the curve 
By observation or estimation, the peak of the curve 
Not applicable 
Note: O optimistic value, P = pessimistic value, ML = most
likely value. 

It is useful to compare the more common distributions under the
conditions of identical estimates. Figure 26 provides the illustration. Rules of thumb can
be inferred from this illustration:
In addition to the estimates given above, there are a couple of
exact statistics about the Normal distribution that are handy to keep in
mind:

68.3% of the values of a Normal distribution fall within
±1σ of the mean value.

95.4% of the values of a Normal distribution fall within
±2σ of the mean value, and this figure goes up to
99.7% for ±3σ of the mean value.

A process quality interpretation of 99.7% is that there are
three errors per thousand events. If software coding were the object of the
error measurement, then "three errors per thousand lines of code" probably would
not be acceptable. At ±6σ, the error rate is so
small, 99.9998%, it is more easily spoken of in terms of "two errors per million
events," about 1,000 times better than "3σ". ^{[20]}
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