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\right)\right] \\[1ex]
&= \Phi{\left(\frac{\ln x -\mu}{\sigma} \right)} \end{align} 
| quantile = \begin{align}
&\exp\left( \mu + \sqrt{2\sigma^2}\operatorname{erf}^{-1}(2 p - 1) \right) \\1ex & = \exp(\mu + \sigma \Phi^{-1}(p)) \end{align} 
| mean =  \exp\left( \mu + \frac{\sigma^2}{2} \right) 
| median =  \exp( \mu ) 
| mode =  \exp\left( \mu - \sigma^2 \right) 
| variance =  \left[ \exp(\sigma^2) - 1 \right] \exp\left( 2 \mu + \sigma^2\right ) 
| skewness =  \left[ \exp\left( \sigma^2 \right) + 2 \right] \sqrt{\exp(\sigma^2) - 1 }
| kurtosis =  \exp\left( 4 \sigma^2 \right) + 2 \exp\left( 3 \sigma^2 \right) + 3 \exp\left( 2\sigma^2 \right) - 6 
| entropy =  \log_2 \left( \sqrt{2\pi e} \, \sigma e^{ \mu } \right) 
| mgf = defined only for numbers with a  real part, see text
| char = representation  \sum_{n=0}^{\infty} \frac{ {\left(i t\right)}^n }{ n! }e^{ n \mu + n^2 \sigma^2/2}  is asymptotically divergent, but adequate for most numerical purposes
| fisher =  \frac{1}{\sigma^2} \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} 
| moments =  \mu = \ln \operatorname{E}[X] - \frac{1}{2} \ln\left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right),

\sigma = \sqrt{ \ln \left( \frac{ \operatorname{Var}X }{ \operatorname{E}X^2 } + 1 \right) } 
| ES = \begin{align}
&\frac{ e^{ \mu + \frac{ \sigma^2 }{2}} }{ 2p } \left[ 1 + \operatorname{erf} \left( \frac{ \sigma }{ \sqrt{2} } + \operatorname{erf}^{-1}(2p-1) \right) \right] \\[0.5ex]
&= \frac{e^{ \mu + \frac{ \sigma^2 }{2}}}{1-p} \left[1 - \Phi(\Phi^{-1}(p) - \sigma)\right]
\end{align} 
}}

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a whose is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal distribution. Equivalently, if has a normal distribution, then the exponential function of , , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and sciences, as well as , and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics).

The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after . The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.

A log-normal process is the statistical realization of the multiplicative product of many independent , each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate —for which the mean and variance of are specified.

Definitions
Generation and parameters
Let Z be a standard normal variable, and let \mu and \sigma be two real numbers, with Then, the distribution of the random variable

X = e^{\mu + \sigma Z}

is called the log-normal distribution with parameters \mu and These are the (or ) and standard deviation of the variable's natural , not the expectation and standard deviation of X itself.

This relationship is true regardless of the base of the logarithmic or exponential function: If \log_a X is normally distributed, then so is \log_b X for any two positive numbers Likewise, if e^Y is log-normally distributed, then so is where

In order to produce a distribution with desired mean \mu_X and variance one uses \mu = \ln \frac{ \mu_X^2 }{ \sqrt{ \mu_X^2 + \sigma_X^2 } } and

Alternatively, the "multiplicative" or "geometric" parameters \mu^* = e^\mu and \sigma^* = e^\sigma can be used. They have a more direct interpretation: \mu^* is the of the distribution, and \sigma^* is useful for determining "scatter" intervals, see below.

Probability density function
A positive random variable X is log-normally distributed (i.e., if the natural logarithm of X is normally distributed with mean \mu and variance

\ln X \sim \mathcal{N}(\mu,\sigma^2)

Let \Phi and \varphi be respectively the cumulative probability distribution function and the probability density function of the \mathcal{N}( 0, 1 ) standard normal distribution, then we have that the probability density function of the log-normal distribution is given by:

\begin{align} f_X(x) & = \frac{d}{dx} \Pr\nolimits_X\left \\6pt & = \frac{d}{dx} \Pr\nolimits_X\left \\6pt & = \frac{d}{dx} \Phi{\left( \frac{ \ln x -\mu }{ \sigma } \right)} \\6pt & = \varphi{\left( \frac{\ln x - \mu} \sigma \right)} \frac{d}{dx} \left( \frac{ \ln x - \mu }{ \sigma }\right) \\6pt & = \varphi{\left( \frac{ \ln x - \mu }{ \sigma } \right)} \frac{ 1 }{ \sigma x } \\6pt & = \frac{ 1 }{ x \sigma\sqrt{2 \pi } } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right) ~. \end{align}

