Library / Probability distributions / PRD-37
Probability distributions
Logistic threshold: interquartile range
Width between the 25th and 75th percentiles of a logistic distribution, about 2.197 s.
- Ref
- PRD-37
- Rev
- v1
- Field
- Probability distributions
- Result
- IQR [MPa]
The interquartile range 2s ln 3 is the width between the 25th and 75th percentiles, about 2.197 s. It is a robust measure of the width of the transition zone and does not depend on the tails.
Width between the 25th and 75th percentiles of a logistic distribution, about 2.197 s.
Calculator
- 25th percentile, relative to m−3.296 MPa
- 75th percentile, relative to m3.296 MPa
- Standard deviation, πs/√35.441 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.
- The middle half of the thresholds lies within a band 6.592 MPa wide, centred on the location.
- The 25th and 75th percentiles sit 3.296 MPa either side of m.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Scale s | 0.5 to 10 MPa | 1.099 | 21.97 | 20.87 | +10% | +10 % |
Sweep of Scale s
| Scale s (MPa) | Interquartile range (MPa) |
|---|---|
| 0.5 | 1.099 |
| 1.688 | 3.708 |
| 2.875 | 6.317 |
| 4.063 | 8.926 |
| 5.25 | 11.54 |
| 6.438 | 14.14 |
| 7.625 | 16.75 |
| 8.813 | 19.36 |
| 10 | 21.97 |
Sweep one variable
Variables and how each one contributes
Interquartile range
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 +10 %, from 6.592 to 7.251 MPa.
Worked example with the values above
- Natural log of 3: ln 3 = 1.0986.
- Multiply by 2s = 6 MPa.
- IQR = 6.5917 MPa.
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
The inputs are parameters, usually fitted to data, and their estimation error carries into the result. The probability or density is a smooth function of the parameters, so the sweep chart shows how sensitive it is: where the curve is steep, a small error in a parameter makes a large difference. The formula treats the parameters as exact, so it gives no confidence interval on its own.
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