Lognormal Distribution Calculator - PDF, CDF, Mean, and Quantile

A lognormal distribution calculator turns the two lognormal parameters μ and σ into every probability and summary measure that a continuous right-skewed positive variable might need. Use it for quantities that are always positive and have a long right tail, such as failure times in reliability, income, or monthly maximum rainfall, and read the PDF, CDF, survival, mean, median, mode, variance, skewness, and quantile from the same pair of inputs.

• • Reliability and failure-time analysis: Estimate the probability that a component lasts longer than a service interval, or read the median time to failure, when the times are right-skewed and strictly positive.
• • Finance and income modeling: Compare the mean, median, and 95th percentile of a portfolio return, an income distribution, or a stock price modeled as lognormal.
• • Hydrology and environmental data: Translate a measured mean and standard deviation of the natural log of rainfall or flow into the long-tail probabilities used in flood-risk planning.
• • Homework and textbook checks: Verify a worked example in an introductory statistics or probability course by reading the same mean, variance, PDF, and CDF the textbook reports.

A positive random variable X is lognormal if and only if ln X is normally distributed with mean μ and standard deviation σ. That fact powers every formula in the calculator.

This calculator does not simulate datasets or fit μ and σ from a list of observations. For those tasks, use a dedicated fitting or simulation tool.

Because a lognormal variable becomes a normal variable after a log transform, the Normal Distribution Calculator is the natural pair to revisit when you need the bell-curve probability behind any lognormal calculation.

The calculator reads μ and σ, builds the closed-form common measures, and evaluates whichever at-a-point outputs the user supplies. When x is given it returns the PDF, CDF, survival, and the standardized z-score; when p is given it returns the matching quantile from the inverse standard normal CDF.

• μ (mu): Scale parameter, the mean of ln X. Any real number, but does not equal the mean of X.
• σ (sigma): Shape parameter, the standard deviation of ln X. Strictly positive.
• x: Optional positive observation. Drives the PDF, CDF, survival, and z-score when supplied.
• p: Optional probability in (0, 1). Drives the inverse CDF, also called the quantile function.
• z = (ln x − μ) / σ: Standardized normal score that links the lognormal calculation to the standard normal distribution.

The CDF is the standard normal CDF evaluated at the standardized z-score, so the calculator uses an Abramowitz and Stegun error-function approximation, plus a Beasley-Springer-Moro inverse normal routine for the quantile branch.

All outputs are dimensionless. The mean, median, mode, standard deviation, and quantile share the units of x, while the PDF, CDF, survival, and skewness are pure numbers.

According to Wikipedia Log-normal Distribution, the lognormal PDF is 1 / (x σ √(2π)) · exp(−(ln x − μ)² / (2σ²)), the mean is exp(μ + σ²/2), the median is exp(μ), the mode is exp(μ − σ²), and the variance is (exp(σ²) − 1) · exp(2μ + σ²)

The standardized z-score that drives the lognormal PDF and CDF is the same statistic the Z-Score Calculator returns, so the two outputs should agree when you feed the same (ln x − μ) and σ into both.

Four ideas separate a lognormal calculation from a black-box PDF and let the user interpret the result on real data.

These four ideas are also why the lognormal distribution is the right tool for the right-skewed, strictly positive data that the normal distribution cannot fit.

For a sanity check, read the median and the mean at the same time. If the mean is much larger than the median, σ is large.

For waiting times and other positive, right-skewed phenomena, the Exponential Distribution Calculator is the closest single-parameter alternative, and a side-by-side comparison helps you see how μ shapes the mean differently from a fixed rate λ.

Enter the two parameters μ and σ, then add an optional x or p to read whichever at-a-point quantity the workflow needs.

1. 1 Set μ and σ: Type the scale parameter μ (the mean of ln X) and the shape parameter σ. σ must be strictly positive.
2. 2 Read the common measures: The mean, median, mode, variance, standard deviation, and skewness update automatically from μ and σ alone.
3. 3 Enter an optional x for PDF, CDF, and survival: Type any positive x to read the PDF, the cumulative probability P(X ≤ x), the survival P(X > x), and the z-score.
4. 4 Enter an optional p for the quantile: Type a probability p in (0, 1) to read the matching quantile x such that P(X ≤ x) = p.
5. 5 Reset to defaults: Press reset to return μ, σ, x, and p to 0, 1, 1, and 0.5.

Suppose μ = 0, σ = 1, x = 1, and p = 0.975. The mean is 1.6487, the median is 1, the variance is 4.6708, the PDF is 0.3989, the CDF is 0.5, and the 97.5th percentile is 7.0991. The gap between the mean and the 97.5th percentile is the visual fingerprint of the lognormal right tail.

When you have sample data and need to estimate σ for the lognormal fit, the Standard Deviation Calculator helps you back out the underlying standard deviation of ln X from the raw observations.

An all-in-one lognormal distribution calculator removes the need to switch between separate PDF, CDF, quantile, and summary tools.

• • Closed-form common measures: Read the mean, median, mode, variance, standard deviation, and skewness from μ and σ alone.
• • At-a-point probabilities: Type any positive x to see the PDF, CDF, survival, and the underlying z-score in the same panel.
• • Inverse CDF for tail quantiles: Type a probability p to read the matching quantile, useful for service levels, flood return periods, and 95th or 99th percentile planning.
• • Log-space friendly inputs: μ and σ are the natural inputs in a lognormal model, so the user does not have to convert from raw-scale means and standard deviations first.
• • Auditable z-score: The standardized z-score (ln x − μ) / σ is returned alongside the lognormal probability, which lets the user compare with a normal distribution reference.

The benefit is that the lognormal distribution is one of the few continuous distributions whose PDF, CDF, quantile, and common measures all have closed forms in elementary functions, so a single deterministic calculator can cover every question a course or analyst is likely to ask.

Once μ and σ are in hand, feed them into a downstream tool such as a Monte Carlo sample, a reliability prediction, or a portfolio simulation.

For discrete positive counts such as trial numbers and counts of events, the Geometric Distribution Calculator is a useful contrast that shows when the discrete cousin is a better model than the continuous lognormal distribution.

The shape of every output changes with μ and σ, and a few practical caveats apply when the data does not match the lognormal assumption.

• • The calculator does not fit μ and σ from a sample, run a goodness-of-fit test, or simulate datasets.
• • The formulas assume the underlying normal distribution for ln X is the correct model. If the histogram of the logged data has heavy tails, the Weibull or generalized gamma distribution may fit better.
• • When σ is small (below about 0.2), the lognormal distribution looks almost like a normal distribution clipped to x > 0, and the gap between mean, median, and mode is hard to detect in small samples.

Read the standardized z-score together with the lognormal probability whenever the result is sensitive to the tail. A z-score above 2 means the right tail is doing most of the work.

For finance, reliability, and hydrology, the choice between lognormal and a heavier-tailed alternative is usually decided by the empirical kurtosis of the logged data.

According to NIST/SEMATECH e-Handbook of Statistical Methods, the lognormal distribution is defined for positive x, and its parameters μ and σ are the mean and standard deviation of ln X, not of X

According to Omni Calculator Log-normal Distribution, the lognormal distribution is widely used to model failure times in reliability, income and stock prices in finance, and monthly maximum rainfall in hydrology