Cumulative distribution function
The cumulative distribution function is

F_X(x) = \Phi{\left( \frac{\ln x - \mu} \sigma \right)}

where \Phi is the cumulative distribution function of the standard normal distribution (i.e.,

This may also be expressed as follows:

\frac{1}{2} \left = \frac12 \operatorname{erfc} \left(-\frac{\ln x - \mu}{\sigma\sqrt{2}}\right)

where is the complementary error function.

Multivariate log-normal
If \boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma) is a multivariate normal distribution, then Y_i = \exp(X_i) has a multivariate log-normal distribution. The exponential is applied element-wise to the random vector \boldsymbol X. The mean of \boldsymbol Y is

\operatorname{E}\boldsymbol_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii}} ,

and its covariance matrix is

\operatorname{Var}\boldsymbol_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij}} - 1\right) .

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution.

Characteristic function and moment generating function
All moments of the log-normal distribution exist and

\operatorname{E}X^n = e^{n\mu+n^2\sigma^2/2}

This can be derived by letting z = \tfrac{\ln x - \mu}{\sigma} - n \sigma within the integral. However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero.

(2025). 9781118122372, Wiley.
Indeed, the expected value \operatorname{E}e^{t is not defined for any positive value of the argument t, since the defining integral diverges.

The characteristic function \operatorname{E}e^{i is defined for real values of , but is not defined for any complex value of that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor diverges:

\sum_{n=0}^\infty \frac}

where W is the Lambert W function. This approximation is derived via an asymptotic method, but it stays sharp all over the domain of convergence of \varphi.

Properties
Probability in different domains
The probability content of a log-normal distribution in any arbitrary domain can be computed to desired precision by first transforming the variable to normal, then numerically integrating using the ray-trace method. ( Matlab code)

Probabilities of functions of a log-normal variable
Since the probability of a log-normal can be computed in any domain, this means that the cdf (and consequently pdf and inverse cdf) of any function of a log-normal variable can also be computed. ( Matlab code)

Geometric or multiplicative moments
The of the log-normal distribution is \operatorname{GM}X = e^\mu = \mu^*. It equals the median. The geometric or multiplicative standard deviation is \operatorname{GSD}X = e^{\sigma} = \sigma^*.

By analogy with the arithmetic statistics, one can define a geometric variance, \operatorname{GVar}X = e^{\sigma^2}, and a geometric coefficient of variation, \operatorname{GCV}X = e^{\sigma} - 1, has been proposed. This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of \operatorname{CV} itself (see also Coefficient of variation).

Note that the geometric mean is smaller than the arithmetic mean. This is due to the AM–GM inequality and is a consequence of the logarithm being a . In fact,

\operatorname{E}X = e^{\mu + \frac12 \sigma^2} = e^{\mu} \cdot \sqrt{e^{\sigma^2}} = \operatorname{GM}X \cdot \sqrt{\operatorname{GVar}X}.

In finance, the term e^{-\sigma^2/2} is sometimes interpreted as a convexity correction. From the point of view of stochastic calculus, this is the same correction term as in Itō's lemma for geometric Brownian motion.

Arithmetic moments
For any real or complex number , the -th moment of a log-normally distributed variable is given by \operatorname{E}X^n = e^{n\mu + \frac{1}{2}n^2\sigma^2}.

