Let ${\displaystyle Z}$ be a standard normal variable, and let ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ be two real numbers, with ${\displaystyle \sigma >0}$. Then, the distribution of the random variable

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

is called the log-normal distribution with parameters ${\displaystyle \mu }$ and ${\displaystyle \sigma }$. These are the expected value (or mean) and standard deviation of the variable's natural logarithm, ${\displaystyle \ln X}$, not the expectation and standard deviation of ${\displaystyle X}$ itself.

This relationship is true regardless of the base of the logarithmic or exponential function: If ${\displaystyle \log _{a}X}$ is normally distributed, then so is ${\displaystyle \log _{b}X}$ for any two positive numbers ${\displaystyle a,b\neq 1}$. Likewise, if ${\displaystyle e^{Y}}$ is log-normally distributed, then so is ${\displaystyle a^{Y}}$, where ${\displaystyle 0<a\neq 1}$.

In order to produce a distribution with desired mean ${\displaystyle \mu _{X}}$ and variance ${\displaystyle \sigma _{X}^{2}}$, one uses ${\displaystyle \mu =\ln {\frac {\mu _{X}^{2}}{\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}}}}}$ and ${\displaystyle \sigma ^{2}=\ln \left(1+{\frac {\sigma _{X}^{2}}{\mu _{X}^{2}}}\right)}$.

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

A positive random variable ${\displaystyle X}$ is log-normally distributed (i.e., ${\textstyle X\sim \operatorname {Lognormal} \left(\mu ,\sigma ^{2}\right)}$), if the natural logarithm of ${\displaystyle X}$ is normally distributed with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$:

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

Let ${\displaystyle \Phi }$ and ${\displaystyle \varphi }$ be respectively the cumulative probability distribution function and the probability density function of the ${\displaystyle {\mathcal {N}}(0,1)}$ standard normal distribution, then we have that^[2][4] the probability density function of the log-normal distribution is given by:

$${\displaystyle {\begin{aligned}f_{X}(x)&={\frac {d}{dx}}\Pr \nolimits _{X}\left[X\leq x\right]\\[6pt]&={\frac {d}{dx}}\Pr \nolimits _{X}\left[\ln X\leq \ln x\right]\\[6pt]&={\frac {d}{dx}}\Phi {\left({\frac {\ln x-\mu }{\sigma }}\right)}\\[6pt]&=\varphi {\left({\frac {\ln x-\mu }{\sigma }}\right)}{\frac {d}{dx}}\left({\frac {\ln x-\mu }{\sigma }}\right)\\[6pt]&=\varphi {\left({\frac {\ln x-\mu }{\sigma }}\right)}{\frac {1}{\sigma x}}\\[6pt]&={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left(-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right)~.\end{aligned}}}$$

The cumulative distribution function is

$${\displaystyle F_{X}(x)=\Phi {\left({\frac {\ln x-\mu }{\sigma }}\right)}}$$

where ${\displaystyle \Phi }$ is the cumulative distribution function of the standard normal distribution (i.e., ${\displaystyle \operatorname {\mathcal {N}} (0,1)}$).

This may also be expressed as follows:^[2]

$${\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)}$$

where erfc is the complementary error function.

If ${\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})}$ is a multivariate normal distribution, then ${\displaystyle Y_{i}=\exp(X_{i})}$ has a multivariate log-normal distribution.^[6][7] The exponential is applied element-wise to the random vector ${\displaystyle {\boldsymbol {X}}}$. The mean of ${\displaystyle {\boldsymbol {Y}}}$ is

$${\displaystyle \operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},}$$

and its covariance matrix is

$${\displaystyle \operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}\left(e^{\Sigma _{ij}}-1\right).}$$

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

All moments of the log-normal distribution exist and

$${\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +n^{2}\sigma ^{2}/2}}$$

This can be derived by letting ${\textstyle z={\tfrac {\ln x-\mu }{\sigma }}-n\sigma }$ within the integral. However, the log-normal distribution is not determined by its moments.^[8] This implies that it cannot have a defined moment generating function in a neighborhood of zero.^[9] Indeed, the expected value ${\displaystyle \operatorname {E} [e^{tX}]}$ is not defined for any positive value of the argument ${\displaystyle t}$, since the defining integral diverges.

