Log-normal Distribution Calculator

Log-normal Distribution Calculator

For parameters \(\mu\) and \(\sigma\) (with \(\sigma>0\)), the PDF is:

$$ f(x) = \frac{1}{x\,\sigma\sqrt{2\pi}} \exp\Bigl(-\frac{(\ln x – \mu)^2}{2\sigma^2}\Bigr),\quad x>0. $$

and the CDF is:

$$ F(x) = \frac{1}{2}\Bigl[1 + \operatorname{erf}\Bigl(\frac{\ln x – \mu}{\sigma\sqrt{2}}\Bigr)\Bigr],\quad x>0. $$

Step 1: Enter Parameters

Enter a real number (e.g., 0)

Enter a positive number (e.g., 1)

Enter a number greater than 0 (e.g., 2)

Log-normal Distribution: $$ f(x) = \frac{1}{x\,\sigma\sqrt{2\pi}} \exp\Bigl(-\frac{(\ln x – \mu)^2}{2\sigma^2}\Bigr),\quad x>0. $$

Understanding the Log-normal Distribution

Definition of Log-Normal Distribution

A log-normal distribution is a statistical distribution where the logarithm of random variables is normally distributed. This means that if you take the natural logarithm of the data values, the distribution of these logarithmic values forms a normal (bell-shaped) curve.

Understanding Normal and Lognormal Distributions

A normal distribution, commonly known as the bell curve, describes data where values cluster symmetrically around the mean. Approximately 68% of data points fall within one standard deviation of the mean, and about 95% within two standard deviations.

Unlike the normal distribution, the log-normal distribution is not symmetrical but positively skewed, meaning it has a long tail on the right side. Data following a log-normal distribution originate from variables whose natural logarithms are normally distributed.

The primary mathematical difference is that log-normal distributions represent exponential or multiplicative growth processes, while normal distributions typically represent additive processes. The base of the logarithm commonly used in log-normal calculations is Euler's number (e ≈ 2.718), but other bases can also adjust the distribution's shape.

Applications and Uses of Log-Normal Distribution in Finance

Log-normal distributions effectively model variables that cannot be negative, making them especially useful in finance. For instance, stock prices and investment returns, which cannot realistically drop below zero, are better represented by a log-normal distribution. Analyzing financial data using log-normal distributions helps investors and analysts predict potential stock prices, measure risk, and understand compounded returns accurately.

Notation Lognormal⁡( μ,σ2 ) {\displaystyle \ \operatorname {Lognormal} \left(\ \mu ,\,\sigma ^{2}\ \right)\ }
Parameters μ∈( −∞,+∞ ) {\displaystyle \ \mu \in (\ -\infty ,+\infty \ )\ } (logarithm of location),
 σ>0 {\displaystyle \ \sigma >0\ } (logarithm of scale)
Support x∈( 0,+∞ ) {\displaystyle \ x\in (\ 0,+\infty \ )\ }
PDF 1 xσ2π   exp⁡(−(ln⁡x−μ )22σ2){\displaystyle \ {\frac {1}{\ x\sigma {\sqrt {2\pi \ }}\ }}\ \exp \left(-{\frac {\left(\ln x-\mu \ \right)^{2}}{2\sigma ^{2}}}\right)}
CDF  1 2[1+erf⁡( ln⁡x−μ σ2 )]=Φ(ln⁡(x)−μσ){\displaystyle \ {\frac {\ 1\ }{2}}\left[1+\operatorname {erf} \left({\frac {\ \ln x-\mu \ }{\sigma {\sqrt {2\ }}}}\right)\right]=\Phi \left({\frac {\ln(x)-\mu }{\sigma }}\right)}
Quantile exp⁡(μ+2σ2erf−1⁡(2p−1)) {\displaystyle \ \exp \left(\mu +{\sqrt {2\sigma ^{2}}}\operatorname {erf} ^{-1}(2p-1)\right)\ }

