Log Normal Distribution Calculator

Work out log normal distribution instantly with clear inputs, formula shown and shareable results.

About Log Normal Distribution Calculator

X is log-normal when ln X is normal, so probabilities are found by standardising the logarithm. The distribution is right-skewed, which is why its mean exceeds its median.

Formulas

Log-normal CDF

P(X ≤ x) = Φ((ln x − μ)/σ)

Mean and median

mean = e^(μ + σ²/2), median = e^μ

Worked example

With the default values below, Log Normal Distribution Calculator returns P(X ≤ x) of 0.6692168064. Change any input above and every figure here is recalculated in your browser.

Inputs used

Mean of ln X (μ): 3
Standard deviation of ln X (σ)minimum 0: 0.5
Evaluate at xminimum 0: 25

Results

P(X ≤ x): 0.6692168064
Probability density at x: 0.0289993915
Mean e^(μ + σ²/2): 22.759895
Median e^μ: 20.085537

Step by step

1Values used
2Log-normal CDF
3Mean and median
4P(X ≤ x)= 0.6692168064
5Probability density at x= 0.0289993915
6Mean e^(μ + σ²/2)= 22.759895
7Median e^μ= 20.085537

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For μ = 3, σ = 0.5 at x = 25?

z = (3.2189 − 3)/0.5 = 0.4377, so P(X ≤ 25) ≈ 0.6692. The median is e³ ≈ 20.09 and the mean about 22.76.

Where does it arise?

Wherever effects multiply rather than add: incomes, particle sizes, stock prices and time-to-failure data.