Library / Probability distributions / PRD-18

Probability distributions

Lognormal measurement: variance

Variance of a lognormal quantity on its original scale, in squared units.

Ref: PRD-18
Rev: v1
Field: Probability distributions
Result: Var[X] [µm²]

V a r [X] = m^{2} e^{s^{2}} (e^{s^{2}} - 1)

The variance on the original scale, m² e^(s²)(e^(s²) − 1), is in squared units. It grows very fast with the log spread s because the upper tail stretches, so small changes in s have large effects.

Variance of a lognormal quantity on its original scale, in squared units. It grows very fast as the log spread s increases.

Calculator

Solve for

Var[X]VarianceComputed

µm²

mMedian m

µm

150

sLog spread s

0.11.5

Standard deviation6.039 µm
Mean11.33 µm
Coefficient of variation, √(e^(s²) − 1)0.5329

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.

Variance (µm²) current value 36.47 µm²

sLog spread sKey driver

1.028,050

mMedian m

0.365912

The standard deviation is 6.039 µm against a mean of 11.33 µm.
The coefficient of variation depends only on s, here 0.533: scatter scales with the size of the quantity.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (µm²)	Local step	Change in result
s Log spread s	0.1 to 1.5	1.015	8,053	8,052	+10%	+31.1 %
m Median m	1 to 50 µm	0.3647	911.7	911.4	+10%	+21 %

Sweep of Log spread s

Log spread s	Variance (µm²)
0.1	1.015
0.275	8.473
0.45	27.48
0.625	70.63
0.8	170
0.975	410.7
1.15	1,033
1.325	2,770
1.5	8,053

Sweep one variable

Vary

Variance (µm²)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Var[X]

Variance

Result · µm²

What the equation gives you. Everything else feeds it.

Median m

µm · typical 1 to 50

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 +21 %, from 36.47 to 44.13 µm².

Log spread s

· typical 0.1 to 1.5

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 +31.1 %, from 36.47 to 47.8 µm².

Worked example with the values above

e^(s²) = e^0.25 = 1.284, and e^(s²) − 1 = 0.28403.
m² = 100.
Var = 100 × 1.284 × 0.28403 = 36.47 µm².

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

Related formulas

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