Log-normal Distribution Variance Calculator

Log‑normal Distribution Variance Calculator

For a log‑normal distribution with parameters \(\mu\) and \(\sigma\) (where \(\ln(X) \sim N(\mu, \sigma^2)\)), the variance is: $$ \operatorname{Var}(X) = \Bigl(\exp\left(\sigma^2\right)-1\Bigr)\exp\Bigl(2\mu+\sigma^2\Bigr). $$

* Enter the location parameter \(\mu\) and the scale parameter \(\sigma\) (with \(\sigma > 0\)).

Step 1: Enter Parameters

e.g., 0

e.g., 1 (must be > 0)

How It Works

The log‑normal distribution is the distribution of a random variable whose logarithm is normally distributed.

If \(\ln(X) \sim N(\mu, \sigma^2)\), then the mean of \(X\) is: $$ E[X] = \exp\!\Bigl(\mu + \frac{\sigma^2}{2}\Bigr), $$ and the variance is: $$ \operatorname{Var}(X) = \Bigl(\exp\left(\sigma^2\right)-1\Bigr)\exp\Bigl(2\mu+\sigma^2\Bigr). $$

Enter the parameters \(\mu\) and \(\sigma\) to compute the variance.

Formula: \( \operatorname{Var}(X) = \Bigl(\exp\left(\sigma^2\right)-1\Bigr)\exp\Bigl(2\mu+\sigma^2\Bigr) \)

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