Library / Probability distributions / PRD-15
Probability distributions
Lognormal measurement: probability density
Probability density of a positive quantity whose natural logarithm is normally distributed, written with its median m and a dimensionless log spread s.
- Ref
- PRD-15
- Rev
- v1
- Field
- Probability distributions
- Result
- f [1/µm]
The lognormal density describes a positive quantity whose natural logarithm is normal. It is zero at the origin, rises to a peak below the median, and has a long upper tail. The median m is in physical units and the log spread s is dimensionless; the density has units of 1/x. Use x > 0.
Probability density of a positive quantity whose natural logarithm is normally distributed, written with its median m and a dimensionless log spread s.
Calculator
- Median10 µm
- Most probable value (mode), m e^(−s²)7.788 µm
- Mean, m e^(s²/2)11.33 µm
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.
- At 12 µm the density is 0.06221 per µm, so a 1 µm slice around it holds about 6.22 % of the population.
- Median 10 µm, mode 7.788 µm and mean 11.33 µm appear in that order: the skew pulls the mean to the right.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/µm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| x Threshold x | 0.5 to 50 µm | 2.558e-8 | 0.08888 | 0.08888 | +10% | −16.7 % |
| s Log spread s | 0.1 to 1.5 | 0.022 | 0.1102 | 0.08823 | +10% | −8.04 % |
| m Median m | 1 to 50 µm | 2.88e-7 | 0.06632 | 0.06632 | +10% | +5.27 % |
Sweep of Threshold x
| Threshold x (µm) | Probability density (1/µm) |
|---|---|
| 0.5 | 2.558e-8 |
| 6.688 | 0.08631 |
| 12.88 | 0.05454 |
| 19.06 | 0.01821 |
| 25.25 | 0.005682 |
| 31.44 | 0.00184 |
| 37.63 | 6.329e-4 |
| 43.81 | 2.315e-4 |
| 50 | 8.976e-5 |
Sweep one variable
Variables and how each one contributes
Probability density
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 +5.27 %, from 0.06221 to 0.06549 1/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 −8.04 %, from 0.06221 to 0.05721 1/µm.
Threshold x
The threshold, in µm, at which the result is read. Moves the point at which the density or the fraction is evaluated. Raising it by 10 % (to 13.2) moves the result by −16.7 %, from 0.06221 to 0.05181 1/µm.
Worked example with the values above
- Standardised log: ln(x/m)/s = ln(12/10)/0.5 = 0.36464.
- Exponent: −½ × (0.36464)² = −0.066482.
- f = e^(that) / (x s √(2π)) = 0.062214 per 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
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
- 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