Library / Probability distributions / PRD-19
Probability distributions
Lognormal measurement: most probable value
Value at which the lognormal density peaks.
- Ref
- PRD-19
- Rev
- v1
- Field
- Probability distributions
- Result
- Mode [µm]
The mode m e^(−s²) is the value at which the density peaks. It always lies below the median, which lies below the mean: mode < median < mean for every s > 0. For narrow distributions the three are close; for wide ones the mode is far below the mean.
Value at which the lognormal density peaks. It always lies below the median, which lies below the mean.
Calculator
- Median10 µm
- Mean11.33 µm
- Mode divided by median, e^(−s²)0.7788
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 most common size is 7.788 µm, 22.1 % below the median of 10 µm.
- Reporting the mode, median and mean together tells a reader how skewed the population is.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (µm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| m Median m | 1 to 50 µm | 0.7788 | 38.94 | 38.16 | +10% | +10 % |
| s Log spread s | 0.1 to 1.5 | 1.054 | 9.9 | 8.847 | +10% | −5.11 % |
Sweep of Log spread s
| Log spread s | Most probable value (µm) |
|---|---|
| 0.1 | 9.9 |
| 0.275 | 9.272 |
| 0.45 | 8.167 |
| 0.625 | 6.766 |
| 0.8 | 5.273 |
| 0.975 | 3.865 |
| 1.15 | 2.665 |
| 1.325 | 1.728 |
| 1.5 | 1.054 |
Sweep one variable
Variables and how each one contributes
Most probable value
What the equation gives you. Everything else feeds it.
Median m
The median of the quantity, in µm: half the population is below it. The logarithm of the quantity is normal with mean ln m. Stretches the whole distribution in proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 7.788 to 8.567 µm.
Log spread s
The standard deviation of the natural logarithm, a plain number. Roughly, s = 0.5 means a typical factor of about 1.65 either side of the median. Controls the spread and the length of the upper tail; the effect is steep at large s. Raising it by 10 % (to 0.55) moves the result by −5.11 %, from 7.788 to 7.39 µm.
Worked example with the values above
- Square the log spread: s² = 0.25.
- Exponentiate the negative: e^(−0.25) = 0.7788.
- Mode = 10 × 0.7788 = 7.788 µm.
Assumptions and limits
- The quantity is strictly positive and its logarithm is normally distributed.
- The median m and the log spread s come from data on the same population.
Common mistakes
- Using the standard deviation of the raw data as s: s is the standard deviation of ln x.
- Using log base 10 values in place of natural logs: s changes by a factor of ln 10.
- Quoting the median as the average when the mean is larger.
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
- Particle and droplet size distributions, concentrations of contaminants, and material strengths and fatigue lives.
- Repair times and other positive, right-skewed engineering quantities.
Maths you need
- Natural logarithm: the inverse of e to a power
- Normal distribution: the bell curve and its cumulative function