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable are respectively given by:

\begin{align} 

\operatorname{E}[X] & = e^{\mu + \tfrac{1}{2}\sigma^2}, \\[4pt]
\operatorname{E}[X^2] & = e^{2\mu + 2\sigma^2}, \\[4pt]
\operatorname{Var}[X] & = \operatorname{E}[X^2] - \operatorname{E}[X]^2
= {\left(\operatorname{E}[X]\right)}^2 \left(e^{\sigma^2} - 1\right) \\[2pt]
&= e^{2\mu + \sigma^2} \left(e^{\sigma^2} - 1\right), \\[4pt]
\operatorname{SD}[X] & = \sqrt{\operatorname{Var}[X]}
= \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1} \\[2pt]
&= e^{\mu + \tfrac{1}{2}\sigma^2} \sqrt{e^{\sigma^2} - 1},
\end{align}

The arithmetic coefficient of variation \operatorname{CV}X is the ratio \tfrac{\operatorname{SD}X}{\operatorname{E}X}. For a log-normal distribution it is equal to \operatorname{CV}X = \sqrt{e^{\sigma^2} - 1}. This estimate is sometimes referred to as the "geometric CV" (GCV),Sawant, S.; Mohan, N. (2011) "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS" , PharmaSUG2011, Paper PO08 due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

The parameters and can be obtained, if the arithmetic mean and the arithmetic variance are known:

\begin{align} = \ln \frac{\operatorname{E}X^2}{\sqrt{\operatorname{Var}X + \operatorname{E}X^2}}, \\1ex 

\sigma^2 &= \ln \frac{\operatorname{E}[X^2]}{\operatorname{E}[X]^2}
= \ln \left(1 + \frac{\operatorname{Var}X}{\operatorname{E}X^2}\right). 
\end{align}

A probability distribution is not uniquely determined by the moments for . That is, there exist other distributions with the same set of moments. In fact, there is a whole family of distributions with the same moments as the log-normal distribution.

Mode, median, quantiles
The mode is the point of global maximum of the probability density function. In particular, by solving the equation (\ln f)'=0, we get that:

\operatorname{Mode}X = e^{\mu - \sigma^2}.

Since the log-transformed variable Y = \ln X has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of X are

q_X(\alpha) = \exp\left\mu = \mu^* (\sigma^*)^{q_\Phi(\alpha)},

where q_\Phi(\alpha) is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,

(2025). 9780632047635, Wiley-Blackwell.
print edition. Online eBook

\operatorname{Med}X = e^\mu = \mu^* ~.

Partial expectation
The partial expectation of a random variable X with respect to a threshold k is defined as

g(k) = \int_k^\infty x \, f_X(x)\, dx .

Alternatively, by using the definition of conditional expectation, it can be written as g(k) = \operatorname{E}X\mid \Pr(X>k). For a log-normal random variable, the partial expectation is given by:

\begin{align} g(k) &= \int_k^\infty x f_X(x)\, dx \\1ex &= e^{\mu+\tfrac{1}{2} \sigma^2}\, \Phi{\left(\frac{\mu-\ln k}{\sigma} - \sigma\right)} \end{align}

where \Phi is the normal cumulative distribution function. The derivation of the formula is provided in the . The partial expectation formula has applications in and , it is used in solving the partial differential equation leading to the Black–Scholes formula.

Conditional expectation
The conditional expectation of a log-normal random variable X—with respect to a threshold k—is its partial expectation divided by the cumulative probability of being in that range:

\begin{align} \operatorname{E}X\mid & = e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left\frac{\ln}}{\Phi {\left\frac{\ln}} \\8pt \operatorname{E}X &= e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{\Phi {\left\frac{\mu}}{1 - \Phi {\left\frac{\ln}} \\8pt \operatorname{E}X\mid] &= e^{\mu +\frac{\sigma^2}{2}} \cdot \frac{ 

\Phi{\left[\frac{\ln k_2 - \mu}{\sigma} - \sigma \right]} - \Phi{\left[\frac{\ln k_1 - \mu}{\sigma} - \sigma\right]}
}{ 
\Phi \left[\frac{\ln k_2 - \mu}{\sigma}\right]-\Phi \left[\frac{\ln k_1 - \mu}{\sigma}\right]
} \end{align}

Alternative parameterizations
In addition to the characterization by \mu, \sigma or \mu^*, \sigma^*, here are multiple ways how the log-normal distribution can be parameterized. , the knowledge base and ontology of probability distributions lists seven such forms:

