Can you add task P80s? Why the sum overshoots
Eight task P80s add up to $58,551, but the project’s P80 is $51,765: that sum sits at P98.7. Why percentiles don’t add, and what correlation changes.
Short answer No. Percentiles do not add. Overruns on some tasks are offset by underruns on others, so the sum of eight task P80s lands at about the 98.7th percentile of the total, not the 80th. It is right only if every task overruns together.
- Sum of task P80s
- $58,551what the shortcut gives
- Project P80
- $51,765what the simulation gives
- The sum sits at
- P98.7of the simulated total
- Price difference
- $9,049at a 25% margin
The shortcut, and why teams use it
Ask each lead for a safe number on their task, a P80 or just “the number I would not be embarrassed by”, add them up, and call the total the project’s P80. It is easy to do in a spreadsheet, it feels prudent, and it is wrong in a way that costs money. Here is a small project to test it on: eight tasks, each a three-point effort estimate in days, at a blended $800 a day. The P50 and P80 columns are each task simulated on its own.
| Task | Days (low / likely / high) | Likely cost | P50 | P80 |
|---|---|---|---|---|
| Scoping & requirements | 3 / 5 / 10 | $4,000 | $4,304 | $5,305 |
| Data model | 2 / 4 / 9 | $3,200 | $3,504 | $4,505 |
| API build | 6 / 10 / 20 | $8,000 | $8,608 | $10,610 |
| Admin screens | 5 / 8 / 16 | $6,400 | $6,905 | $8,469 |
| Customer screens | 6 / 9 / 18 | $7,200 | $7,810 | $9,506 |
| Integrations | 4 / 8 / 22 | $6,400 | $7,414 | $9,948 |
| Testing & fixes | 4 / 6 / 14 | $4,800 | $5,410 | $6,805 |
| Deploy & handover | 2 / 3 / 7 | $2,400 | $2,705 | $3,403 |
| Sum of the eight | $42,400 | $46,659 | $58,551 |
Inputs. Each task costs its days times $800. Beta-PERT (λ = 4), tasks independent, 100,000 simulated runs with a fixed seed. Illustrative inputs, not benchmarks. Figures are rounded for display, so recomputing from the rounded values can differ by a dollar or two.
Five ways to total the same eight tasks
Now simulate all eight together, 100,000 runs, and ask where each shortcut lands in the resulting distribution. The last column is the share of simulated outcomes at or below that total, so it is the confidence the total actually buys you.
| Way of totalling | Total | Confidence it buys |
|---|---|---|
| Sum of the “most likely” costs | $42,400 | 10.6% |
| Sum of the task P50s | $46,659 | 39.6% |
| Sum of the task averages | $48,000 | 51.4% |
| Simulated project P80 | $51,765 | 80.0% |
| Sum of the task P80s | $58,551 | 98.7% |
Figure 1. Simulated total cost of the eight tasks. Blue bars are the cheapest 80% of outcomes, rust the rest.
The red line, the sum of the task P80s, sits beyond almost every outcome. The grey line, the sum of the likely costs, sits where only about one run in nine comes in under it.
Three separate things are going on. The sum of the likely costs is far too low, because every task is skewed towards overruns and the most likely value is not the average. The sum of the P50s is also too low, for the same reason: medians of skewed tasks add up to less than the median of the total. The sum of the averages is the one total that does add exactly, and it lands near the middle at $48,000. The sum of the P80s overshoots in the other direction.
Why percentiles do not add
Averages add. Spread does not. For independent tasks the variances add, so the spread of the total grows with the square root of the sum of squares, not with the sum. Here the eight task standard deviations add up to $11,974, but the standard deviation of the simulated total is only $4,509. Adding eight P80s stacks eight individual “80th percentile cushions” on top of the average; the project only has the spread of one combined distribution.
In the same run a few tasks overrun, a few come in early, and the early ones pay for some of the late ones. The sum of the P80s quietly assumes all eight overrun together. That is a coincidence you should not expect, and it is the reason the total is so far out in the tail.
