Library / Probability distributions / PRD-07
Probability distributions
Triangular estimate: expected value
Mean of a triangular distribution: the plain average of the lower bound, the most likely value and the upper bound.
- Ref
- PRD-07
- Rev
- v1
- Field
- Probability distributions
- Result
- E[X] [days]
The mean of a triangular distribution is the plain average of its three defining values. It is pulled above the most likely value when the upper bound is farther from m than the lower bound is, which is the usual case for durations and costs.
Mean of a triangular distribution: the plain average of the lower bound, the most likely value and the upper bound.
Calculator
- Most likely value (mode)3 days
- Mean minus mode1.333 days
- Midrange, (a + b)/25 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.
- With a most likely value of 3 days and a range of 1 to 9, the expected value is 4.333 days.
- Each of the three values has the same weight of one third, so stretching the upper bound moves the mean by a third of the change.
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 | 2.5 | 8 | 5.5 | +10% | +6.92 % |
| a Lower bound a | −5 to 2.5 days | 2.333 | 4.833 | 2.5 | +0.15 days | +1.15 % |
| m Most likely value m | 1.5 to 8.5 days | 3.833 | 6.167 | 2.333 | +10% | +2.31 % |
Sweep of Upper bound b
| Upper bound b (days) | Expected value (days) |
|---|---|
| 3.5 | 2.5 |
| 5.563 | 3.188 |
| 7.625 | 3.875 |
| 9.688 | 4.563 |
| 11.75 | 5.25 |
| 13.81 | 5.938 |
| 15.88 | 6.625 |
| 17.94 | 7.313 |
| 20 | 8 |
Sweep one variable
Variables and how each one contributes
Expected value
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 +0.769 %, from 4.333 to 4.367 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 +2.31 %, from 4.333 to 4.433 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 +6.92 %, from 4.333 to 4.633 days.
Worked example with the values above
- Add the three values: a + m + b = 1 + 3 + 9 = 13.
- Divide by 3.
- E[X] = 4.3333 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