Lognormal Distribution Calculator (Mean, Median, Mode, Probability)
Mean, median, mode and standard deviation of a lognormal variable, the μ and σ behind it, and the chance that it stays at or below any value.
Results
Chance that X is at or below x
57.82
%
Mean of X
3.0802
Median of X
2.7183
Mode of X (the peak)
2.1170
Standard deviation of X
1.6416
μ (mean of ln X)
1.0000
σ (SD of ln X)
0.5000
Geometric standard deviation (e^σ)
1.6487
What this tool does
A variable is lognormal when its logarithm is normal — incomes, particle sizes, repair times, rainfall and stock prices tend to follow it, because many small multiplicative effects pile up. The curve is skewed: mode < median < mean, and the mean can be far above the typical value. The most common trap is feeding the mean and SD of X into formulas that expect those of ln X; choose which pair you have and the calculator converts, using σ² = ln(1 + (SD ÷ mean)²).
Formula
mean = e^(μ + σ²/2) · median = e^μ · mode = e^(μ − σ²) · SD = mean √(e^(σ²) − 1) · P(X ≤ x) = Φ((ln x − μ) ÷ σ)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
md | The two numbers you have are | — |
a | First number (μ, or the mean of X) | — |
b | Second number (σ, or the SD of X) | — |
x | Value x to look up | — |
CDF | Chance that X is at or below x | % |
MEAN | Mean of X | — |
MED | Median of X | — |
MODE | Mode of X (the peak) | — |
SD | Standard deviation of X | — |
MU | μ (mean of ln X) | — |
SG | σ (SD of ln X) | — |
GSD | Geometric standard deviation (e^σ) | — |
Worked example
- The two numbers you have are1
- First number (μ, or the mean of X)1
- Second number (σ, or the SD of X)0.5
- Value x to look up3
- Chance that X is at or below x57.82 %
- Mean of X3.0802
- Median of X2.7183
- Mode of X (the peak)2.1170
- Standard deviation of X1.6416
- μ (mean of ln X)1.0000
- σ (SD of ln X)0.5000
- Geometric standard deviation (e^σ)1.6487
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.