Library / Probability distributions / PRD-36
Probability distributions
Logistic threshold: variance
Variance of the logistic distribution, in squared units.
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- PRD-36
- Rev
- v1
- Field
- Probability distributions
- Result
- Var[X] [MPa²]
The variance π²s²/3 converts the fitted logistic scale into squared units. The standard deviation is πs/√3, about 1.81 s: a fitted logistic scale is therefore smaller than the standard deviation of the same data.
Variance of the logistic distribution, in squared units. Its standard deviation is πs/√3, about 1.81 s.
Calculator
- Standard deviation, πs/√35.441 MPa
- Interquartile range, 2s ln 36.592 MPa
- Peak density, 1/(4s)0.08333 1/MPa
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 MPa gives a standard deviation of 5.441 MPa.
- The variance rises with the square of the scale.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 MPa | 0.8225 | 329 | 328.2 | +10% | +21 % |
Sweep of Scale s
| Scale s (MPa) | Variance (MPa²) |
|---|---|
| 0.5 | 0.8225 |
| 1.688 | 9.368 |
| 2.875 | 27.19 |
| 4.063 | 54.3 |
| 5.25 | 90.68 |
| 6.438 | 136.3 |
| 7.625 | 191.3 |
| 8.813 | 255.5 |
| 10 | 329 |
Sweep one variable
Variables and how each one contributes
Variance
What the equation gives you. Everything else feeds it.
Scale s
The scale, in MPa: it sets how gradual the transition is. About 4.4 scales separate the 10 % and 90 % points. Flattens the S-curve and widens the transition. Raising it by 10 % (to 3.3) moves the result by +21 %, from 29.61 to 35.83 MPa².
Worked example with the values above
- Square the scale: s² = 9 MPa².
- Multiply by π²/3 = 3.2899.
- Var = 29.609 MPa².
Assumptions and limits
- The threshold follows a logistic distribution: symmetric, with slightly heavier tails than a normal curve.
- The location and scale come from a fit to pass-fail or threshold data.
Common mistakes
- Confusing the scale s with the standard deviation: the logistic standard deviation is πs/√3.
- Evaluating the sigmoid with a naive exponential at extreme arguments, which can overflow; use a stable form.
- Using the fitted curve far outside the range of the data.
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
- Dose-response and pass-fail curves, and the sigmoid of logistic regression.
- Strength and fatigue thresholds where the chance of failure rises smoothly with load.
Maths you need
- Exponential function: e raised to a power
- Sigmoid curve: an S-shaped transition from 0 to 1