The characteristic function ${\displaystyle \operatorname {E} [e^{itX}]}$ is defined for real values of t, but is not defined for any complex value of t 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.^[10] In particular, its Taylor formal series diverges:

$${\displaystyle \sum _{n=0}^{\infty }{\frac {{\left(it\right)}^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}$$

However, a number of alternative divergent series representations have been obtained.^{[10][11][12][13]}

A closed-form formula for the characteristic function ${\displaystyle \varphi (t)}$ with ${\displaystyle t}$ in the domain of convergence is not known. A relatively simple approximating formula is available in closed form, and is given by^[14]

$${\displaystyle \varphi (t)\approx {\frac {\exp \left(-{\frac {W^{2}(-it\sigma ^{2}e^{\mu })+2W(-it\sigma ^{2}e^{\mu })}{2\sigma ^{2}}}\right)}{\sqrt {1+W{\left(-it\sigma ^{2}e^{\mu }\right)}}}}}$$

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

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.^[15] (Matlab code)

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.^[15] (Matlab code)

The geometric or multiplicative mean of the log-normal distribution is ${\displaystyle \operatorname {GM} [X]=e^{\mu }=\mu ^{*}}$. It equals the median. The geometric or multiplicative standard deviation is ${\displaystyle \operatorname {GSD} [X]=e^{\sigma }=\sigma ^{*}}$.^[16][17]

By analogy with the arithmetic statistics, one can define a geometric variance, ${\displaystyle \operatorname {GVar} [X]=e^{\sigma ^{2}}}$, and a geometric coefficient of variation,^[16] ${\displaystyle \operatorname {GCV} [X]=e^{\sigma }-1}$, 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 ${\displaystyle \operatorname {CV} }$ 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 concave function. In fact,^[18]

$${\displaystyle \operatorname {E} [X]=e^{\mu +{\frac {1}{2}}\sigma ^{2}}=e^{\mu }\cdot {\sqrt {e^{\sigma ^{2}}}}=\operatorname {GM} [X]\cdot {\sqrt {\operatorname {GVar} [X]}}.}$$

In finance, the term ${\displaystyle e^{-\sigma ^{2}/2}}$ 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.

For any real or complex number n, the n-th moment of a log-normally distributed variable X is given by^[4] $${\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +{\frac {1}{2}}n^{2}\sigma ^{2}}.}$$

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable X are respectively given by:^[2]

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

The arithmetic coefficient of variation ${\displaystyle \operatorname {CV} [X]}$ is the ratio ${\displaystyle {\tfrac {\operatorname {SD} [X]}{\operatorname {E} [X]}}}$. For a log-normal distribution it is equal to^[3] $${\displaystyle \operatorname {CV} [X]={\sqrt {e^{\sigma ^{2}}-1}}.}$$ This estimate is sometimes referred to as the "geometric CV" (GCV),^[19][20] 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 μ and σ can be obtained, if the arithmetic mean and the arithmetic variance are known:

$${\displaystyle {\begin{aligned}\mu &=\ln {\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {E} [X^{2}]}}}=\ln {\frac {\operatorname {E} [X]^{2}}{\sqrt {\operatorname {Var} [X]+\operatorname {E} [X]^{2}}}},\\[1ex]\sigma ^{2}&=\ln {\frac {\operatorname {E} [X^{2}]}{\operatorname {E} [X]^{2}}}=\ln \left(1+{\frac {\operatorname {Var} [X]}{\operatorname {E} [X]^{2}}}\right).\end{aligned}}}$$

A probability distribution is not uniquely determined by the moments E[Xⁿ] = e^{nμ + ⁠1/2⁠n2σ2} for n ≥ 1. That is, there exist other distributions with the same set of moments.^[4] In fact, there is a whole family of distributions with the same moments as the log-normal distribution.^{[citation needed]}

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

$${\displaystyle \operatorname {Mode} [X]=e^{\mu -\sigma ^{2}}.}$$

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

$${\displaystyle q_{X}(\alpha )=\exp \left[\mu +\sigma q_{\Phi }(\alpha )\right]=\mu ^{*}(\sigma ^{*})^{q_{\Phi }(\alpha )},}$$

where ${\displaystyle q_{\Phi }(\alpha )}$ is the quantile of the standard normal distribution.

