Lognormal Distribution Calculator

Enter the lognormal parameters mu and sigma and a value x to compute the probability density, cumulative probability, upper tail probability, mean, median, mode, and variance, with curve charts.

Input

Enter the lognormal parameters mu and sigma and the value x you want to evaluate. The density, lower and upper tail probabilities, and more are computed.

Value x (x positive)

Enter a positive value at which to evaluate the probability.

mu (mean of the log)

Mean of the normal distribution that ln X follows.

sigma (standard deviation of the log)

Standard deviation of the normal distribution that ln X follows. Enter a positive value.

Result

Lower tail probability P that X is at most 2

75.5891 percent

Upper tail probability

24.4109 percent

Probability density PDF

0.15687402

Mean

1.64872127

Median

Mode

0.36787944

Variance

4.67077427

Probability density function chart

Cumulative distribution function chart

How it works

A lognormal distribution describes a variable X whose natural logarithm ln X follows a normal distribution with mean mu and standard deviation sigma. The value x must be positive.
The probability density function is f(x) = 1 / (x sigma √(2π)) × exp(−(ln x − mu)^2 / (2 sigma^2)).
The cumulative distribution function (lower tail probability) is F(x) = Φ((ln x − mu) / sigma), where Φ is the standard normal CDF. This tool evaluates Φ accurately using the error function erf.
The upper tail probability is 1 − F(x).
The mean is exp(mu + sigma^2 / 2), the median is exp(mu), the mode is exp(mu − sigma^2), and the variance is (exp(sigma^2) − 1) × exp(2 mu + sigma^2).
Note that mu and sigma are parameters of the underlying normal distribution of ln X, not the mean and standard deviation of X itself.

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