Log-normal distribution

\right)\right] \\[1ex]

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

\end{align}</math>

| quantile = <math>\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}</math>

| mean = <math> \exp\left( \mu + \frac{\sigma^2}{2} \right) </math>

| median = <math> \exp( \mu ) </math>

| mode = <math> \exp\left( \mu - \sigma^2 \right) </math>

| variance = <math> \left[ \exp(\sigma^2) - 1 \right] \exp\left( 2 \mu + \sigma^2\right ) </math>

| skewness = <math> \left[ \exp\left( \sigma^2 \right) + 2 \right] \sqrt{\exp(\sigma^2) - 1 }</math>

| kurtosis = <math> \exp\left( 4 \sigma^2 \right) + 2 \exp\left( 3 \sigma^2 \right) + 3 \exp\left( 2\sigma^2 \right) - 6 </math>

| entropy = <math> \log_2 \left( \sqrt{2\pi e} \, \sigma e^{ \mu } \right) </math>

| mgf = defined only for numbers with a real part, see text

| char = representation <math> \sum_{n=0}^{\infty} \frac{ {\left(i t\right)}^n }{ n! }e^{ n \mu + n^2 \sigma^2/2} </math> is asymptotically divergent, but adequate for most numerical purposes

| fisher = <math> \frac{1}{\sigma^2} \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} </math>

| moments = <math> \mu = \ln \operatorname{E}[X] - \frac{1}{2} \ln\left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right),</math>

<br/>

<math> \sigma = \sqrt{ \ln \left( \frac{ \operatorname{Var}[X] }{ \operatorname{E}[X]^2 } + 1 \right) } </math>

| ES = <math>\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}</math>

<!-- end "infobox" -->

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm 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 engineering sciences, as well as medicine, economics 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 Francis Galton.

Definitions

Generation and parameters

Let <math> Z </math> be a standard normal variable, and let <math>\mu</math> and <math>\sigma</math> be two real numbers, with Then, the distribution of the random variable

<math display="block"> X = e^{\mu + \sigma Z} </math>

is called the log-normal distribution with parameters <math>\mu</math> and These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of <math> X </math> itself.

thumb|The cumulative distribution function and the corresponding probability density function of a log-normal distribution plotted on a semi-log plot. The median value is equal to 1, and the modal value (the peak of the probability density function) is equal to 1/e. In this case, 1/e corresponds to roughly the 16th percentile, 1 corresponds to the median (50th percentile), and e corresponds to roughly the 84th percentile.

thumb|A log-normal distribution on a semi-log plot with a median of 1, whose ~84th percentile value is 10 times the median, whose ~16th percentile value is 1/10th the median, and whose modal value (~.005) is at roughly the 1st percentile. The peak density of values around the mode is high, but the distribution would possess a tall, narrow peak, and a long right tail on a linear-linear plot.

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

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

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

Probability density function

A positive random variable <math> X </math> is log-normally distributed (i.e., if the natural logarithm of <math> X </math> is normally distributed with mean <math> \mu</math> and variance

<math display="block"> \ln X \sim \mathcal{N}(\mu,\sigma^2)</math>

Let <math> \Phi </math> and <math> \varphi </math> be respectively the cumulative probability distribution function and the probability density function of the <math> \mathcal{N}( 0, 1 ) </math> standard normal distribution, then we have that The exponential is applied element-wise to the random vector <math>\boldsymbol X</math>. The mean of <math>\boldsymbol Y</math> is

<math display="block">\operatorname{E}[\boldsymbol Y]_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii ,</math>

and its covariance matrix is

<math display="block">\operatorname{Var}[\boldsymbol Y]_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij - 1\right) . </math>

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

<math display="block">\operatorname{E}[X^n] = e^{n\mu+n^2\sigma^2/2}\,.</math>

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. Indeed, the expected value <math>\operatorname{E}[e^{t X}]</math> is not defined for any positive value of the argument <math>t</math>, since the defining integral diverges.