Specifically, the median of a log-normal distribution is equal to its multiplicative mean,^[21]

$${\displaystyle \operatorname {Med} [X]=e^{\mu }=\mu ^{*}~.}$$

The partial expectation of a random variable ${\displaystyle X}$ with respect to a threshold ${\displaystyle k}$ is defined as

$${\displaystyle g(k)=\int _{k}^{\infty }x\,f_{X}(x)\,dx.}$$

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

$${\displaystyle {\begin{aligned}g(k)&=\int _{k}^{\infty }xf_{X}(x)\,dx\\[1ex]&=e^{\mu +{\tfrac {1}{2}}\sigma ^{2}}\,\Phi {\left({\frac {\mu -\ln k}{\sigma }}+\sigma \right)}\end{aligned}}}$$

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

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

$${\displaystyle {\begin{aligned}\operatorname {E} [X\mid X<k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi {\left[{\frac {\ln k-\mu }{\sigma }}-\sigma \right]}}{\Phi {\left[{\frac {\ln k-\mu }{\sigma }}\right]}}}\\[8pt]\operatorname {E} [X\mid X\geq k]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi {\left[{\frac {\mu -\ln k}{\sigma }}+\sigma \right]}}{1-\Phi {\left[{\frac {\ln k-\mu }{\sigma }}\right]}}}\\[8pt]\operatorname {E} [X\mid X\in [k_{1},k_{2}]]&=e^{\mu +{\frac {\sigma ^{2}}{2}}}\cdot {\frac {\Phi {\left[{\frac {\ln k_{2}-\mu }{\sigma }}-\sigma \right]}-\Phi {\left[{\frac {\ln k_{1}-\mu }{\sigma }}-\sigma \right]}}{\Phi \left[{\frac {\ln k_{2}-\mu }{\sigma }}\right]-\Phi \left[{\frac {\ln k_{1}-\mu }{\sigma }}\right]}}\end{aligned}}}$$

In addition to the characterization by ${\displaystyle \mu ,\sigma }$ or ${\displaystyle \mu ^{*},\sigma ^{*}}$, here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions^[22][23] lists seven such forms:

• LogNormal1(μ,σ) with mean, μ, and standard deviation, σ, both on the log-scale ^[24] $${\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}\right]}$$
• LogNormal2(μ,υ) with mean, μ, and variance, υ, both on the log-scale $${\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {v}})={\frac {1}{x{\sqrt {v}}{\sqrt {2\pi }}}}\exp \left[-{\frac {(\ln x-\mu )^{2}}{2v}}\right]}$$
• LogNormal3(m,σ) with median, m, on the natural scale and standard deviation, σ, on the log-scale^[24] $${\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma }})={\frac {1}{x\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\sigma ^{2}}}\right]}$$
• LogNormal4(m,cv) with median, m, and coefficient of variation, cv, both on the natural scale $${\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {cv}})={\frac {1}{x{\sqrt {\ln(cv^{2}+1)}}{\sqrt {2\pi }}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln(cv^{2}+1)}}\right]}$$
• LogNormal5(μ,τ) with mean, μ, and precision, τ, both on the log-scale^[25] $${\displaystyle P(x;{\boldsymbol {\mu }},{\boldsymbol {\tau }})={\sqrt {\frac {\tau }{2\pi }}}{\frac {1}{x}}\exp \left[-{\frac {\tau }{2}}(\ln x-\mu )^{2}\right]}$$
• LogNormal6(m,σ_g) with median, m, and geometric standard deviation, σ_g, both on the natural scale^[26] $${\displaystyle P(x;{\boldsymbol {m}},{\boldsymbol {\sigma _{g}}})={\frac {1}{x{\sqrt {2\pi }}\,\ln \sigma _{g}}}\exp \left[-{\frac {\ln ^{2}(x/m)}{2\ln ^{2}(\sigma _{g})}}\right]}$$
• LogNormal7(μ_N,σ_N) with mean, μ_N, and standard deviation, σ_N, both on the natural scale^[27] $${\displaystyle P(x;{\boldsymbol {\mu _{N}}},{\boldsymbol {\sigma _{N}}})={\frac {1}{x{\sqrt {2\pi \ln \left(1+\sigma _{N}^{2}/\mu _{N}^{2}\right)}}}}\exp \left[-{\frac {\left(\ln x-\ln {\frac {\mu _{N}}{\sqrt {1+\sigma _{N}^{2}/\mu _{N}^{2}}}}\right)^{2}}{2\ln \left(1+{\frac {\sigma _{N}^{2}}{\mu _{N}^{2}}}\right)}}\right]}$$

Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM^[28] and PopED.^[29] 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 ${\displaystyle \operatorname {LN2} (\mu ,v)\to \operatorname {LN7} (\mu _{N},\sigma _{N})}$ following formulas hold ${\textstyle \mu _{N}=\exp(\mu +v/2)}$ and ${\textstyle \sigma _{N}=\exp(\mu +v/2){\sqrt {\exp(v)-1}}}$.