  • with , , and standard deviation, , both on the log-scale Forbes et al. Probability Distributions (2011), John Wiley & Sons, Inc. P(x;\boldsymbol\mu,\boldsymbol\sigma) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp\left-\frac{(\ln
  • with mean, , and variance, , both on the log-scale P(x;\boldsymbol\mu,\boldsymbol {v}) = \frac{1}{x \sqrt{v} \sqrt{2 \pi}} \exp\left-\frac{(\ln
  • with , , on the natural scale and standard deviation, , on the log-scale P(x;\boldsymbol m,\boldsymbol \sigma) =\frac{1}{x \sigma \sqrt{2 \pi}} \exp\left-\frac{\ln^2(x/m)}{2
  • with median, , and coefficient of variation, , both on the natural scale P(x;\boldsymbol m,\boldsymbol {cv}) = \frac{1}{x \sqrt{\ln(cv^2+1)} \sqrt{2 \pi}} \exp\left-\frac{\ln^2(x/m)}{2\ln(cv^2+1)}\right
  • with mean, , and precision, , both on the log-scaleLunn, D. (2012). The BUGS book: a practical introduction to Bayesian analysis. Texts in statistical science. CRC Press. P(x;\boldsymbol\mu,\boldsymbol \tau) = \sqrt{\frac{\tau}{2 \pi}} \frac{1}{x} \exp\left-\frac{\tau}{2}(\ln
  • with median, , and geometric standard deviation, , both on the natural scale
P(x;\boldsymbol m,\boldsymbol {\sigma_g}) = \frac{1}{x \sqrt{2 \pi} \, \ln\sigma_g} \exp\left-\frac{\ln^2(x/m)}{2
  • with mean, , and standard deviation, , both on the natural scale P(x;\boldsymbol {\mu_N},\boldsymbol {\sigma_N}) = \frac{1}{x \sqrt{2 \pi \ln\left(1+\sigma_N^2/\mu_N^2\right)}} \exp\left-\frac{\left(

Examples for re-parameterization
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM and PopED.The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University. The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition \operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N) following formulas hold \mu_N = \exp(\mu+v/2) and \sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}.

For the transition \operatorname{LN7}(\mu_N, \sigma_N) \to \operatorname{LN2}(\mu, v) following formulas hold \mu = \ln \mu_N - \frac{1}{2} v and v = \ln(1+\sigma_N^2/\mu_N^2).

All remaining re-parameterisation formulas can be found in the specification document on the project website.ProbOnto website, URL: http://probonto.org

Multiple, reciprocal, power
  • Multiplication by a constant: If X \sim \operatorname{Lognormal}(\mu, \sigma^2) then a X \sim \operatorname{Lognormal}( \mu + \ln a, \sigma^2) for a > 0.
  • Reciprocal: If X \sim \operatorname{Lognormal}(\mu, \sigma^2) then \tfrac{1}{X} \sim \operatorname{Lognormal}(-\mu, \sigma^2).
  • Power: If X \sim \operatorname{Lognormal}(\mu, \sigma^2) then X^a \sim \operatorname{Lognormal}(a\mu, a^2 \sigma^2) for a \neq 0.

Multiplication and division of independent, log-normal random variables
If two independent, log-normal variables X_1 and X_2 are multiplied divided, the product ratio is again log-normal, with parameters \mu = \mu_1 + \mu_2 and where

More generally, if X_j \sim \operatorname{Lognormal} (\mu_j, \sigma_j^2) are n independent, log-normally distributed variables, then Y = \prod_{j=1}^n X_j \sim \operatorname{Lognormal} \Big( \sum_{j=1}^n\mu_j, \sum_{j=1}^n \sigma_j^2 \Big).

Multiplicative central limit theorem
The geometric or multiplicative mean of n independent, identically distributed, positive random variables X_i shows, for n \to \infty, approximately a log-normal distribution with parameters \mu = \operatorname{E}\ln and \sigma^2 = \operatorname{var}\ln/n, assuming \sigma^2 is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of \ln X_i to all have finite variance and satisfy the other conditions of any of the many variants of the central limit theorem.

This is commonly known as Gibrat's law.

Heavy-tailness of the Log-Normal
Whether a Log-Normal can be considered or not a true heavy-tail distribution is still debated. The main reason is that its variance is always finite, differently from what happen with certain Pareto distributions, for instance. However a recent study has shown how it is possible to create a Log-Normal distribution with infinite variance using Robinson Non-Standard Analysis.

Other
A set of data that arises from the log-normal distribution has a symmetric (see also Lorenz asymmetry coefficient).

The harmonic H, geometric G and arithmetic A means of this distribution are related; such relation is given by

H = \frac{G^2} A.

Log-normal distributions are infinitely divisible, but they are not stable distributions, which can be easily drawn from.

