Library / Probability distributions / PRD-08
Probability distributions
Triangular estimate: variance
Variance of a triangular distribution, in squared units.
- Ref
- PRD-08
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [days²]
The variance measures spread about the triangular mean and is a symmetric function of the three values. It is zero only when a = m = b, and it grows with the square of the range. Add the variances of independent tasks to get the variance of their total.
Variance of a triangular distribution, in squared units. It grows with the width of the range and is smallest when the three values coincide.
Calculator
- Standard deviation1.7 days
- Range, b − a8 days
- Mean of the distribution4.333 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.
- The spread is 1.7 days (standard deviation) around a mean of 4.333 days.
- The upper bound dominates: stretching b from 9 to 11 days raises the variance far more than the same change in a.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (days²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Upper bound b | 3.5 to 20 days | 0.2917 | 18.17 | 17.88 | +10% | +25.8 % |
| a Lower bound a | −5 to 2.5 days | 2.181 | 8.222 | 6.042 | +0.15 days | −2.84 % |
| m Most likely value m | 1.5 to 8.5 days | 2.667 | 3.347 | 0.6806 | +10% | −2.13 % |
Sweep of Upper bound b
| Upper bound b (days) | Variance (days²) |
|---|---|
| 3.5 | 0.2917 |
| 5.563 | 0.8717 |
| 7.625 | 1.924 |
| 9.688 | 3.45 |
| 11.75 | 5.448 |
| 13.81 | 7.919 |
| 15.88 | 10.86 |
| 17.94 | 14.28 |
| 20 | 18.17 |
Sweep one variable
Variables and how each one contributes
Variance
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 −1.9 %, from 2.889 to 2.834 days².
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 −2.13 %, from 2.889 to 2.827 days².
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 +25.8 %, from 2.889 to 3.634 days².
Worked example with the values above
- Squares: a² + m² + b² = 91.
- Cross terms: am + ab + mb = 39.
- Var = (91 − 39) / 18 = 2.8889 days².
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