PERT scheduling: if activity times are given as to (optimistic), tm (most likely), and tp (pessimistic), what is the variance of the activity duration used for network analysis?

Introduction / Context:
Program Evaluation and Review Technique (PERT) models activity duration as a beta distribution using three time estimates: optimistic (to), most likely (tm), and pessimistic (tp). PERT converts these to an expected time and a variance to analyze critical paths and project completion risk.

Given Data / Assumptions:

• Activity time estimates: to, tm, tp.
• Standard PERT (original) parameters are used.
• Independence between activities is assumed for variance summation along paths.

Concept / Approach:
PERT expected duration te = (to + 4 tm + tp) / 6. The variance is derived from the assumed beta distribution shape using the range (tp − to). The canonical PERT variance expression is Var = ((tp − to) / 6)^2, which reflects greater uncertainty when the pessimistic-optimistic spread is large.

Step-by-Step Solution:
Recall PERT mean: te = (to + 4 tm + tp) / 6.Recall PERT variance: σ^2 = ((tp − to) / 6)^2.Identify the provided choices and select the canonical formula.

Verification / Alternative check:
Numerical checks: if tp − to is zero (no uncertainty), variance becomes zero, which is logical. Larger spreads proportionally increase variance.

Why Other Options Are Wrong:

• Using (tp − tm) or (tm − to) ignores the full range of uncertainty.
• ((tp + to − 2 tm) / 6)^2 is not a standard PERT variance.
• (tp − to)^2 omits the 1/36 scaling and overstates variance by 36×.

Common Pitfalls:
Confusing the mean formula with the variance; forgetting the division by 6 before squaring.

Final Answer:
((tp - to) / 6)^2