Enter x, μ, and σ, then click Calculate.

The lognormal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. If Y = ln(X) follows a normal distribution with mean μ and standard deviation σ, then X follows a lognormal distribution. This distribution is characterized by positive skewness and is bounded below by zero, making it suitable for modeling quantities that cannot be negative, such as stock prices, income distributions, asset prices, failure times, and environmental concentrations. The probability density function (PDF) is f(x) = (1/(x·σ·√(2π))) · exp(-(ln(x)-μ)²/(2σ²)). The cumulative distribution function (CDF) is derived from the standard normal CDF: F(x) = Φ((ln(x)-μ)/σ). The lognormal distribution has important properties: the mean is exp(μ + σ²/2), the median is exp(μ), and the mode is exp(μ - σ²). This calculator computes CDF, PDF, mean, median, mode, variance, and standard deviation for given parameters.

1. Enter the Value (x) – the quantile for which you want the probability.
2. Enter the Mean of ln(x) (μ) – the mean of the underlying normal distribution.
3. Enter the Std Dev of ln(x) (σ) – the standard deviation (must be > 0).
4. Click Calculate Probability to compute all statistics.
5. The result shows CDF, PDF, mean, median, mode, variance, and standard deviation.
6. Click Reset to restore default example values.