PERT Statistics

What Is PERT?

PERT (Program Evaluation and Review Technique) is a three-point estimation method that produces a weighted expected value and standard deviation from optimistic, most likely, and pessimistic estimates.

The Formulas

Expected = (Optimistic + 4 × Most_Likely + Pessimistic) / 6
SD = (Pessimistic - Optimistic) / 6

The "4×" weighting of the most likely value reflects that outcomes cluster around the middle more than the extremes.

Log-Normal Weighting (v0.5.0)

Deep validation (KS goodness-of-fit test, n=84k tasks across 4 size bands) showed that actual software effort follows a log-normal distribution, not the beta distribution that traditional PERT assumes. Log-normal captures the heavy right tail — the reality that overruns can be dramatic while underruns are bounded.

The skill uses the geometric mean as the most-likely value instead of the arithmetic midpoint:

most_likely = sqrt(total_min × total_max)          # geometric mean
pert_expected = (total_min + 4 × most_likely + total_max) / 6
pert_sd = (total_max - total_min) / 6

The geometric mean is always less than or equal to the arithmetic mean (when min ≠ max), which shifts the expected value slightly toward the minimum. This reflects the right-skewed nature of effort distributions — the most likely outcome is below the arithmetic midpoint, with a longer tail toward overruns.

Why this matters: The old beta assumption underestimated tail risk. For an L task estimated at 5-15 hours, the arithmetic midpoint is 10 hours, but the geometric mean is ~8.7 hours — a better reflection of where actual outcomes cluster.

See deep validation findings for the KS test results per size band.

Confidence Bands

From the standard deviation, we get confidence intervals:

Band	Formula	Meaning
68%	Expected ± 1 SD	The actual will fall in this range about 2/3 of the time
95%	Expected ± 2 SD	The actual will fall in this range about 19/20 of the time
99.7%	Expected ± 3 SD	Almost certainly within this range

Example

For a task estimated at min=2 hrs, max=8 hrs:

Geometric Mean = sqrt(2 × 8) = 4.0
Expected = (2 + 4×4.0 + 8) / 6 = 26/6 = 4.33 hrs
SD = (8 - 2) / 6 = 1.0 hr

68% band: 3.3 - 5.3 hrs
95% band: 2.3 - 6.3 hrs

Compare with the old arithmetic midpoint approach:

Midpoint = (2 + 8) / 2 = 5.0
Old Expected = (2 + 4×5.0 + 8) / 6 = 30/6 = 5.0 hrs

The log-normal weighting gives a lower expected value (4.33 vs 5.0), better reflecting where actual outcomes cluster.

Why PERT Over Raw Ranges?

Raw Range	PERT
"2-8 hours"	"Expected ~4.3 hrs (68%: 3.3-5.3 hrs)"
Which number do I plan with?	Plan with 4.3, budget for 5-6
Wide range feels unhelpful	Single number with explicit uncertainty

PERT gives stakeholders a single number to plan with plus transparent uncertainty, rather than forcing them to guess which end of the range to use.

In the Output

Every estimate includes:

PERT Expected: 4.3 hrs (most likely outcome)
Standard Deviation: ±1.0 hr
68% Confidence: 3.3 - 5.3 hrs
95% Confidence: 2.3 - 6.3 hrs
Committed (80%): 6.1 hrs