Lognormal Distribution Calculator
Enter the lognormal parameters mu and sigma (the mean and standard deviation of the underlying normal distribution ln(X)), then supply a value x. The calculator returns the probability density (PDF), the cumulative probability P(X at or below x), the survival probability P(X above x), the quantile for any probability p, and all standard statistical measures including mean, median, mode, variance, and skewness. Results update instantly as you type.
What is the lognormal distribution?
A random variable X follows a lognormal distribution when its natural logarithm, ln(X), is normally distributed. Because the logarithm is only defined for positive numbers, lognormal random variables are always positive, making this distribution ideal for quantities that are inherently non-negative: incomes, stock prices, particle sizes, rainfall amounts, and biological measurements such as blood concentrations. The distribution is characterised by two parameters: mu (the mean of ln X) and sigma (the standard deviation of ln X). These are not the mean and standard deviation of X itself; the actual mean of X is exp(mu + sigma^2 / 2), which is always larger than the median exp(mu).
PDF, CDF and survival probability explained
The probability density function (PDF) gives the relative likelihood of X taking a specific value and is given by f(x) = exp(-(ln(x) - mu)^2 / (2 * sigma^2)) / (x * sigma * sqrt(2*pi)) for x > 0. The cumulative distribution function (CDF) gives P(X at or below x) = Phi((ln(x) - mu) / sigma), where Phi is the standard normal CDF. The survival (or complementary CDF) is P(X > x) = 1 - CDF, the probability that X exceeds the value x. Because the lognormal CDF reduces to the normal CDF on a log scale, standard normal tables and the usual z-score machinery apply directly after the log transformation.
Mean, median, mode and skewness
The three central-tendency measures of a lognormal distribution all differ, reflecting its right skew. The mode, the most probable value, is exp(mu - sigma^2), always below the median exp(mu), which is always below the mean exp(mu + sigma^2 / 2). As sigma increases, the gap between mode and mean widens rapidly. Variance is (exp(sigma^2) - 1) * exp(2*mu + sigma^2), and skewness is (exp(sigma^2) + 2) * sqrt(exp(sigma^2) - 1), which grows sharply with sigma. A small sigma (say 0.25) gives a nearly symmetric shape; a large sigma (say 1.5) produces a heavily right-skewed distribution with a thin, elongated tail.
Practical applications of the lognormal distribution
The lognormal distribution appears across many fields because multiplicative processes naturally produce it. In finance, stock prices and asset returns over long horizons are approximately lognormal under the Black-Scholes framework. In environmental science, pollutant concentrations and particle sizes follow lognormal patterns. In reliability engineering, failure times of components subject to wear or fatigue are often modelled as lognormal. In economics, income and wealth distributions are frequently lognormal within a cohort. In medicine, drug half-lives and antibody titres in serology are lognormally distributed. The key signal that a lognormal model may fit is a histogram that is right-skewed on the original scale but symmetric after a log transformation.
Common lognormal percentiles by sigma
| Sigma | 5th pct | 25th pct (Q1) | Median | 75th pct (Q3) | 95th pct | 99th pct |
|---|---|---|---|---|---|---|
| 0.25 | 0.664 | 0.847 | 1 | 1.181 | 1.505 | 1.779 |
| 0.5 | 0.441 | 0.717 | 1 | 1.395 | 2.266 | 3.08 |
| 0.75 | 0.293 | 0.606 | 1 | 1.65 | 3.411 | 5.318 |
| 1 | 0.193 | 0.513 | 1 | 1.948 | 5.18 | 9.488 |
| 1.25 | 0.128 | 0.435 | 1 | 2.301 | 7.824 | 17.377 |
| 1.5 | 0.085 | 0.368 | 1 | 2.718 | 11.794 | 31.796 |
Percentile values relative to the median (exp(mu) = 1 when mu = 0). Values scale by exp(mu) for other mu.
Frequently asked questions
What is the difference between mu and the mean of the lognormal distribution?
Mu is the mean of ln(X), the log-transformed variable, not the mean of X itself. The actual expected value (mean) of X is exp(mu + sigma^2 / 2), which is always larger than exp(mu). For example, if mu = 0 and sigma = 1, the mean of X is exp(0.5) approximately 1.649, while exp(mu) = 1 is the median. The gap grows with sigma: a larger shape parameter pulls the mean further above the median.
How do I find mu and sigma from a sample of data?
Take the natural logarithm of each positive data point. The sample mean of those log-values is the estimate of mu, and the sample standard deviation of those log-values is the estimate of sigma. This method of moments estimate is the same as the maximum-likelihood estimate for the lognormal distribution. You can then plug mu and sigma into this calculator to compute probabilities and percentiles for the fitted distribution.
What does the 95th percentile of the lognormal tell me?
The 95th percentile is the value x such that 95% of the distribution lies at or below x and only 5% lies above it. Use the quantile mode of this calculator: set p = 0.95 to find that value. In practice this is useful for setting safety thresholds - for example, a pollutant concentration that is exceeded on only 5% of days - or for pricing options at a specified confidence level.
Why is the lognormal distribution always right-skewed?
Because the variable is constrained to be positive (it is bounded below by zero but has no upper bound), and because multiplicative random effects compound over time, the tail on the right extends much farther than the tail on the left. Mathematically, skewness equals (exp(sigma^2) + 2) * sqrt(exp(sigma^2) - 1), which is always positive for any sigma > 0, confirming permanent right skew. Larger sigma means heavier tails and higher skewness.
How does the lognormal distribution relate to the normal distribution?
If X is lognormal with parameters mu and sigma, then Y = ln(X) is exactly normal with mean mu and standard deviation sigma. This means every lognormal probability question can be answered by first taking the log of x, computing a z-score z = (ln(x) - mu) / sigma, and then looking up the standard normal CDF. This calculator performs that transformation automatically.
What is the mode of the lognormal distribution and when does it equal zero?
The mode is exp(mu - sigma^2), the single peak of the PDF. For large sigma (specifically sigma > sqrt(mu) when mu > 0, or any sigma > 0 when mu <= 0), the mode can be very close to zero, meaning the distribution is very steeply peaked near zero with a long right tail. As sigma approaches zero, the distribution collapses toward a spike at exp(mu), and the mode, median and mean all converge.