Library / Probability distributions / PRD-17
Probability distributions
Lognormal measurement: arithmetic mean
Arithmetic mean of a lognormal quantity from its median and log spread.
- Ref
- PRD-17
- Rev
- v1
- Field
- Probability distributions
- Result
- E[X] [µm]
The arithmetic mean is the median times e^(s²/2), so it always exceeds the median, and the gap widens quickly as the log spread s grows. It is the mean you need for totals and loads, while the median describes a typical item.
Arithmetic mean of a lognormal quantity from its median and log spread. The mean always exceeds the median because the long upper tail pulls it up.
Calculator
- Median10 µm
- Mean divided by median, e^(s²/2)1.133
- Most probable value (mode)7.788 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.
- The mean of 11.33 µm is 13.3 % above the median of 10 µm.
- At a log spread of 1 the mean would be 65 % above the median, so using the median as the average would badly understate totals.
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 | 1.133 | 56.66 | 55.52 | +10% | +10 % |
| s Log spread s | 0.1 to 1.5 | 10.05 | 30.8 | 20.75 | +10% | +2.66 % |
Sweep of Log spread s
| Log spread s | Arithmetic mean (µm) |
|---|---|
| 0.1 | 10.05 |
| 0.275 | 10.39 |
| 0.45 | 11.07 |
| 0.625 | 12.16 |
| 0.8 | 13.77 |
| 0.975 | 16.09 |
| 1.15 | 19.37 |
| 1.325 | 24.06 |
| 1.5 | 30.8 |
Sweep one variable
Variables and how each one contributes
Arithmetic mean
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 11.33 to 12.46 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 +2.66 %, from 11.33 to 11.63 µm.
Worked example with the values above
- Square the log spread and halve it: s²/2 = 0.125.
- Exponentiate: e^0.125 = 1.1331.
- E[X] = 10 × 1.1331 = 11.331 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