Free Lognormal Distribution Calculator

What Is a Lognormal Distribution?

A lognormal distribution (also called log‑normal) is a continuous probability distribution that applies to positive random variables only. Its defining property is that a variable $X$ is lognormally distributed when its natural logarithm $\ln(X)$ follows a normal distribution. Equivalently, if $Y$ is normally distributed with mean $\mu$ and standard deviation $\sigma$, then $e^{Y}$ follows a lognormal distribution. Because the exponential function always produces a positive output, every lognormal variable is strictly greater than zero.

Parameters of the Lognormal Distribution

The distribution is characterized by two log normal distribution parameters:

• $\mu$ – the scale parameter (mean of the underlying normal distribution).
• $\sigma$ – the shape parameter (standard deviation of the underlying normal distribution).

It is important to remember that $\mu$ and $\sigma$ are not the mean and standard deviation of $X$ itself; they describe the normally distributed logarithm. The actual mean, variance, and other measures of $X$ are derived from these parameters using exponential transformations.

Probability Density Function (PDF) and Cumulative Distribution Function (CDF)

The probability density function of the lognormal distribution for $x > 0$ is:

$$f(x) = \frac{1}{x\,\sigma\,\sqrt{2\pi}} \exp\!\left( -\frac{(\ln x - \mu)^2}{2\sigma^2} \right)$$

The cumulative distribution function (CDF) gives the probability that $X \leq x$ and is expressed using the standard normal CDF $\Phi$:

$$F(x) = \Phi\!\left( \frac{\ln x - \mu}{\sigma} \right)$$

This direct link between normal and lognormal distributions makes all lognormal PDF CDF computations straightforward when using a log normal probability calculator.

Quantile Function

The lognormal quantile function (inverse CDF) for a probability $p \in [0,1)$ is:

$$Q(p) = \exp\!\left( \mu + \sigma \cdot \Phi^{-1}(p) \right)$$

This function is essential for generating random samples from the lognormal distribution and for determining thresholds given a cumulative probability.

Mean, Variance, and Other Key Measures

The lognormal mean variance and other common measures are expressed directly in terms of $\mu$ and $\sigma$. For a lognormal variable $X$:

• Mean: $\displaystyle \mathbb{E}[X] = e^{\mu + \sigma^2/2}$
• Median: $\displaystyle \text{Med}[X] = e^{\mu}$
• Mode: $\displaystyle \text{Mode}[X] = e^{\mu - \sigma^2}$
• Variance: $\displaystyle \text{Var}[X] = \left(e^{\sigma^2} - 1\right) e^{2\mu + \sigma^2}$
• Skewness: $\displaystyle \text{Skew}[X] = \left(e^{\sigma^2} + 2\right) \sqrt{e^{\sigma^2} - 1}$

These closed-form formulas allow instant computation once the log normal distribution parameters are known.

Applications of the Lognormal Distribution

The lognormal distribution appears in many real‑world contexts where variables are positive and result from multiplicative processes. Typical applications include:

• Reliability engineering – modeling failure times of mechanical or electronic components.
• Finance and economics – stock prices, asset returns, income distributions.
• Hydrology – monthly maximum daily rainfall or river discharge.
• Internet analytics – comment lengths, time spent reading a post, network traffic per unit time.

In each case, the assumption that the data are lognormally distributed provides a realistic fit for right‑skewed, positive data.

How to Use This Lognormal Distribution Calculator

This free online statistics tool offers six distinct computation modes:

1. Select a mode – choose from pdf, cdf, quantile, sample generator, common measures (mean/variance), or skewness.
2. Enter the parameters – input the scale parameter $\mu$ and the shape parameter $\sigma$.
3. Provide additional input – for pdf/cdf, enter the value $x$; for quantile, enter the probability $p$; for sample generation, specify the sample size.
4. Get instant results – the calculator returns the requested quantity and, where applicable, displays a plot.

Whether you need to evaluate the lognormal PDF CDF, compute the lognormal quantile function, or derive the lognormal mean variance for a given set of parameters, this tool streamlines the entire process.