Lognormal Distribution in Business Statistics

In business statistics, Lognormal Distribution is an important probability distribution model used to characterize data with positive values that show right-skewed patterns. It is widely applied in real-world scenarios such as stock prices, income distribution, resource reserves and social media metrics, making it valuable for risk assessment, portfolio optimization and decision-making in finance, economics and resource management.

• Lognormal distribution is a way to describe the likelihood of different values for a variable.
• A variable X follows a lognormal distribution if its natural logarithm, ln(X), follows a normal distribution.
• It is used to model variables that are always positive and tend to be right-skewed, unlike the symmetric normal distribution.
• If X has a lognormal distribution with parameters \mu and \sigma it is denoted as X \sim \operatorname{LogN}(\mu, \sigma^2) where \mu and \sigma represent the mean and standard deviation of ln(X)

Probability Density Function (PDF) of Lognormal Distribution

The probability density function (PDF) for the lognormal distribution depends on two parameters, μ (mean) and σ (standard deviation), for x values greater than 0. When we take the logarithm of our lognormal data, μ represents the mean and σ is the standard deviation of this transformed data.

f(x)=\frac{1}{xσ√2π}e^\frac{-1}{2}(\frac{logx-μ}{σ})^2, for ~0<x<\infty

Where:

• \mu represents the mean or the location parameter.
• \sigma represents the standard deviation or the shape parameter.
• x is the value for which is required to find the probability density.
• e is mathematical constants.

Lognormal Distribution Curve

• It is right-skewed, meaning it tilts to the right.
• The curve begins at zero, rises to its peak and then declines.
• The degree of skewness increases as the standard deviation (σ) rises, keeping the mean (μ) constant.
• μ represents the mean of natural logarithms of the data.
• σ represents the standard deviation of natural logarithms of the data.
• When σ is much larger than 1, the curve rises steeply at the start, peaks early and then falls rapidly, resembling an exponential curve.
• In this distribution, μ acts as more of a scale parameter, unlike the normal distribution where it serves as a location parameter.

Mean and Variance of Lognormal Distribution

Mean (\mu)

• The mean (\mu) of a lognormal distribution is not simply the mean of the original data it is the mean of the natural logarithm of the data.
• Then, find the mean of these natural logarithms. Mathematically, \mu is the average of ln(x), where x represents the original data.
• This mean does not equal the median or mode of the original data, since the lognormal distribution is asymmetric

μ=e^{μ+{\frac{1}{2}σ^2}}

Where:

• \mu represents the mean of the natural logarithm of the data.
• \sigma represents the standard deviation of the natural logarithm of the data.
• e is the mathematical constant approximately equal to 2.71828.

Variance (\sigma^{2})

• The variance (\sigma^{2}) of a lognormal distribution is similarly calculated from the natural logarithms of the data.
• The standard deviation of the natural logarithm of the data is \sigma. To get the variance, square this standard deviation that will result in \sigma^{2}.
• The variance formula involves both \sigma and \mu.
• The variance of the lognormal distribution helps describe how data points are dispersed around the mean of the natural logarithm of the data.

σ^2=(e^{σ^2}-1)e^{2μ+σ^2}

Where:

• \sigma: represent standard deviation of the natural logarithm of the data.
• \mu: mean of the natural logarithm of the data.
• e: mathematical constant, approximately equal to 2.71828.

Examples of Lognormal Distribution

Example 1: The daily website visitors of a small blog follow a lognormal distribution where the underlying normal distribution (of ln X) has mean \mu= 3.91 and standard deviation \sigma = 0.1. Calculate the variance of the daily website visitors (X).

Solution:

To find the variance \sigma^{2} we will use the formula for the variance of a lognormal distribution:

Var(X)=(e^{σ^2}-1)e^{2μ+σ^2}

Accordng to the given information, we have:

• μ = 3.91
• σ = 0.1

putting these values in the formula we get,

Var(X)=(e^{0.1^2}-1)·e^{2(3.91)+0.1^2}

Var(X)=(e^{0.01}-1)·e^{(2(3.91)+0.01)}

Var(X)=(1.01005-1)·e^{7.83}

{Var}(X)= 0.01005 \times 2520.5\approx 25.33

The variance of daily website visitors is approximately 25.33

Example 2: The population of a village follows a lognormal distribution with a median population of 1,000 and a geometric standard deviation of 1.2. Calculate the mean (average) population of the village.

Solution:

For a lognormal distribution, the parameters μ and σ (of the underlying normal distribution of ln X) are derived from the median and geometric standard deviation as follows:

\mu=ln(median)=ln(1000)=6.908

\sigma=ln(geometric standard deviation)=ln(1.2)=0.182

To find the mean E(X), we use the formula for the mean of a lognormal distribution:

E(X)=e^{μ}⋅\ e^{\frac{σ^2}{2}}

Putting these values in the formula, we get:

E(X)=e^{6.908}⋅\ e^{\frac{0.182^2}{2}}

E(X)=e^{6.908+0.01656}

E(X)=e^{6.9246}\approx 1016.7

The mean population of the village is approximately 1,016.7.

Applications

• Stock Prices and Asset Returns: Widely used in finance to model stock prices and asset returns, which often exhibit right-skewed patterns and cannot fall below zero.
• Income Distribution: Used in economics to model income distribution across populations, helping analysts understand how income is spread across different groups.
• Resource Reserves: Applied in mining and petroleum industries to model the distribution of natural resource reserves, such as oil, gas and mineral deposits.
• Online Reading and Engagement Time: The time users spend reading articles or engaging with online content often follows a lognormal distribution, useful for content creators and marketers.
• Social Media Comment Length: The length of comments on social media platforms can be modeled using a lognormal distribution, aiding content moderation and engagement analysis.
• Task Completion Times: Time taken to complete certain tasks, such as solving a Rubik's Cube, often follows a lognormal distribution, useful for performance analysis and prediction.

Difference Between Normal Distribution and Lognormal Distribution

Here we compare Normal Distribution and Lognormal Distribution