Compute P(X ≤ x), mean, median, mode, variance, and standard deviation
This calculator computes the cumulative probability \( P(X \le x) \), mean, median, mode, variance, and standard deviation for a log-normal distribution.
Important: the parameters \( \mu \) and \( \sigma \) refer to the underlying normal distribution of \( \ln(X) \), not to the mean and standard deviation of \( X \) itself. The mean of \( X \) is \( e^{\mu + \sigma^2/2} \), which is always larger than the median \( e^\mu \) because the log-normal distribution is right-skewed.
The probability density function of the log-normal distribution is:
\[ f(x) = \frac{1}{x\sigma\sqrt{2\pi}}\, e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}, \quad x > 0 \]
The cumulative probability \( P(X \le x) \) can be expressed via the error function:
\[ F_X(x) = \frac{1}{2}\left(1 + \operatorname{Erf}\!\left(\frac{\ln x - \mu}{\sigma\sqrt{2}}\right)\right) \]where \( \operatorname{Erf}(x) \) is the error function. Equivalently, \( F_X(x) = \Phi\!\left(\frac{\ln x - \mu}{\sigma}\right) \), where \( \Phi \) is the standard normal CDF.
| Quantity | Formula |
|---|---|
| Mean | \( e^{\mu + \sigma^2/2} \) |
| Median | \( e^{\mu} \) |
| Mode | \( e^{\mu - \sigma^2} \) |
| Variance | \( (e^{\sigma^2} - 1)\, e^{2\mu + \sigma^2} \) |
| Standard deviation | \( \sqrt{(e^{\sigma^2} - 1)\, e^{2\mu + \sigma^2}} \) |
The mean is always greater than the median for a log-normal distribution (because of the long right tail), and the median is always greater than the mode. For small \( \sigma \) the three nearly coincide.
Enter \( \mu \) (any real number), \( \sigma > 0 \), and \( x > 0 \).
Cumulative probability.
\[ F_X(2) = \Phi\!\left(\frac{\ln 2 - 3}{2}\right) = \Phi\!\left(\frac{0.6931 - 3}{2}\right) = \Phi(-1.1534) \approx 0.1244 \]Descriptive statistics.
Observation. The mean (≈148) is much larger than the median (≈20) — the signature of a right-skewed distribution.
For \( \mu = 0 \), \( \sigma = 1 \), find \( P(X \le 1) \).
(The median of the log-normal with \( \mu = 0 \) is \( e^0 = 1 \).)
For \( \mu = 1 \), \( \sigma = 0.5 \), compute the mean, median, and mode.
Observe the ordering: mode < median < mean, as expected for a right-skewed distribution.
What happens to the mean, median, and mode of a log-normal distribution as \( \sigma \to 0 \)?
As \( \sigma \to 0 \):
All three converge to \( e^\mu \) — the distribution becomes a point mass at \( e^\mu \). For small but positive \( \sigma \), the three are nearly identical; the right-skew becomes more pronounced as \( \sigma \) increases.
The distribution of a positive random variable \(X\) whose logarithm \( \ln(X) \) is normally distributed. Its PDF is \( f(x) = \dfrac{1}{x\sigma\sqrt{2\pi}}\, e^{-(\ln x - \mu)^2/(2\sigma^2)} \) for \(x > 0\). Used to model incomes, stock prices, and biological measurements.
They are the mean and standard deviation of the underlying normal variable \( \ln(X) \), not of \(X\) itself. The mean of \(X\) is \( e^{\mu + \sigma^2/2} \).
The log-normal distribution is right-skewed with a long upper tail. The mean \( e^{\mu + \sigma^2/2} \) is pulled upward by extreme large values, while the median \( e^\mu \) splits the distribution in half. The gap grows with \( \sigma \).
The median is \( e^\mu \). It represents the middle value (50% below, 50% above) and is often preferred to the mean as a "typical" value because it's not pulled up by the right tail.
Last updated: October 20, 2026