The Analyst Who Added Standard Deviations

The addition is wrong. Variances add; standard deviations do not.

Correct method:

• Each activity: σ = 2 → variance = 4
• Path variance = 4 + 4 + 4 + 4 + 4 = 20
• Path σ = √20 = 4.47 days

Correct probability for day 55:

• z = (55 − 50) / 4.47 = 1.12
• P ≈ 87%

The analyst's 69% understates confidence by about 18 points. Summing σ overstates path uncertainty whenever there is more than one activity - and the error grows with the number of activities: two activities would give 4.0 instead of 2.83, five give 10.0 instead of 4.47, ten would give 20.0 instead of 6.32. The root-sum-square rule reflects partial offsetting: independent activities do not all hit their pessimistic values in the same run, which is precisely the diversification the naive sum ignores.

Consequence in this case: the analyst's number triggered a crash decision that the schedule did not need. Fix the arithmetic before spending the money.

Recommend the activity 3 risk response, and do not crash activity 1.

Activity 3 dominates the variance:

• σ₃ = (P − O) / 6 = (20 − 4) / 6 = 2.67; variance ≈ 7.11
• The other four activities have σ ≤ 8/6 = 1.33, variance ≤ 1.78 each, about 7.1 combined at most
• So activity 3 alone accounts for roughly half the path variance

If the response cuts P from 20 to 12: σ₃ = (12 − 4)/6 = 1.33, variance 1.78 - path variance drops from about 14.2 to 8.9, path σ from about 3.8 to about 3.0.

Crashing activity 1 by 2 days:

• Moves the mean from 50 to 48 → z = (55 − 48)/4.47 = 1.57 → about 94%
• It also costs 3× more ($18,000 vs $6,000) and buys the schedule nothing durable: it shifts the mean but leaves a fat pessimistic tail on activity 3, so the downside scenario is unchanged.

Judgement:

• If the goal is a date, crashing buys probability fastest and most predictably - so if day 55 were a hard external commitment and the response could not be delivered in time, crashing wins on speed.
• If the goal is a robust plan, the variance reduction is the better buy: cheaper, it shrinks the tail instead of hiding it, and it addresses where the uncertainty actually lives.
• Best of all, and what the numbers support: do the $6,000 response, re-run the probability, and only crash if the recomputed figure is still below tolerance. Crashing to fix a 69% that was really 87% is spending against a spreadsheet error.

Because the output is a probability at a date, not a date - and the two cuts of this same data give very different management messages.

• Day 50 (the mean) → z = 0 → 50%. A coin flip. Half the time the plan is late.
• Day 52 (μ + 0.5σ) → z = 0.45 → about 67%.
• Day 55 (sponsor's target) → z = 1.12 → about 87%.
• Day 59 (μ + 2σ) → z = 2.0 → about 98%.

All four are true statements about the same path. Choosing one is a risk-tolerance decision owned by the sponsor, not a calculation owned by the scheduler. The scheduler's job is to present the ladder and say what each rung costs.

Three caveats to attach to any number quoted from this model:

1. It assumes independence. The five activities are sequential and almost certainly share a team; correlation widens the tail and lowers every figure above.
2. It assumes the critical path is the only path that matters. A near-critical path with a wide range can govern - the 87% is only as good as the path enumeration behind it.
3. It is only as good as the three-point inputs. The formula transforms judgement; it does not create it. Activity 3's P = 20 is doing most of the work here, and if that number is a guess, so is the 87%.

Suggested wording for the sponsor: 'Eighty-seven percent at day 55 with current estimates, assuming the conversion risk is not addressed. If we fund the $6,000 response, the band tightens. Which confidence level do you want to sign up to, and at what date?'