The characteristic function <math>\operatorname{E}[e^{i t X}]</math> 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 formal series diverges:

<math display="block">\sum_{n=0}^\infty \frac}</math>

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

Properties

Geometric or multiplicative moments

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

By analogy with the arithmetic statistics, one can define a geometric variance, <math>\operatorname{GVar}[X] = e^{\sigma^2}</math>, and a geometric coefficient of variation,

<math display="block">\operatorname{E}[X] = e^{\mu + \frac12 \sigma^2} = e^{\mu} \cdot \sqrt{e^{\sigma^2 = \operatorname{GM}[X] \cdot \sqrt{\operatorname{GVar}[X]}.</math>

In finance, the term <math>e^{-\sigma^2/2}</math> 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 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:

<math display="block">\begin{align}

\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{align}</math>

A probability distribution is not uniquely determined by the moments for . That is, there exist other distributions with the same set of moments.

<math display="block">\operatorname{Med}[X] = e^\mu = \mu^* ~.</math>

Partial expectation

The partial expectation of a random variable <math>X</math> with respect to a threshold <math>k</math> is defined as

<math display="block"> g(k) = \int_k^\infty x \, f_X(x)\, dx . </math>

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

<math display="block">\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} </math>

where <math>\Phi</math> 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.

Conditional expectation

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

<math display="block">\begin{align}

\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{align}</math>

Alternative parameterizations

In addition to the characterization by <math>\mu, \sigma</math> or <math>\mu^*, \sigma^*</math>, here are multiple ways how the log-normal distribution can be parameterized. ProbOnto, the knowledge base and ontology of probability distributions lists seven such forms: thumb|400px|Overview of parameterizations of the log-normal distributions.

• with mean, , and standard deviation, , both on the log-scale <math display="block">P(x;\boldsymbol\mu,\boldsymbol\sigma) = \frac{1}{x \sigma \sqrt{2 \pi \exp\left[-\frac{(\ln x - \mu)^2}{2 \sigma^2}\right]</math>
• with mean, , and variance, , both on the log-scale <math display="block">P(x;\boldsymbol\mu,\boldsymbol {v}) = \frac{1}{x \sqrt{v} \sqrt{2 \pi \exp\left[-\frac{(\ln x - \mu)^2}{2 v}\right]</math>
• with median, , on the natural scale and standard deviation, , on the log-scale <math display="block">P(x;\boldsymbol\mu,\boldsymbol \tau) = \sqrt{\frac{\tau}{2 \pi \frac{1}{x} \exp\left[-\frac{\tau}{2}(\ln x-\mu)^2\right]</math>
• with median, , and geometric standard deviation, , both on the natural scale <math display="block">

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]</math>

• with mean, , and standard deviation, , both on the natural scale <math display="block">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]</math>

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 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 <math>\operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N)</math> following formulas hold <math display="inline">\mu_N = \exp(\mu+v/2) </math> and <math display="inline">\sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}</math>.

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

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

Multiple, reciprocal, power

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

Multiplication and division of independent, log-normal random variables

If two independent, log-normal variables <math>X_1</math> and <math>X_2</math> are multiplied [divided], the product [ratio] is again log-normal, with parameters <math>\mu = \mu_1 + \mu_2</math> and where

More generally, if <math>X_j \sim \operatorname{Lognormal} (\mu_j, \sigma_j^2)</math> are <math>n</math> independent, log-normally distributed variables, then <math display="inline">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).</math>

<span class="anchor" id="Multiplicative Central Limit Theorem"></span>Multiplicative central limit theorem

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

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of <math>\ln X_i</math> 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.

Other

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

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

<math display="block">H = \frac{G^2} A.</math>

Log-normal distributions are infinitely divisible, Its probability density function at the neighborhood of 0 has been characterized) is obtained by matching the mean and variance of another log-normal distribution: <math display="block">\begin{align}

\sigma^2_Z &= \ln\!\left[ \frac{\sum_j e^{2\mu_j+\sigma_j^2} \left(e^{\sigma_j^2} - 1\right)}

</math>

Method of moments

When the individual values <math>x_1, x_2, \ldots, x_n</math> are not available, but the sample's mean <math>\bar x</math> 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 <math>\operatorname{E}[X]</math> and variance <math>\operatorname{Var}[X]</math> for <math>\mu</math> and <math>\sigma</math>:

<math display="block"> \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}</math>

Other estimators

Other estimators also exist, such as Finney's UMVUE estimator, the "Approximately Minimum Mean Squared Error Estimator", the "Approximately Unbiased Estimator" and "Minimax Estimator", also "A Conditional Mean Squared Error Estimator", and other variations as well.