Related distributions
  • If X \sim \mathcal{N}(\mu, \sigma^2) is a normal distribution, then \exp(X) \sim \operatorname{Lognormal}(\mu, \sigma^2).
  • If X \sim \operatorname{Lognormal}(\mu, \sigma^2) is distributed log-normally, then \ln X \sim \mathcal{N}(\mu, \sigma^2) is a normal random variable.
  • Let X_j \sim \operatorname{Lognormal}(\mu_j, \sigma_j^2) be independent log-normally distributed variables with possibly varying \sigma and \mu parameters, and Y = \sum_{j = 1}^n X_j. The distribution of Y has no closed-form expression, but can be reasonably approximated by another log-normal distribution Z at the right tail. Its probability density function at the neighborhood of 0 has been characterized and it does not resemble any log-normal distribution. A commonly used approximation due to L.F. Fenton (but previously stated by R.I. Wilkinson and mathematically justified by Marlow) is obtained by matching the mean and variance of another log-normal distribution: \begin{align} 
\sigma^2_Z &= \ln\!\left[ \frac{\sum_j e^{2\mu_j+\sigma_j^2} \left(e^{\sigma_j^2} - 1\right)}

Method of moments
When the individual values x_1, x_2, \ldots, x_n are not available, but the sample's mean \bar x and standard deviation s Https://math.stackexchange.com/q/4355343< /ref> \begin{align} \mu &= \ln \frac{ \bar x} {\sqrt{1+\widehat\sigma^2/\bar x^2} } , \\1ex \sigma^2 &= \ln\left(1 + {\widehat\sigma^2} / \bar x^2 \right). \end{align}

Other estimators
Other estimators also exist, such as Finney's estimator,Finney, D. J. "On the distribution of a variate whose logarithm is normally distributed." Supplement to the Journal of the Royal Statistical Society 7.2 (1941): 155–161. the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator",Longford, Nicholas T. "Inference with the lognormal distribution." Journal of Statistical Planning and Inference 139.7 (2009): 2329–2340. also "A Conditional Mean Squared Error Estimator",Zellner, Arnold. "Bayesian and non-Bayesian analysis of the log-normal distribution and log-normal regression." Journal of the American Statistical Association 66.334 (1971): 327–330. and other variations as well.Tang, Qi. "Comparison of different methods for estimating log-normal means". MS thesis. East Tennessee State University, 2014. Https://dc.etsu.edu/cgi/viewcontent.cgi?article=3728&context=etd#page=12.13 pdf]Kwon, Yeil. "An alternative method for estimating lognormal means." Communications for Statistical Applications and Methods 28.4 (2021): 351–368. link

Interval estimates
The most efficient way to obtain interval estimates when analyzing log-normally distributed data consists of applying the well-known methods based on the normal distribution to logarithmically transformed data and then to back-transform results if appropriate.

Prediction intervals
A basic example is given by prediction intervals: For the normal distribution, the interval \mu-\sigma,\mu+\sigma contains approximately two thirds (68%) of the probability (or of a large sample), and \mu-2\sigma,\mu+2\sigma contain 95%. Therefore, for a log-normal distribution,
  • \mu^*/\sigma^*,\mu^*\cdot\sigma^*=\mu^* contains 2/3, and
  • \mu^*/(\sigma^*)^2,\mu^*\cdot(\sigma^*)^2 = \mu^* contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

Confidence interval for eμ
Using the principle, note that a confidence interval for \mu is \widehat\mu, where \mathop{se} = \widehat\sigma / \sqrt{n} is the standard error and q is the 97.5% quantile of a t distribution with n-1 degrees of freedom. Back-transformation leads to a confidence interval for \mu^* = e^\mu (the median), is: \widehat\mu^* with \operatorname{sem}^*=(\widehat\sigma^*)^{1/\sqrt{n}}

Confidence interval for
The literature discusses several options for calculating the confidence interval for \mu (the mean of the log-normal distribution). These include bootstrap as well as various other methods.Olsson, Ulf. "Confidence intervals for the mean of a log-normal distribution." Journal of Statistics Education 13.1 (2005). pdf Https://stats.stackexchange.com/q/33395< /ref>

