Library / Probability distributions / PRD-09
Probability distributions
Triangular estimate: planning percentile
Value that a triangular quantity stays below with cumulative probability q: the inverse of the cumulative distribution.
- Ref
- PRD-09
- Rev
- v1
- Field
- Probability distributions
- Result
- xq [days]
The planning percentile is the inverse of the cumulative distribution: the value that the quantity stays below with probability q. The lower branch applies up to q = (m − a)/(b − a), the probability at the most likely value; the upper branch applies above it. Use it to pick a value you will meet with a stated confidence.
Value that a triangular quantity stays below with cumulative probability q: the inverse of the cumulative distribution. Use it to read off a planning value such as the 80th percentile. The first form applies while q is at or below (m − a)/(b − a), the second above it.
Calculator
- Probability at the most likely value0.25
- Median (q = 0.5)4.101 days
- 90th percentile6.809 days
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- There is a 75 % chance that the value does not exceed 5.536 days. Planning to that value means accepting a 25 % risk of overrun.
- The mean is 4.333 days, so planning at the mean leaves a much larger risk of overrun than planning at a high percentile.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (days) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Upper bound b | 3.5 to 20 days | 2.936 | 11.01 | 8.077 | +10% | +8.05 % |
| q Cumulative probability q | 0.01 to 0.99 | 1.4 | 8.307 | 6.907 | +10% | +10.2 % |
| m Most likely value m | 1.5 to 8.5 days | 5.127 | 7.708 | 2.581 | +10% | +1.58 % |
| a Lower bound a | −5 to 2.5 days | 4.417 | 5.878 | 1.46 | +0.15 days | +0.589 % |
Sweep of Cumulative probability q
| Cumulative probability q | Value at cumulative probability q (days) |
|---|---|
| 0.01 | 1.4 |
| 0.1325 | 2.456 |
| 0.255 | 3.02 |
| 0.3775 | 3.534 |
| 0.5 | 4.101 |
| 0.6225 | 4.743 |
| 0.745 | 5.501 |
| 0.8675 | 6.478 |
| 0.99 | 8.307 |
Sweep one variable
Variables and how each one contributes
Value at cumulative probability q
What the equation gives you. Everything else feeds it.
Lower bound a
The lower bound of the estimate, in days: the least you think the quantity could be. Moves the lower end of the range; the density is zero below it. Raising it by 10 % (to 1.1) moves the result by +0.392 %, from 5.536 to 5.558 days.
Most likely value m
The most likely value, in days, where the density peaks. It must lie between a and b. Moves the peak and shifts the mean towards it. Raising it by 10 % (to 3.3) moves the result by +1.58 %, from 5.536 to 5.624 days.
Upper bound b
The upper bound of the estimate, in days: the most you think it could be. Moves the upper end; a long upper tail raises the mean and the variance. Raising it by 10 % (to 9.9) moves the result by +8.05 %, from 5.536 to 5.982 days.
Cumulative probability q
The cumulative probability you want the value for, between 0 and 1 (0.8 means an 80 % chance of not being exceeded). A higher confidence level pushes the planning value towards the upper bound. Raising it by 0.05 (to 0.8) moves the result by +6.61 %, from 5.536 to 5.902 days.
Worked example with the values above
- Find the crossover: (m − a)/(b − a) = 0.25; q = 0.75 is above it.
- Upper branch: b − √((1 − q)(b − a)(b − m)) with the root of 12.
- xq = 5.5359 days.
Assumptions and limits
- The quantity lies between a and b, with a < m < b, and the density falls linearly either side of the most likely value.
- The three values are honest estimates: the bounds are the limits you believe, not worst cases you would never see.
Common mistakes
- Choosing bounds that are too narrow: people tend to be overconfident, so the real range is usually wider than the first guess.
- Using the most likely value as the expected value: with a long upper tail the mean is above the mode.
- Adding the most likely values of several tasks to get a total: add the means instead, and add the variances for the spread.
Statistics and uncertainty
A mean, variance or percentile is calculated from fitted parameters, and the uncertainty in those parameters passes straight through. A distribution fitted to a small sample can have parameters that are far from the truth, especially in the tails and for the variance, so quote the result together with the sample size it rests on.
Where it is used
- Three-point estimates of task durations and costs in project planning and risk analysis.
- A simple input distribution for Monte Carlo analysis when only a minimum, a best guess and a maximum are known.
Maths you need
- Probability density: area under the curve gives probability
- Quadratic equations: solving for the percentile