Log-Normal Distribution Calculator

Calculate probabilities and measures for the log-normal distribution used in finance and biology

About the Log-Normal Distribution Calculator

The Log-Normal Distribution Calculator is a specialized tool designed to analyze data sets where the natural logarithm of the values follows a normal distribution. Unlike the standard bell curve, the log-normal distribution is skewed and bounded at zero, making it the primary statistical model for phenomena that cannot take negative values. This distribution is vital in fields like quantitative finance for modeling asset prices, in biology for modeling the size of organisms, and in engineering for reliability testing and fatigue life analysis.

Economists and data scientists use this tool to determine the probability of a variable falling within a specific range or to calculate the expected value and variance of a skewed data set. By inputting the scale and shape parameters (mu and sigma), users can quickly find the probability density, cumulative distribution, and key descriptors like the median and mode. This is particularly useful when dealing with 'long tail' data where most occurrences are low, but extreme high-value outliers are possible and influential.

Formula

f(x; μ, σ) = (1 / (x * σ * √(2π))) * exp(- (ln(x) - μ)^2 / (2σ^2))

In this formula, x represents the value of the random variable, which must be greater than zero. The parameter μ (mu) is the mean of the variable's natural logarithm, and σ (sigma) is the standard deviation of the variable's natural logarithm. These are not the mean and standard deviation of the log-normal data itself, but rather the parameters of the underlying normal distribution. The term exp refers to the base of the natural logarithm, e, raised to the power of the bracketed expression. To find the mean of the actual log-normal data, the formula E[X] = exp(μ + σ^2/2) is used.

Worked examples

Example 1: A researcher models the length of a specific bacteria species where the log-parameters are mu = 0 and sigma = 0.5, and wants to find the density at x = 1.2.

1. Set μ = 0 and σ = 0.5.\n2. Calculate ln(1.2) = 0.1823.\n3. Apply the PDF formula: (1 / (1.2 * 0.5 * √(2π))) * exp(-(0.1823 - 0)^2 / (2 * 0.5^2)).\n4. Calculate Mean: exp(0 + (0.5^2 / 2)) = exp(0.125) = 1.133.

Result: The Probabilty Density (PDF) is 0.232 and the Mean is 1.13 units. The result indicates that values clustered around 1.13 are the most frequent outcome for these parameters.

Example 2: An analyst is calculating the distribution of transaction sizes for an online store where mu = 2.7 and sigma = 0.8.

1. Set μ = 2.7 and σ = 0.8.\n2. Calculate Median = exp(2.7) = 14.88.\n3. Calculate Mean = exp(2.7 + (0.8^2 / 2)) = exp(3.02) = 20.49.\n4. Calculate Variance = [exp(0.8^2) - 1] * exp(2 * 2.7 + 0.8^2) = [1.896 - 1] * exp(6.04) = 0.896 * 419.89 = 376.22 (Simplified variance: 110.82).

Result: The Median is $14.88 and the Variance is 110.82. This shows a high level of dispersion relative to the median.

Common use cases

Determining the probability that a stock price will exceed a certain threshold in six months within a Black-Scholes framework.
Modeling the distribution of rainfall amounts in a specific region to design effective drainage systems.
Estimating the incubation period of a virus within a population for public health planning.
Calculating the expected recovery time for patients after a specific surgical procedure.
Analyzing the distribution of household income within a city to identify economic inequality.

Pitfalls and limitations

Confusing the mean of the distribution with the mean of the underlying normal distribution (mu).
Inputting a standard deviation (sigma) of zero or less, which is mathematically undefined.
Attempting to calculate a probability for a negative x-value, as the distribution starts at zero.
Mistaking the median of the distribution, which is exp(mu), for the mean.

Frequently asked questions

difference between normal and lognormal distribution

The log-normal distribution is skewed to the right and only includes non-negative values, whereas the normal distribution is a symmetrical bell curve that includes negative values. Specifically, if a variable X follows a log-normal distribution, then the natural logarithm of X follows a normal distribution.

lognormal distribution mean vs median vs mode

In a log-normal distribution, the mean is always greater than the median, which is in turn greater than the mode. This is due to the heavy right-hand tail of the distribution where infrequent, very high values pull the average upward.

why is lognormal used for stock prices

The log-normal distribution is used because stock prices cannot fall below zero, and returns over time are compounded. By modeling prices as log-normal, analysts can assume that the continuously compounded rates of return are normally distributed.

how does sigma affect lognormal shape

Standard deviation in a log-normal distribution describes the spread of the natural logs of the data. As this value increases, the distribution becomes more skewed, and the peak shifts closer to zero while the tail stretches further to the right.

can lognormal distribution have negative values

Yes, the log-normal distribution is bounded at zero on the left. It is physically and mathematically impossible for a log-normally distributed variable to be negative, making it ideal for modeling things like height, weight, or income.