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 <math>[\mu-\sigma,\mu+\sigma]</math> contains approximately two thirds (68%) of the probability (or of a large sample), and <math>[\mu-2\sigma,\mu+2\sigma]</math> contain 95%. Therefore, for a log-normal distribution,

• <math>[\mu^*/\sigma^*,\mu^*\cdot\sigma^*]=[\mu^* {}^\times\!\!/ \sigma^*]</math> contains 2/3, and
• <math>[\mu^*/(\sigma^*)^2,\mu^*\cdot(\sigma^*)^2] = [\mu^* {}^\times\!\!/ (\sigma^*)^2]</math> 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 e^μ

Using the principle, note that a confidence interval for <math>\mu</math> is <math>[\widehat\mu \pm q \cdot \widehat\mathop{se}]</math>, where <math>\mathop{se} = \widehat\sigma / \sqrt{n}</math> 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 <math>\mu^* = e^\mu</math> (the median), is:

<math display="block">[\widehat\mu^* {}^\times\!\!/ (\operatorname{sem}^*)^q]</math> with <math>\operatorname{sem}^*=(\widehat\sigma^*)^{1/\sqrt{n</math>

Confidence interval for

The literature discusses several options for calculating the confidence interval for <math>\mu</math> (the mean of the log-normal distribution). These include bootstrap as well as various other methods.

The Cox Method{e^{\mu_2 + \sigma_2^2 /2

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

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:

<math display="block">\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)</math>

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

This means that

<math display="block">(\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)</math>

Based on the above, standard confidence intervals can be constructed (using a Pivotal quantity) as: <math>(\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)} } </math>

And since confidence intervals are preserved for monotonic transformations, we get that:

<math>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)}</math>

As desired.

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

\end{align} </math>

<math display="block">\begin{align}

\operatorname{E}\left[ \frac{\widehat \operatorname{E}(X_1)}{\widehat \operatorname{E}(X_2)} \right]

&= \operatorname{E}\left[\exp\left(\left(\widehat \mu_1 - \widehat \mu_2\right) + \tfrac{1}{2} \left(S_1^2 - S_2^2\right)\right)\right] \\

&\approx \exp\left[{(\mu_1 - \mu_2) + \frac{1}{2}(\sigma_1^2 - \sigma_2^2) + \frac{1}{2}\left( \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) }\right]

\end{align} </math>

Extremal principle of entropy to fix the free parameter σ

In applications, <math>\sigma</math> 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

<math display="block">\sigma = \frac 1 \sqrt{6} </math>

This value can then be used to give some scaling relation between the inflexion point and maximum point of the log-normal distribution.

The value <math display="inline">\sigma = 1 \big/ \sqrt{6}</math> 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, reference ranges 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.

• Users' dwell time on online articles (jokes, news etc.) follows a log-normal distribution.
• The length of chess 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.
• Incubation period of diseases.
• Diameters of banana leaf spots, powdery mildew on barley.
• 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 pharmacokinetic variables, such as C_max, 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.
• 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% confidence belt based on the binomial distribution.
• 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 economics, there is evidence that the income of 97–99% of the population is distributed log-normally. (The distribution of higher-income individuals follows a Pareto distribution).
• If an income distribution follows a log-normal distribution with standard deviation <math>\sigma</math>, then the Gini coefficient, commonly used to evaluate income inequality, can be computed as <math>G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)</math> where <math>\operatorname{erf}</math> is the error function, since <math> G = 2 \Phi{\left(\frac{\sigma }{\sqrt{2\right)} - 1</math>, where <math>\Phi(x)</math> is the cumulative distribution function of a standard normal distribution.
• In finance, 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 that log-Lévy distributions, which possess heavy tails, would be a more appropriate model, in particular for the analysis for stock market crashes. Indeed, stock price distributions typically exhibit a fat tail. The fat tailed distribution of changes during stock market crashes invalidate the assumptions of the central limit theorem.
• In scientometrics, 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.
• 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 shadowing, is often modeled as a log-normal distribution.
• Particle size distributions produced by comminution with random impacts, such as in ball milling.
• The file size distribution of publicly available audio and video data files (MIME types) 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.
• in physical testing when the test produces a time-to-failure of an item under specified conditions, the data is often best analyzed using a lognormal distribution.

See also

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

Notes

References

Further reading

External links

• The normal distribution is the log-normal distribution