For the transition ${\displaystyle \operatorname {LN7} (\mu _{N},\sigma _{N})\to \operatorname {LN2} (\mu ,v)}$ following formulas hold ${\textstyle \mu =\ln \mu _{N}-{\frac {1}{2}}v}$ and ${\textstyle v=\ln(1+\sigma _{N}^{2}/\mu _{N}^{2})}$.

All remaining re-parameterisation formulas can be found in the specification document on the project website.^[30]

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

If two independent, log-normal variables ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are multiplied [divided], the product [ratio] is again log-normal, with parameters ${\displaystyle \mu =\mu _{1}+\mu _{2}}$ [${\displaystyle \mu =\mu _{1}-\mu _{2}}$] and ${\displaystyle \sigma }$, where ${\displaystyle \sigma ^{2}=\sigma _{1}^{2}+\sigma _{2}^{2}}$.

More generally, if ${\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}$ are ${\displaystyle n}$ independent, log-normally distributed variables, then ${\textstyle Y=\prod _{j=1}^{n}X_{j}\sim \operatorname {Lognormal} {\Big (}\sum _{j=1}^{n}\mu _{j},\sum _{j=1}^{n}\sigma _{j}^{2}{\Big )}.}$

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

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of ${\displaystyle \ln X_{i}}$ 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.

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.^[31]

A set of data that arises from the log-normal distribution has a symmetric Lorenz curve (see also Lorenz asymmetry coefficient).^[32]

The harmonic ${\displaystyle H}$, geometric ${\displaystyle G}$ and arithmetic ${\displaystyle A}$ means of this distribution are related;^[33] such relation is given by

$${\displaystyle H={\frac {G^{2}}{A}}.}$$

Log-normal distributions are infinitely divisible,^[34] but they are not stable distributions, which can be easily drawn from.^[35]

• If ${\displaystyle X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ is a normal distribution, then ${\displaystyle \exp(X)\sim \operatorname {Lognormal} (\mu ,\sigma ^{2}).}$
• If ${\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}$ is distributed log-normally, then ${\displaystyle \ln X\sim {\mathcal {N}}(\mu ,\sigma ^{2})}$ is a normal random variable.
• Let ${\displaystyle X_{j}\sim \operatorname {Lognormal} (\mu _{j},\sigma _{j}^{2})}$ be independent log-normally distributed variables with possibly varying ${\displaystyle \sigma }$ and ${\displaystyle \mu }$ parameters, and ${\textstyle Y=\sum _{j=1}^{n}X_{j}}$. The distribution of ${\displaystyle Y}$ has no closed-form expression, but can be reasonably approximated by another log-normal distribution ${\displaystyle Z}$ at the right tail.^[36] Its probability density function at the neighborhood of 0 has been characterized^[35] 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^[37]) is obtained by matching the mean and variance of another log-normal distribution: $${\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[{\frac {\sum _{j}e^{2\mu _{j}+\sigma _{j}^{2}}\left(e^{\sigma _{j}^{2}}-1\right)}{{\left(\sum _{j}e^{\mu _{j}+\sigma _{j}^{2}/2}\right)}^{2}}}+1\right],\\[1ex]\mu _{Z}&=\ln \!\left[\sum _{j}e^{\mu _{j}+\sigma _{j}^{2}/2}\right]-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}$$ In the case that all ${\displaystyle X_{j}}$ have the same variance parameter ${\displaystyle \sigma _{j}=\sigma }$, these formulas simplify to $${\displaystyle {\begin{aligned}\sigma _{Z}^{2}&=\ln \!\left[\left(e^{\sigma ^{2}}-1\right){\frac {\sum _{j}e^{2\mu _{j}}}{{\left(\sum _{j}e^{\mu _{j}}\right)}^{2}}}+1\right],\\[1ex]\mu _{Z}&=\ln \!\left[\sum _{j}e^{\mu _{j}}\right]+{\frac {\sigma ^{2}}{2}}-{\frac {\sigma _{Z}^{2}}{2}}.\end{aligned}}}$$