=exp⁡(μ+σΦ−1(p)){\displaystyle =\exp(\mu +\sigma \Phi ^{-1}(p))}
Mean exp⁡( μ+σ22 ) {\displaystyle \ \exp \left(\ \mu +{\frac {\sigma ^{2}}{2}}\ \right)\ }
Median exp⁡( μ ) {\displaystyle \ \exp(\ \mu \ )\ }
Mode exp⁡( μ−σ2 ) {\displaystyle \ \exp \left(\ \mu -\sigma ^{2}\ \right)\ }
Variance [ exp⁡(σ2)−1 ] exp⁡(2 μ+σ2) {\displaystyle \ \left[\ \exp(\sigma ^{2})-1\ \right]\ \exp \left(2\ \mu +\sigma ^{2}\right)\ }
Skewness [ exp⁡(σ2)+2 ]exp⁡(σ2)−1{\displaystyle \ \left[\ \exp \left(\sigma ^{2}\right)+2\ \right]{\sqrt {\exp(\sigma ^{2})-1\;}}}
Excess kurtosis exp⁡(4 σ2)+2 exp⁡(3 σ2)+3 exp⁡(2σ2)−6 {\displaystyle \ \exp \left(4\ \sigma ^{2}\right)+2\ \exp \left(3\ \sigma ^{2}\right)+3\ \exp \left(2\sigma ^{2}\right)-6\ }
Entropy log2⁡( 2π  σ eμ+12 ) {\displaystyle \ \log _{2}\left(\ {\sqrt {2\pi \ }}\ \sigma \ e^{\mu +{\tfrac {1}{2}}}\ \right)\ }
MGFdefined only for numbers with a
non-positive real part, see text
CFrepresentation  ∑n=0∞ (i t)n n!e nμ+n2σ2/2 {\displaystyle \ \sum _{n=0}^{\infty }{\frac {\ (i\ t)^{n}\ }{n!}}e^{\ n\mu +n^{2}\sigma ^{2}/2}\ }
is asymptotically divergent, but adequate
for most numerical purposes
Fisher information (1 σ2 002 σ2 ) {\displaystyle \ {\begin{pmatrix}{\frac {1}{\ \sigma ^{2}\ }}&0\\0&{\frac {2}{\ \sigma ^{2}\ }}\end{pmatrix}}\ }
Method of moments μ=log⁡(E⁡[X]   Var⁡[X]   E⁡[X]2 +1  ) ,{\displaystyle \ \mu =\log \left({\frac {\operatorname {\mathbb {E} } [X]\ }{\ {\sqrt {{\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ }}\ }}\right)\ ,}

 σ=log⁡( Var⁡[X]   E⁡[X]2 +1 ) {\displaystyle \ \sigma ={\sqrt {\log \left({\frac {\ \operatorname {Var} [X]~~}{\ \operatorname {\mathbb {E} } [X]^{2}\ }}+1\ \right)\ }}}
Expected shortfall 12eμ+σ22 1+erf⁡(σ 2  +erf−1⁡(2p−1)) p{\displaystyle \ {\frac {1}{2}}e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {\ 1+\operatorname {erf} \left({\frac {\sigma }{\ {\sqrt {2\ }}\ }}+\operatorname {erf} ^{-1}(2p-1)\right)\ }{p}}}[1]

 =eμ+σ2211−p(1−Φ(Φ−1(p)−σ)){\displaystyle \ =e^{\mu +{\frac {\sigma ^{2}}{2}}}{\frac {1}{1-p}}(1-\Phi (\Phi ^{-1}(p)-\sigma ))}

Available Log-normal Distribution Calculators

Quickly and accurately analyze log-normal distributions with these specialized calculators. Ideal for statisticians, researchers, students, and professionals needing precise distribution analysis for their data.

Log-normal Distribution Variance Calculator

Calculate the variance of a log-normal distribution effortlessly using its known parameters.

Log-normal Distribution Sample Generator

Generate random samples from a log-normal distribution to support simulations or statistical analyses.

Log-normal Distribution Mode Calculator

Quickly find the mode (most frequent value) of a log-normal distribution using specific parameters.

Log-normal Distribution Median Calculator

Efficiently compute the median value of a log-normal distribution.

Log-normal Distribution Mean Calculator

Precisely determine the mean of log-normal distributions based on provided parameters.

Log-normal Distribution Inverse CDF Calculator

Calculate inverse cumulative distribution values (quantiles) for a specific probability in a log-normal distribution.

Log-normal Distribution CDF Calculator

Easily find cumulative probabilities for values within a log-normal distribution.

Log-normal Distribution PDF Calculator

Accurately compute the probability density function (PDF) for given values in a log-normal distribution.

How to Use These Calculators:

  1. Enter Required Parameters: Input necessary values such as mean, standard deviation, or probabilities.
  2. Select or Confirm Settings: Adjust calculator settings as required.
  3. Click Calculate: Receive clear and immediate statistical outcomes.

Frequently Asked Questions:

  • What is a log-normal distribution? It describes positively skewed data whose logarithms follow a normal distribution.
  • When should I use log-normal distribution calculators? These calculators are ideal for analyzing skewed data sets, commonly used in finance, biology, and reliability studies.
  • How do I interpret the results? Each calculator provides easy-to-understand results, facilitating the interpretation of variance, mean, median, mode, cumulative probabilities, and PDFs.

Explore each calculator's detailed page for further insights, comprehensive guides, and practical examples.

Bookmark or share this page for convenient access to powerful statistical analysis tools.

Related Calculators