lognormal calculator - Aaron Graves, PhDude Replica

Interactive Lognormal Calculator

Use this tool to compute summary statistics, density, cumulative probability, interval probability, and quantiles for a lognormal distribution where ln(X) ~ Normal(μ, σ²).

μ (mean of ln(X))

σ (std dev of ln(X), must be > 0)

x value (for PDF and CDF, must be > 0)

p value (for quantile, between 0 and 1)

a (lower bound, > 0)

b (upper bound, > a)

Enter values and click Calculate to see results.

Convert arithmetic mean and standard deviation to μ and σ

Useful when your data are in original units instead of log-space.

Arithmetic Mean E[X]

Arithmetic Std Dev SD[X]

What is a lognormal distribution?

A random variable is lognormal if its natural logarithm is normally distributed. In plain language, values are always positive, often right-skewed, and can vary over orders of magnitude. This pattern appears in finance, reliability engineering, environmental measurements, and biology.

If Y = ln(X) is normal with mean μ and standard deviation σ, then X is lognormal. The parameters μ and σ belong to log-space, not the original units of X.

When a lognormal model is useful

Data are strictly positive (no zeros or negatives).
The histogram is right-skewed with a long upper tail.
Multiplicative effects dominate (growth factors, compounding, cascading uncertainty).
Taking logs makes the data look roughly symmetric and bell-shaped.

How this calculator works

Core formulas

Given parameters μ and σ (σ > 0):

PDF: f(x) = 1 / [xσ√(2π)] · exp(- (ln(x)-μ)² / (2σ²))
CDF: F(x) = Φ((ln(x)-μ)/σ)
Mean: E[X] = exp(μ + σ²/2)
Median: exp(μ)
Mode: exp(μ - σ²)
Variance: (exp(σ²) - 1)exp(2μ + σ²)
Quantile: Q(p) = exp(μ + σΦ^-1(p))

Interval probability

The probability for a range [a, b] is computed as P(a ≤ X ≤ b) = F(b) - F(a). This is useful for tolerance windows, pricing bands, and risk limits.

Practical interpretation tips

Mean vs median: In skewed distributions, the mean is often noticeably larger than the median.
Large σ means heavy right tail: rare high values can dominate averages.
Quantiles are often more intuitive: for example, the 95th percentile tells you a high-but-plausible upper bound.

Common mistakes to avoid

Using non-positive values (lognormal requires x > 0).
Confusing arithmetic mean/std with μ/σ in log-space.
Forgetting that small parameter changes can strongly affect upper-tail probabilities.
Interpreting the mean as “typical” in very skewed data. Median is often a better typical value.

Quick workflow

Enter μ and σ (or convert from arithmetic mean and SD).
Choose x for point estimates (PDF/CDF).
Choose a and b for interval probability.
Choose p for percentile/quantile.
Click Calculate and interpret tail behavior carefully.

FAQ

Can I use base-10 logs instead of natural logs?

This calculator uses natural logs (ln). If your model is in log10, convert parameters before use.

What happens if σ is very small?

The distribution becomes tightly concentrated near exp(μ), approaching a near-deterministic positive value.

Does this help with investment returns?

It can approximate gross return factors and some price models, but real markets may violate assumptions (fat tails, regime shifts, jumps). Use with judgment.