The Cox Method proposes to plug-in the estimators \widehat \mu = \frac {\sum_i \ln x_i}{n}, \qquad S^2 = \frac {\sum_i \left( \ln x_i - \widehat \mu \right)^2} {n-1}

and use them to construct approximate confidence intervals in the following way: \mathrm{CI}(\operatorname{E}(X)) : \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)

We know that .}} Also, \widehat \mu is a normal distribution with parameters: \widehat \mu \sim N\left(\mu, \frac{\sigma^2}{n}\right)

S^2 has a chi-squared distribution, which is approximately normally distributed (via CLT), with parameters: Hence,

Since the sample mean and variance are independent, and the sum of normally distributed variables is also normal, we get that: \widehat \mu + \frac{S^2}{2} \dot \sim N\left(\mu + \frac{\sigma^2}{2}, \frac{\sigma^2}{n} + \frac{\sigma^4}{2(n-1)}\right) Based on the above, standard confidence intervals for \mu + \frac{\sigma^2}{2} can be constructed (using a ) as: \hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)} } And since confidence intervals are preserved for monotonic transformations, we get that: \mathrm{CI}\left(\operatorname{E}X = e^{\mu + \frac{\sigma^2}{2}}\right): \exp\left(\hat \mu + \frac{S^2}{2} \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S^2}{n} + \frac{S^4}{2(n-1)}} \right)

As desired.

Olsson 2005, proposed a "modified Cox method" by replacing z_{1-\frac{\alpha}{2}} with t_{n-1, 1-\frac{\alpha}{2}}, which seemed to provide better coverage results for small sample sizes.

Confidence interval for comparing two log normals
Comparing two log-normal distributions can often be of interest, for example, from a treatment and control group (e.g., in an A/B test). We have samples from two independent log-normal distributions with parameters (\mu_1, \sigma_1^2) and (\mu_2, \sigma_2^2), with sample sizes n_1 and n_2 respectively.

Comparing the medians of the two can easily be done by taking the log from each and then constructing straightforward confidence intervals and transforming it back to the exponential scale.

\mathrm{CI}(e^{\mu_1-\mu_2}): \exp\left(\hat \mu_1 - \hat \mu_2 \pm z_{1-\frac{\alpha}{2}} \sqrt{\frac{S_1^2}{n} + \frac{S_2^2}{n} } \right)

These CI are what's often used in epidemiology for calculation the CI for and . Confidence Intervals for Risk Ratios and Odds Ratios The way it is done there is that we have two approximately Normal distributions (e.g., p1 and p2, for RR), and we wish to calculate their ratio.

However, the ratio of the expectations (means) of the two samples might also be of interest, while requiring more work to develop. The ratio of their means is:

= e^{(\mu_1 - \mu_2) + \frac{1}{2} \left(\sigma_1^2 - \sigma_2^2\right)}

Plugin in the estimators to each of these parameters yields also a log normal distribution, which means that the Cox Method, discussed above, could similarly be used for this use-case:

\mathrm{CI}\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \sigma_1^2 / 2}}{e^{\mu_2 + \sigma_2^2 / 2}} \right): \exp\left(\left(\hat \mu_1 - \hat \mu_2 + \tfrac{1}{2}S_1^2 - \tfrac{1}{2}S_2^2\right) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)

To construct a confidence interval for this ratio, we first note that \hat \mu_1 - \hat \mu_2 follows a normal distribution, and that both S_1^2 and S_2^2 has a chi-squared distribution, which is approximately normally distributed (via CLT, with the relevant parameters).

This means that (\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \sim N\left((\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2), \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} + \frac{\sigma_1^4}{2(n_1-1)} + \frac{\sigma_2^4}{2(n_2-1)} \right)

Based on the above, standard confidence intervals can be constructed (using a ) as: (\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } And since confidence intervals are preserved for monotonic transformations, we get that: CI\left( \frac{\operatorname{E}(X_1)}{\operatorname{E}(X_2)} = \frac{e^{\mu_1 + \frac{\sigma_1^2}{2}}}{e^{\mu_2 + \frac{\sigma_2^2}{2}}} \right):e^{\left((\hat \mu_1 - \hat \mu_2 + \frac{1}{2}S_1^2 - \frac{1}{2}S_2^2) \pm z_{1-\frac{\alpha}{2}} \sqrt{ \frac{S_1^2}{n_1} + \frac{S_2^2}{n_2} + \frac{S_1^4}{2(n_1-1)} + \frac{S_2^4}{2(n_2-1)} } \right)}

As desired.

