Lognormal Distribution Calculator (PDF, CDF, mean, median, mode)

Understanding the lognormal distribution calculator — concept and principle

This calculator gives probabilities and summary statistics for a lognormal distribution, where ln X is normal with mean μ and standard deviation σ. It returns P(X ≤ x), P(X > x), the density, the mean, median, mode, variance, standard deviation and the z-score of ln x. Use it for incomes, particle sizes, repair times, stock prices and other positive, right-skewed quantities.

The long right tail pulls the mean above the median, and the median above the mode.

How it is calculated

P(X ≤ x) = Φ((ln x − μ) ÷ σ)

Density f(x) = exp(−(ln x − μ)² ÷ (2σ²)) ÷ (xσ√(2π)), for x > 0

Median = e^μ; mean = e^(μ + σ²/2); mode = e^(μ − σ²)

Variance = (e^(σ²) − 1)·e^(2μ + σ²)

μ and σ describe ln X, not X itself.

The principle

When a quantity grows by many small random percentage changes, its logarithm is a sum of many small random terms, which tends toward a normal distribution. So the quantity itself is lognormal: never negative, with a long right tail. The skew shows in the summary statistics: the mode is below the median, which is below the mean. The median, e^μ, is often the most useful single number.

Worked example

Default inputs: μ = 3, σ = 0.5, x = 25.

z = (ln 25 − 3) ÷ 0.5 = (3.2189 − 3) ÷ 0.5 = 0.4378.
P(X ≤ 25) = Φ(0.4378) = 0.669217; P(X > 25) = 0.330783.
Density f(25) = 0.0289994.
Median = e³ = 20.0855; mean = e^3.125 = 22.7599; mode = e^2.75 = 15.6426.
Variance = 147.129; standard deviation = 12.1297.

Each point is the calculator re-run with only “μ (mean of ln X)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

μ (mean of ln X)	P(X ≤ x)
0.75	1
1.5	0.999707
2.25	0.973673
3	0.669217
3.75	0.144061
4.5	0.00519985
6	1.3316e-8

How to read the result

About two thirds of values are 25 or less. The mean (22.8) is pulled above the median (20.1) by the long right tail, a typical sign of skewed data such as incomes. If you only know the mean and standard deviation of X, you must first convert them to μ and σ of ln X.

Common mistakes

Entering the mean of X as μ instead of the mean of ln X.
Using a normal distribution for strongly skewed positive data.
Expecting the mean and median to be equal.
Taking the logarithm in base 10 instead of the natural log.

Keep in mind: A lognormal model is an assumption. Check it against your data, for example with a histogram of ln x.

Frequently asked questions

What is a lognormal distribution?

The distribution of X when ln X is normally distributed.

What is the median of a lognormal distribution?

e^μ.

Why is the mean larger than the median?

The long right tail pulls the mean up.

Sources: NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.9 Lognormal Distribution

Checked 9 Nov 2026. NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.9 Lognormal Distribution (density with shape σ and scale m, μ = log m; CDF Φ(ln x ÷ σ); mean e^(σ²/2), median m, mode e^(−σ²) and standard deviation √(e^(σ²)(e^(σ²) − 1)) for the standard form, scaled here by m = e^μ).

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