Library / Probability distributions / PRD-41
Probability distributions
Gumbel maxima: variance
Variance of Gumbel block maxima, in squared units.
- Ref
- PRD-41
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [m²/s²]
The variance π²s²/6 depends only on the scale s, in squared units. The standard deviation is πs/√6, about 1.28 s. Use it for the variation of block maxima across comparable blocks.
Variance of Gumbel block maxima, in squared units. It depends only on the scale s.
Calculator
- Standard deviation, πs/√63.848 m/s
- Mean, relative to the location, 0.5772 s1.732 m/s
- Skewness (always the same)1.14
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.
- A scale of 3 m/s gives a standard deviation of 3.848 m/s between block maxima.
- The variance rises with the square of the scale: doubling s quadruples it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m²/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 m/s | 0.4112 | 164.5 | 164.1 | +10% | +21 % |
Sweep of Scale s
| Scale s (m/s) | Variance (m²/s²) |
|---|---|
| 0.5 | 0.4112 |
| 1.688 | 4.684 |
| 2.875 | 13.6 |
| 4.063 | 27.15 |
| 5.25 | 45.34 |
| 6.438 | 68.17 |
| 7.625 | 95.64 |
| 8.813 | 127.7 |
| 10 | 164.5 |
Sweep one variable
Variables and how each one contributes
Variance
What the equation gives you. Everything else feeds it.
Scale s
The scale, in m/s: it sets the spread of the block maxima. Widens the distribution and lengthens the upper tail. Raising it by 10 % (to 3.3) moves the result by +21 %, from 14.8 to 17.91 m²/s².
Worked example with the values above
- Square the scale: s² = 9 (m/s)².
- Multiply by π²/6 = 1.6449.
- Var = 14.804 m²/s².
Assumptions and limits
- Each value is the maximum of many roughly independent observations in a block of equal length, such as a year.
- The location and scale come from a fit to block maxima of the same kind.
Common mistakes
- Mixing blocks of different lengths in one fit.
- Treating the probability as a return period: a non-exceedance probability F corresponds to a return period of 1/(1 − F) blocks.
- Extrapolating far beyond the data, where the choice of tail model dominates.
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
- Design wind speeds, flood levels and wave heights from annual maxima.
- Maximum loads and extreme temperatures in structural and thermal design.
Maths you need
- Exponential function: e raised to a power
- Return period: average time between exceedances