For a more accurate approximation, one can use the Monte Carlo method to estimate the cumulative distribution function, the pdf and the right tail.^[38][39] The cdf and pdf of the sum of correlated log-normally distributed random variables can also be approximated by Monte Carlo simulation.^[40]

• If ${\displaystyle X\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}$ then ${\displaystyle X+c}$ is said to have a Three-parameter log-normal distribution with support ${\displaystyle x\in (c,+\infty )}$.^[41] ${\displaystyle \operatorname {E} [X+c]=\operatorname {E} [X]+c}$, ${\displaystyle \operatorname {Var} [X+c]=\operatorname {Var} [X]}$.
• The log-normal distribution is a special case of the semi-bounded Johnson's SU-distribution.^[42]
• If ${\displaystyle X\mid Y\sim \operatorname {Rayleigh} (Y)}$ with ${\displaystyle Y\sim \operatorname {Lognormal} (\mu ,\sigma ^{2})}$, then ${\displaystyle X\sim \operatorname {Suzuki} (\mu ,\sigma )}$ (Suzuki distribution).
• A substitute for the log-normal whose integral can be expressed in terms of more elementary functions^[43] can be obtained based on the logistic distribution to get an approximation for the CDF $${\displaystyle F(x;\mu ,\sigma )=\left[\left({\frac {e^{\mu }}{x}}\right)^{\pi /(\sigma {\sqrt {3}})}+1\right]^{-1}.}$$ This is a log-logistic distribution.

For determining the maximum likelihood estimators of the log-normal distribution parameters μ and σ, we can use the same procedure as for the normal distribution. Note that $${\displaystyle L(\mu ,\sigma )=\prod _{i=1}^{n}{\frac {1}{x_{i}}}\varphi _{\mu ,\sigma }(\ln x_{i}),}$$ where ${\displaystyle \varphi }$ is the density function of the normal distribution ${\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})}$. Therefore, the log-likelihood function is $${\displaystyle \ell (\mu ,\sigma \mid x_{1},x_{2},\ldots ,x_{n})=-\sum _{i}\ln x_{i}+\ell _{N}(\mu ,\sigma \mid \ln x_{1},\ln x_{2},\dots ,\ln x_{n}).}$$

Since the first term is constant with regard to μ and σ, both logarithmic likelihood functions, ${\displaystyle \ell }$ and ${\displaystyle \ell _{N}}$, reach their maximum with the same ${\displaystyle \mu }$ and ${\displaystyle \sigma }$. Hence, the maximum likelihood estimators are identical to those for a normal distribution for the observations ${\displaystyle \ln x_{1},\ln x_{2},\dots ,\ln x_{n})}$, $${\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad {\widehat {\sigma }}^{2}={\frac {\sum _{i}{\left(\ln x_{i}-{\widehat {\mu }}\right)}^{2}}{n}}.}$$

For finite n, the estimator for ${\displaystyle \mu }$ is unbiased, but the one for ${\displaystyle \sigma }$ is biased. As for the normal distribution, an unbiased estimator for ${\displaystyle \sigma }$ can be obtained by replacing the denominator n by n−1 in the equation for ${\displaystyle {\widehat {\sigma }}^{2}}$.

From this, the MLE for the expectancy of x is:^[44] ${\displaystyle {\widehat {\theta }}_{\text{MLE}}={\widehat {\operatorname {E} [X]}}_{\text{MLE}}=e^{{\hat {\mu }}+{{\hat {\sigma }}^{2}}/{2}}}$

When the individual values ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ are not available, but the sample's mean ${\displaystyle {\bar {x}}}$ and standard deviation s is, then the method of moments can be used. The corresponding parameters are determined by the following formulas, obtained from solving the equations for the expectation ${\displaystyle \operatorname {E} [X]}$ and variance ${\displaystyle \operatorname {Var} [X]}$ for ${\displaystyle \mu }$ and ${\displaystyle \sigma }$:^[45] $${\displaystyle {\begin{aligned}\mu &=\ln {\frac {\bar {x}}{\sqrt {1+{\widehat {\sigma }}^{2}/{\bar {x}}^{2}}}},\\[1ex]\sigma ^{2}&=\ln \left(1+{{\widehat {\sigma }}^{2}}/{\bar {x}}^{2}\right).\end{aligned}}}$$

Other estimators also exist, such as Finney's UMVUE estimator,^[46] the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator",^[47] also "A Conditional Mean Squared Error Estimator",^[48] and other variations as well.^[49][50]

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.

