Lognormal Distribution Calculator

Enter the log-normal distribution parameters — μ (mu) and σ (sigma) — along with either an x value or a probability p to solve your problem. Choose between computing P(X < x) or P(X > x), and the calculator returns the corresponding probability or percentile, plus key distribution statistics like the mean, median, and variance.

The mean of the natural logarithm of X. Can be any real number.

The standard deviation of the natural logarithm of X. Must be positive.

Choose whether to find a probability from x, or an x value from a probability.

The value of X. Must be greater than 0. Used when finding P(X < x) or P(X > x).

A probability between 0 and 1. Used when finding the percentile x.

Results

Result

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PDF f(x)

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Distribution Mean

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Distribution Median

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Distribution Variance

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Distribution Std Dev

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Distribution Mode

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Lognormal PDF Curve

Results Table

Frequently Asked Questions

What is a lognormal distribution?

A lognormal distribution describes a random variable X whose natural logarithm ln(X) follows a normal distribution. It is defined for positive values only (x > 0). It is commonly used to model phenomena like stock prices, income distributions, and survival times where values are always positive and right-skewed.

What do μ (mu) and σ (sigma) represent in a lognormal distribution?

In a lognormal distribution X ~ LogN(μ, σ), μ is the mean and σ is the standard deviation of the underlying normal variable ln(X) — not of X itself. The actual mean of X is e^(μ + σ²/2), and the median is e^μ. Always ensure σ > 0.

How do I compute a left-tail probability P(X < x)?

Select 'P(X < x)' from the calculation mode dropdown, enter your μ and σ parameters, and enter the x value. The calculator returns the cumulative probability that the random variable is less than x. This equals Φ((ln(x) − μ) / σ), where Φ is the standard normal CDF.

How do I compute a right-tail probability P(X > x)?

Select 'P(X > x)' from the dropdown and provide x, μ, and σ. The right-tail probability is simply 1 − P(X < x). It represents the probability that the random variable exceeds x.

How do I find a percentile (inverse CDF) for a lognormal distribution?

Select 'Find x from probability p', enter a probability between 0 and 1 in the p field (e.g. 0.8 for the 80th percentile), and provide μ and σ. The calculator returns the x value such that P(X < x) = p. Internally this is computed as x = e^(μ + σ · Φ⁻¹(p)).

What is the probability density function (PDF) of a lognormal distribution?

The PDF is f(x) = [1 / (x · σ · √(2π))] · exp(−(ln(x) − μ)² / (2σ²)) for x > 0. It gives the relative likelihood of the variable taking a specific value. The calculator displays f(x) at your entered x value alongside the probability result.

How are the mean, median, and mode of a lognormal distribution calculated?

For X ~ LogN(μ, σ): the mean is e^(μ + σ²/2), the median is e^μ, and the mode is e^(μ − σ²). The variance is (e^(σ²) − 1) · e^(2μ + σ²). Note that the mean is always greater than the median, which is always greater than the mode for a lognormal distribution.

Can x be zero or negative in a lognormal distribution?

No. A lognormal distribution is only defined for strictly positive values (x > 0). The probability P(X ≤ 0) = 0. If you enter x = 0 or a negative value, the result is undefined. This is one of the key properties that makes the lognormal distribution suitable for modeling prices, sizes, and other inherently positive quantities.

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