Log-Normal Distribution Calculator

Mean

μ

(log scale):

Std Dev

σ

(log scale):

Probability Type:

X Value (

x

Upper Bound (

x_{2}

Output Type: PDF CDF Table

Decimal Places: 2 3 5 8

Clear

Introduction

The log-normal distribution models continuous random variables $X$ that take only positive values and exhibit right-skewed behaviour, arising naturally from multiplicative processes and logarithmic transformations. Analysing this distribution enables the evaluation of probabilities across specified thresholds and supports the study of scale-dependent variation in fields where measurements grow proportionally and remain strictly non-negative.

What this calculator does

The calculator performs probability density and cumulative distribution functions for a log-normal random variable. Users input the mean $μ$ and standard deviation $σ$ on a logarithmic scale, along with a threshold value $x$ . The system generates precise probabilities for "less than", "greater than", or "between" intervals. Outputs include the numerical result, a detailed step-by-step calculation process, and visualisations through PDF or CDF charts and data tables.

Formula used

The probability density function (PDF) and cumulative distribution function (CDF) rely on the natural logarithm of the variable. The CDF is calculated using the error function $\erf$ to determine the area under the curve. Here, $μ$ is the location parameter and $σ$ is the scale parameter of the underlying normal distribution.

f (x, μ, σ) = \frac{1}{x σ \sqrt{2 π}} \exp (- \frac{{(\ln x - μ)}^{2}}{2 σ^{2}})

F (x, μ, σ) = 0.5 (1 + \erf (\frac{\ln x - μ}{σ \sqrt{2}}))

How to use this calculator

1. Enter the logarithmic mean $μ$ and standard deviation $σ$ into the respective fields.
2. Select the desired probability type and provide the threshold value $x$ or interval range.
3. Choose the preferred output format, such as a chart or a distribution table, and set the decimal precision.
4. Execute the calculation to view the resulting probability and the step-by-step transformation process.

Example calculation

Scenario: Environmental scientists are studying the concentration of a specific mineral in soil samples, which follows a log-normal distribution to model positive, skewed concentration levels across a site.

Inputs: Log-scale mean $μ = 0$ , log-scale standard deviation $σ = 1$ , and threshold $x = 1$ .

Working:

Step 1: $z = \frac{\ln (x) - μ}{σ \sqrt{2}}$

Step 2: $z = \frac{\ln (1) - 0}{1 \times \sqrt{2}}$

Step 3: $z = 0$

Step 4: $P = 0.5 \times (1 + \erf (0))$

Result: 0.50

Interpretation: There is a 50% probability that the mineral concentration is less than or equal to 1 unit.

Summary: The result confirms that the median of this specific distribution occurs at the value where the natural logarithm equals the mean.

Understanding the result

The result represents the area under the probability density curve for the specified range. A value of 0.85 indicates an 85% likelihood that the variable falls within that interval. Because the distribution is skewed, the mean of the raw data will typically be higher than the median, reflecting the presence of high-value outliers.

Assumptions and limitations

The primary assumption is that the natural logarithm of the variable is normally distributed. The variable $x$ must be strictly positive, as the logarithm of zero or negative numbers is undefined. The standard deviation $σ$ must also be greater than zero.

Common mistakes to avoid

A frequent error is entering the arithmetic mean of the raw data instead of the mean of the logarithms. Similarly, users may confuse the scale parameter $σ$ with the variance. Ensure that the threshold value $x$ is positive to avoid calculation errors within the logarithmic functions.

Sensitivity and robustness

The output is highly sensitive to changes in $σ$ , as it occupies the denominator and exponent. Small increases in $σ$ can significantly stretch the distribution's tail. Conversely, the calculation is robust for large $x$ values, as the cumulative probability eventually plateaus towards unity, making the tool stable for extreme value analysis.

Troubleshooting

If the results show an error, verify that $x$ is greater than zero and $σ$ is a positive non-zero value. For "between" calculations, ensure the upper bound $x_{2}$ is strictly greater than the lower bound $x_{1}$ . Extreme inputs exceeding 1e12 will be rejected to maintain numerical stability.

Frequently asked questions

Why can I not use a negative X value?

The log-normal distribution is defined only for positive real numbers because the natural logarithm of a non-positive number is undefined in real number mathematics.

What is the difference between the PDF and CDF?

The PDF shows the relative likelihood of the variable taking a specific value, while the CDF represents the total probability that the variable is less than or equal to a value.

What does the error function do?

The error function, or erf, is a special function used to calculate the area under a normal distribution curve, which is essential for determining log-normal probabilities.

Where this calculation is used

In educational settings, this calculation is vital for modelling multiplicative phenomena where growth or changes are proportional to the current size. It appears frequently in environmental science for modelling pollutant concentrations, in social research for income distributions, and in population studies for measuring biological sizes. Students use it to understand how data that cannot be negative, yet is highly skewed, can be standardised and analysed using traditional Gaussian techniques through logarithmic transformation, providing a bridge between linear and non-linear statistical models.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.