A basic example is given by prediction intervals: For the normal distribution, the interval ${\displaystyle [\mu -\sigma ,\mu +\sigma ]}$ contains approximately two thirds (68%) of the probability (or of a large sample), and ${\displaystyle [\mu -2\sigma ,\mu +2\sigma ]}$ contain 95%. Therefore, for a log-normal distribution,

• ${\displaystyle [\mu ^{*}/\sigma ^{*},\mu ^{*}\cdot \sigma ^{*}]=[\mu ^{*}{}^{\times }\!\!/\sigma ^{*}]}$ contains 2/3, and
• ${\displaystyle [\mu ^{*}/(\sigma ^{*})^{2},\mu ^{*}\cdot (\sigma ^{*})^{2}]=[\mu ^{*}{}^{\times }\!\!/(\sigma ^{*})^{2}]}$ contains 95% of the probability. Using estimated parameters, then approximately the same percentages of the data should be contained in these intervals.

Using the principle, note that a confidence interval for ${\displaystyle \mu }$ is ${\displaystyle [{\widehat {\mu }}\pm q\cdot {\widehat {\mathop {se} }}]}$, where ${\displaystyle \mathop {se} ={\widehat {\sigma }}/{\sqrt {n}}}$ 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 ${\displaystyle \mu ^{*}=e^{\mu }}$ (the median), is: $${\displaystyle [{\widehat {\mu }}^{*}{}^{\times }\!\!/(\operatorname {sem} ^{*})^{q}]}$$ with ${\displaystyle \operatorname {sem} ^{*}=({\widehat {\sigma }}^{*})^{1/{\sqrt {n}}}}$

The literature discusses several options for calculating the confidence interval for ${\displaystyle \mu }$ (the mean of the log-normal distribution). These include bootstrap as well as various other methods.^[51][52]

The Cox Method^[a] proposes to plug-in the estimators $${\displaystyle {\widehat {\mu }}={\frac {\sum _{i}\ln x_{i}}{n}},\qquad S^{2}={\frac {\sum _{i}\left(\ln x_{i}-{\widehat {\mu }}\right)^{2}}{n-1}}}$$

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

Olsson 2005, proposed a "modified Cox method" by replacing ${\displaystyle z_{1-{\frac {\alpha }{2}}}}$ with ${\displaystyle t_{n-1,1-{\frac {\alpha }{2}}}}$, which seemed to provide better coverage results for small sample sizes.^{[51]: Section 3.4}

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 ${\displaystyle (\mu _{1},\sigma _{1}^{2})}$ and ${\displaystyle (\mu _{2},\sigma _{2}^{2})}$, with sample sizes ${\displaystyle n_{1}}$ and ${\displaystyle n_{2}}$ 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.

$${\displaystyle \mathrm {CI} (e^{\mu _{1}-\mu _{2}}):\exp \left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n}}+{\frac {S_{2}^{2}}{n}}}}\right)}$$

These CI are what's often used in epidemiology for calculation the CI for relative-risk and odds-ratio.^[55] 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.^[b]

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:

$${\displaystyle {\frac {\operatorname {E} (X_{1})}{\operatorname {E} (X_{2})}}={\frac {e^{\mu _{1}+\sigma _{1}^{2}/2}}{e^{\mu _{2}+\sigma _{2}^{2}/2}}}=e^{(\mu _{1}-\mu _{2})+{\frac {1}{2}}\left(\sigma _{1}^{2}-\sigma _{2}^{2}\right)}}$$

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:

$${\displaystyle \mathrm {CI} \left({\frac {\operatorname {E} (X_{1})}{\operatorname {E} (X_{2})}}={\frac {e^{\mu _{1}+\sigma _{1}^{2}/2}}{e^{\mu _{2}+\sigma _{2}^{2}/2}}}\right):\exp \left(\left({\hat {\mu }}_{1}-{\hat {\mu }}_{2}+{\tfrac {1}{2}}S_{1}^{2}-{\tfrac {1}{2}}S_{2}^{2}\right)\pm z_{1-{\frac {\alpha }{2}}}{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}+{\frac {S_{1}^{4}}{2(n_{1}-1)}}+{\frac {S_{2}^{4}}{2(n_{2}-1)}}}}\right)}$$