It's worth noting that naively using the MLE in the ratio of the two expectations to create a will lead to a consistent, yet biased, point-estimation (we use the fact that the estimator of the ratio is a log normal distribution):

\begin{align} \operatorname{E}\left &= \operatorname{E}\left\exp\left(\left(\widehat \\ &\approx \exp\left{(\mu_1 \end{align}

Extremal principle of entropy to fix the free parameter σ
In applications, \sigma is a parameter to be determined. For growing processes balanced by production and dissipation, the use of an extremal principle of Shannon entropy shows that \sigma = \frac 1 \sqrt{6}

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution. This relationship is determined by the base of natural logarithm, e = 2.718\ldots, and exhibits some geometrical similarity to the minimal surface energy principle. These scaling relations are useful for predicting a number of growth processes (epidemic spreading, droplet splashing, population growth, swirling rate of the bathtub vortex, distribution of language characters, velocity profile of turbulences, etc.). For example, the log-normal function with such \sigma fits well with the size of secondarily produced droplets during droplet impact and the spreading of an epidemic disease.

The value \sigma = 1 \big/ \sqrt{6} is used to provide a probabilistic solution for the Drake equation.

Occurrence and applications
The log-normal distribution is important in the description of natural phenomena. Many natural growth processes are driven by the accumulation of many small percentage changes which become additive on a log scale. Under appropriate regularity conditions, the distribution of the resulting accumulated changes will be increasingly well approximated by a log-normal, as noted in the section above on "Multiplicative Central Limit Theorem". This is also known as Gibrat's law, after Robert Gibrat (1904–1980) who formulated it for companies. If the rate of accumulation of these small changes does not vary over time, growth becomes independent of size. Even if this assumption is not true, the size distributions at any age of things that grow over time tends to be log-normal. Consequently, for measurements in healthy individuals are more accurately estimated by assuming a log-normal distribution than by assuming a symmetric distribution about the mean.

A second justification is based on the observation that fundamental natural laws imply multiplications and divisions of positive variables. Examples are the simple gravitation law connecting masses and distance with the resulting force, or the formula for equilibrium concentrations of chemicals in a solution that connects concentrations of educts and products. Assuming log-normal distributions of the variables involved leads to consistent models in these cases.

Specific examples are given in the following subsections. contains a review and table of log-normal distributions from geology, biology, medicine, food, ecology, and other areas. is a review article on log-normal distributions in neuroscience, with annotated bibliography.

Human behavior
  • The length of comments posted in Internet discussion forums follows a log-normal distribution.
  • Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.
  • The length of games tends to follow a log-normal distribution.
  • Onset durations of acoustic comparison stimuli that are matched to a standard stimulus follow a log-normal distribution.

Biology and medicine
  • Measures of size of living tissue (length, skin area, weight).
    (2025). 9780486611143, London.
  • Incubation period of diseases.Sartwell, Philip E. "The distribution of incubation periods of infectious disease." American journal of hygiene 51 (1950): 310–318.
  • Diameters of banana leaf spots, powdery mildew on barley.
  • For highly communicable epidemics, such as SARS in 2003, if public intervention control policies are involved, the number of hospitalized cases is shown to satisfy the log-normal distribution with no free parameters if an entropy is assumed and the standard deviation is determined by the principle of maximum rate of entropy production.
  • The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth.
  • The normalised RNA-Seq readcount for any genomic region can be well approximated by log-normal distribution.
  • The PacBio sequencing read length follows a log-normal distribution.
  • Certain physiological measurements, such as blood pressure of adult humans (after separation on male/female subpopulations).
  • Several variables, such as Cmax, elimination half-life and the elimination rate constant.
  • In neuroscience, the distribution of firing rates across a population of neurons is often approximately log-normal. This has been first observed in the cortex and striatum and later in hippocampus and entorhinal cortex, and elsewhere in the brain. Also, intrinsic gain distributions and synaptic weight distributions appear to be log-normal as well.
  • Neuron densities in the cerebral cortex, due to the noisy cell division process during neurodevelopment.
  • In operating-rooms management, the distribution of surgery duration.
  • In the size of avalanches of fractures in the cytoskeleton of living cells, showing log-normal distributions, with significantly higher size in cancer cells than healthy ones.