When there are many independent tasks, a spreadsheet shortcut that works well is mean plus 0.84 times the standard deviation of the total: $48,000 + 0.84 × $4,509 ≈ $51,800, against $51,765 simulated. It is an approximation and it leans on the tasks being independent, which brings us to the exception.
When the sum is right: tasks that share a cause
The sum of the P80s is exactly the project P80 in one case: when all tasks are perfectly linked and run long or short together. That is the opposite extreme from independence. Real projects sit between the two. A slow client, one unreliable vendor or one technology you underestimated can push several tasks the same way at once.
Figure 2. Project P50, P80 and P95 as the correlation between every pair of tasks rises from 0 (independent) to 1 (move together). The dashed line is the sum of the eight task P80s.
It meets the project P80 only at the far right, and it is already below the P95 once correlation reaches about 0.2.
| Correlation | Project P80 | Project P95 | Confidence of the sum of task P80s |
|---|---|---|---|
| 0 (independent) | $51,809 | $55,710 | 98.7% |
| 0.2 | $53,620 | $59,534 | 93.5% |
| 0.4 | $55,061 | $62,542 | 88.8% |
| 0.6 | $56,285 | $65,080 | 85.2% |
| 0.8 | $57,416 | $67,499 | 82.4% |
| 1 (move together) | $58,512 | $69,681 | 80.1% |
Inputs. Same eight estimates, same Beta-PERT shapes, 200,000 runs per row, with one common correlation between every pair of tasks (a Gaussian copula). This was computed in a separate script. BidVariance itself samples task durations independently; see the limitations in the methodology.
So the sum of the task P80s is neither an 80% number nor a wrong one. It is an undeclared confidence level somewhere between about 80% and 99%, depending on how tied together your tasks are. Two consequences follow. If you quote from the sum without knowing that, you are pricing at a much higher confidence than you think and will lose bids you did not need to lose. And if your tasks do share a cause, the independent model is the optimistic one at the top of the range: at a correlation of 0.4 the P95 is $62,542, against $55,710 when the tasks are independent.
What to do instead
- Estimate ranges, not padded points. A low, a likely and a high for each task keeps the shape of the uncertainty; padding hides it.
- Simulate the total and read the percentile you intend to quote, rather than adding percentiles. The PERT versus Monte Carlo note shows why a single average is not enough either.
- Name the shared causes. If one event could slow several tasks, such as late client approvals or one vendor, model it as an explicit project-level risk with a probability and a cost, instead of hoping padding covers it. Contingency without double counting covers where it belongs.
- Say which confidence you are quoting at. Your quote is a P80 or a P90 on purpose, not a P99 by accident.
What it does to the price
At a 25% gross margin the price is the cost divided by 0.75. Pricing from the simulated P80 gives $69,019. Pricing from the sum of the task P80s gives $78,068, which is $9,049 (13%) higher. That may be a deliberate choice if you want a very safe quote, but it should be a choice. The same logic runs through margin versus markup and the minimum bid price calculator; for the whole task network, the app simulates the total and reads the percentile for you.
Frequently asked questions
Can you add up P80 estimates?
No. A P80 is a point on a distribution, and the P80 of a total is not the sum of the parts’ P80s. In the worked example eight task P80s sum to $58,551, while the simulated project P80 is $51,765; the sum sits at about the 98.7th percentile of the total.
Is the sum of task P50s the project P50?
No, and it is usually too low. Task costs are skewed towards overruns, so medians add up to less than the median of the total. Here the sum of the P50s is $46,659 against a simulated project P50 of $47,841.
When does the sum of task P80s equal the project P80?
Only when every task is perfectly correlated, meaning they all run long or short together. With independent tasks the sum is far above the project P80; real projects fall in between, so the sum buys a confidence somewhere between about 80% and 99%.
What can I use in a spreadsheet instead of summing percentiles?
For many independent tasks, the project average plus 0.84 times the standard deviation of the total approximates the P80. In the example that gives about $51,800 against $51,765 simulated. It is an approximation and it assumes independence, so simulate the total when the decision matters.
Does BidVariance handle correlated tasks?
BidVariance samples task durations independently. A single risk event can affect one task or the whole project, so shared causes are modelled as explicit project-level risks rather than as a correlation setting.