PERT Calculator: Expected Time, Variance and On-Time Probability
Add your tasks with optimistic, most likely and pessimistic estimates to get the expected duration, the uncertainty and the chance of finishing by a deadline.
Quick answer
PERT expected time = (optimistic + 4 × most likely + pessimistic) ÷ 6. For a task estimated at 4, 6 and 14 days, the expected time is 7 days with a standard deviation of 1.67. Add tasks in series to get a path total, then compare a deadline with that total to read the chance of finishing on time.
How to read your result
The calculator turns three guesses per task into an expected time and a measure of uncertainty, then adds them up along one path.
Expected time (TE): the weighted average duration of a task. In the totals row, it is the sum of the task values, which is the expected length of the whole path.
Standard deviation (σ): the typical size of the miss between the plan and what happens. Task σ is (P − O) ÷ 6. For the path, σ is the square root of the summed variances, not the sum of the σ values.
Variance: σ squared. Variances can be added across independent tasks, which is why the calculator adds these and not the σ values.
Range (two-sided): TE ± z × σ. The 90% range is the span that should contain the real duration about nine times in ten, with the remaining misses split between early and late.
Finish by (one-sided): the duration you can commit to with 80%, 90% or 95% confidence of finishing no later than that. It is always longer than TE, and it is the number to use when you promise a date.
Chance of finishing by the deadline: enter a deadline in the same unit and the calculator returns the probability, using z = (deadline − TE) ÷ σ and the normal curve.
The two tables answer different questions. A range describes where the outcome is likely to land; a finish-by figure describes what you can promise. For example, the 90% range might be 12.6 to 20.1 days, but the duration with 90% confidence of finishing on time is 19.2 days.
PERT formulas
For each task, with optimistic time O, most likely time M and pessimistic time P:
TE = (O + 4M + P) ÷ 6
σ = (P − O) ÷ 6 variance = σ²
For tasks that run one after another on the same path:
Φ is the standard normal cumulative distribution. The calculator evaluates it with the Abramowitz and Stegun polynomial approximation of the error function, which has an error below 0.00001 percentage points. The constants are:
z-values used
Confidence
Range z (two-sided)
Finish-by z (one-sided)
80%
1.282
0.8416
90%
1.645
1.2816
95%
1.960
1.6449
Range: TE ± z × σ. Finish-by duration: TE + z × σ.
Worked examples
Each example uses numbers you can load into the calculator above; the results match the tool exactly.
Example 1 (simple): three tasks in a row, deadline of 20 days
Design, build and test run one after another. The deadline is 20 days.
Task
O
M
P
TE
σ
Variance
Design
2
4
8
4.33
1.00
1.00
Build
5
8
17
9.00
2.00
4.00
Test
2
3
4
3.00
0.33
0.11
Path total
16.33
2.26
5.11
σpath = √5.11 = 2.26
z = (20 − 16.33) ÷ 2.26 = 1.62, so the chance of finishing by day 20 is 94.8%
90% range: 12.61 to 20.05 days. Finish-by durations: 80% at 18.24 days, 90% at 19.23 days, 95% at 20.05 days
These are the calculator's default inputs. The deadline of 20 days sits just inside the 95% finish-by figure, which matches the 94.8% result.
How to calculate PERT expected time and probability by hand
Collect three estimates per task: optimistic (O, everything goes well), most likely (M) and pessimistic (P, serious but realistic problems).
Compute TE for each task: add O, four times M and P, then divide by 6.
Compute σ for each task: subtract O from P and divide by 6. Square it to get the variance.
Add along the path: sum the TE values for the expected duration, and sum the variances for the total variance.
Take the square root of the total variance to get the path standard deviation.
Find z: subtract the path TE from your deadline and divide by the path standard deviation.
Look z up in a standard normal table (or use this calculator) to get the probability of finishing by the deadline.
Assumptions, limits and common mistakes
Assumptions and limits
The probability uses a normal approximation. It improves as the number of tasks on the path grows, and it is rough for a path of one or two tasks.
Tasks are assumed independent. If one cause (for example, a late client) can slow several tasks together, the real spread is larger than shown.
The calculator works on one path of tasks in series. With parallel paths, the chance that the whole project finishes on time is lower than the chance for any single path. Find the longest path first with the critical path calculator, and remember that near-critical paths add risk.
σ = (P − O) ÷ 6 is the standard PERT heuristic. It treats the optimistic and pessimistic values as roughly the extremes of the possible range.
Garbage in, garbage out: the output is only as good as the three estimates.
Common mistakes
Adding the standard deviations of the tasks. Add the variances and take the square root.
Using the same estimate three times or setting P barely above M. This produces a falsely small σ and an over-confident probability.
Treating 50% as a commitment. TE is the middle of the distribution, so you miss it about half the time. Promise a finish-by figure at 80% or 90% instead.
Mixing units. Estimates, deadline and the unit label must all be in the same unit.
Using PERT on parallel work as if it were one chain. Only tasks in sequence belong on one path.
Frequently asked questions
What is the PERT formula for expected time?
Expected time is (O + 4M + P) ÷ 6, where O is the optimistic estimate, M the most likely and P the pessimistic. It is a weighted average that gives the most likely value four times the weight of either extreme.
How do you calculate the standard deviation in PERT?
Task standard deviation is (P − O) ÷ 6. For several tasks on one path, add their variances (the σ values squared) and take the square root of that sum.
How do I calculate the probability of completing a project on time?
Add the expected times and variances along the path, then compute z = (deadline − path TE) ÷ path σ. The probability is the standard normal cumulative distribution at z. Enter your deadline above and the calculator does this for you.
Why divide by 6 in PERT?
PERT approximates a beta distribution. The weights 1, 4 and 1 give TE, and the range from optimistic to pessimistic is treated as about six standard deviations wide, which is why σ is (P − O) ÷ 6.
What is the difference between the range and the finish-by duration?
A range is two-sided: it brackets the likely outcome, with some chance of landing below or above it. A finish-by duration is one-sided: it is the length of time you need so that the chance of finishing no later than that is 80%, 90% or 95%. Use the finish-by figure for commitments.
What does a probability of 50% mean?
The deadline equals the expected time. Under the model, you are as likely to finish late as early.
Can I use PERT for parallel tasks?
Not directly. This calculator treats the tasks as one chain. For a network with parallel branches, identify the critical path first, then use the PERT figures for the tasks on it.
How accurate is the probability?
It is as good as the estimates and the normal approximation. Treat it as a structured way to talk about risk, not a guarantee. The cumulative-distribution calculation itself has an error below 0.00001 percentage points.