Chemistry
  • Particle size distributions and molar mass distributions.
  • The concentration of rare elements in minerals.
  • Diameters of crystals in ice cream, oil drops in mayonnaise, pores in cocoa press cake.

Physical sciences
  • In , the log-normal distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.
    (1994). 9789070754334, International Institute for Land Reclamation and Improvement (ILRI). .
    • The image on the right illustrates an example of fitting the log-normal distribution to ranked annually maximum one-day rainfalls showing also the 90% based on the binomial distribution. CumFreq, free software for distribution fitting
    • The rainfall data are represented by plotting positions as part of a cumulative frequency analysis.
  • In physical oceanography, the sizes of icebergs in the midwinter Southern Atlantic Ocean were found to follow a log-normal size distribution. The iceberg sizes, measured visually and by radar from the F.S. Polarstern in 1986, were thought to be controlled by wave action in heavy seas causing them to flex and break.
  • In atmospheric science, log-normal distributions (or distributions made by combining multiple log-normal functions) have been used to characterize both measurements and models of the sizes and concentrations of many different types of particles, from volcanic ash, to clouds and rain, to airborne microbes. The log-normal distribution is strictly empirical, so more physically-based distributions have been adopted to better understand processes controlling size distributions of particles such as volcanic ash.

Social sciences and demographics
  • In , there is evidence that the of 97–99% of the population is distributed log-normally.Clementi, Fabio; (2005) "Pareto's law of income distribution: Evidence for Germany, the United Kingdom, and the United States", EconWPA (The distribution of higher-income individuals follows a Pareto distribution).
  • If an income distribution follows a log-normal distribution with standard deviation \sigma, then the , commonly used to evaluate income inequality, can be computed as G = \operatorname{erf}\left(\frac{\sigma }{2 }\right) where \operatorname{erf} is the , since G = 2 \Phi{\left(\frac{\sigma }{\sqrt{2}}\right)} - 1, where \Phi(x) is the cumulative distribution function of a standard normal distribution.
  • In , in particular the Black–Scholes model, changes in the logarithm of exchange rates, price indices, and stock market indices are assumed normal (these variables behave like compound interest, not like simple interest, and so are multiplicative). However, some mathematicians such as Benoit Mandelbrot have argued
    (2025). 9780465043552, Basic Books. .
    that log-Lévy distributions, which possess , would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a .Bunchen, P., Advanced Option Pricing, University of Sydney coursebook, 2007 The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.
  • In , the number of citations to journal articles and patents follows a discrete log-normal distribution.
  • City sizes (population) satisfy Gibrat's Law. The growth process of city sizes is proportionate and invariant with respect to size. From the central limit theorem therefore, the log of city size is normally distributed.
  • The number of sexual partners appears to be best described by a log-normal distribution.

Technology
  • In reliability analysis, the log-normal distribution is often used to model times to repair a maintainable system.
    (2025). 9780470979822, John Wiley & Sons.
  • In wireless communication, "the local-mean power expressed in logarithmic values, such as dB or neper, has a normal (i.e., Gaussian) distribution." Also, the random obstruction of radio signals due to large buildings and hills, called , is often modeled as a log-normal distribution.
  • Particle size distributions produced by comminution with random impacts, such as in .
  • The distribution of publicly available audio and video data files () follows a log-normal distribution over five orders of magnitude.
  • File sizes of 140 million files on personal computers running the Windows OS, collected in 1999.
  • Sizes of text-based emails (1990s) and multimedia-based emails (2000s).
  • In computer networks and analysis, log-normal is shown as a good statistical model to represent the amount of traffic per unit time. This has been shown by applying a robust statistical approach on a large groups of real Internet traces. In this context, the log-normal distribution has shown a good performance in two main use cases: (1) predicting the proportion of time traffic will exceed a given level (for service level agreement or link capacity estimation) i.e. link dimensioning based on bandwidth provisioning and (2) predicting 95th percentile pricing.
  • in when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.ASTM D3654, Standard Test Method for Shear Adhesion on Pressure-Sensitive TapeswASTM D4577, Standard Test Method for Compression Resistance of a container Under Constant Load>\

See also
  • Heavy-tailed distribution
  • Log-distance path loss model
  • Modified lognormal power-law distribution

Notes
Further reading
External links

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