Log-Normal Distribution Probability Calculator | PDF, CDF and Moments from μ, σ, x

When ln x follows a normal distribution, the original value x follows a log-normal distribution. Enter μ, σ and x to get the density (PDF), cumulative probability (CDF), plus the mean, median, mode and standard deviation in one panel.

💡 About this tool

Stock returns, incomes, particle diameters, file sizes — they all share a shape: strictly positive and skewed with a long right tail. A plain normal bell curve mishandles that tail and can even assign probability to negative values, which makes no physical sense for a price or a size. The log-normal distribution is the standard fix.

The catch is the arithmetic. The density mixes x, σ and √(2π) inside an exponential, and the cumulative probability needs the error function Φ — the thing you used to look up in a z-table. On top of that, the distribution's parameters μ and σ are not the mean and standard deviation of x. People constantly read μ as "the average" and get burned.

This calculator takes μ, σ and x and lays out the density and cumulative probability alongside the actual moments: mean E[X] = exp(μ + σ²/2), median exp(μ) and mode exp(μ − σ²). Seeing mode < median < mean line up turns the abstract "right skew" into three numbers you can compare at a glance.

🧐 Frequently Asked Questions

Are μ and σ the mean and standard deviation of x? No. μ and σ describe ln x (the natural log of x), not x itself. The mean of x is a separate quantity, exp(μ + σ²/2), shown here as "Mean E[X]". This is the single most common mix-up with log-normal.

What happens if I set μ = 0? The median exp(μ) becomes exp(0) = 1. Changing σ then leaves the median pinned at 1 while the mean and mode move, which is a clean way to isolate what σ does to the spread.

Why don't the mean, median and mode match? Because the distribution is asymmetric. A few large values pull the mean up, giving the ordering mode < median < mean. The larger σ is, the wider that gap grows.

What value should I put in for x? Any number greater than zero — a price, a diameter, a dollar amount, whatever you are modeling. Values of zero or below aren't defined for a log-normal, so the tool clamps them to a small positive minimum.

What does the CDF tell me? The probability that X is at most your x. A CDF of 0.79 means about 79% of draws from that distribution fall at or below x. The percentage form is shown next to it.

📚 Why the log-normal keeps showing up

Here is the intuition that makes the log-normal click: add many small random effects and you get a normal distribution; multiply many small random factors and you get a log-normal one. That multiplicative story was sketched by Galton and McAlister back in 1879, reframed by Kapteyn in 1903, and nailed down by the economist Gibrat in 1931 as the "law of proportionate effect."

It explains why the distribution is everywhere in practice. In quantitative finance it underpins the Black-Scholes model, since prices compound by percentage moves rather than fixed steps. In aerosol and environmental science it describes particle sizes, because sedimentation and coagulation act multiplicatively. Studies even find that the income of roughly 97–99% of a population fits a log-normal curve. Whenever a quantity can only be positive and grows by ratios, this is the curve to reach for — and the reason the plain average so often lies about the data.