About the Log-Normal Distribution
The Log-Normal Distribution Probability Calculator finds the probability that a log-normally distributed value falls above or below a chosen threshold, using the mean and standard deviation of the variable's natural logarithm. It's built for quantities that are always positive and skewed toward large values, such as incomes, asset prices, or particle sizes, where a normal distribution doesn't fit but the logarithm of the data does.
How It Works
Enter the value you want a probability for, along with mu (the mean of the natural log of the underlying variable) and sigma (the standard deviation of that log). The calculator converts the value into a standard normal z-score using mu and sigma, looks up the corresponding cumulative probability, and reports both the probability of falling below and above that value. It also returns the distribution's median and mean, which differ from each other because of the distribution's right skew.
Formula & Methodology
By hand, take the natural log of the value, subtract mu, and divide by sigma to get a standard normal z-score. Look that z-score up in a standard normal table (or compute it with the normal CDF) to get the probability of a value below x. The median of the distribution is simply e raised to mu, while the mean is e raised to mu plus half of sigma squared - always larger than the median because of the distribution's rightward skew.
Examples
Default Threshold Check
With mu = 3.5 and sigma = 0.6, the probability that X falls below 50 works out to about 75.39%, and above 50 to about 24.61%. The distribution's median is about 33.12 and its mean about 39.65, showing the rightward skew.
Income-Style Distribution
For a variable with mu = 11 and sigma = 0.5 (values in the tens of thousands, typical of a log-normal income model), the probability of falling below 100,000 is about 84.75%. The median works out to about 59,874 while the mean is about 67,846, again with mean exceeding median.
Advantages
- Handles distributions that are strictly positive and skewed, a common shape for real-world quantities that a plain normal distribution model gets wrong.
- Returns both directions of the probability, below and above the value, along with the median and mean in a single calculation.
- Uses the same log-transform-to-z-score method that underlies log-normal modeling in finance, biology, and economics.
Common Mistakes
- Entering the mean and standard deviation of the raw data instead of the mean and standard deviation of its natural logarithm.
- Assuming the median and mean of a log-normal distribution sit close together, as they would for a normal distribution, when the mean is systematically pulled upward by the skew.
- Trying to evaluate the distribution at a value of zero or a negative number, which is outside the domain of a log-normal variable.
Edge Cases to Watch For
- The value x must be strictly greater than zero, since the natural log of zero or a negative number is undefined - the calculator returns an error for non-positive inputs.
- Sigma must be greater than zero; a sigma of zero would mean every outcome is identical, making the z-score division undefined.
- Because the distribution is right-skewed, the mean is always greater than the median, so relying on the mean alone to describe a 'typical' value can be misleading.
- Mu and sigma describe the log of the variable, not the variable itself - entering the ordinary mean and standard deviation of the raw values in these fields will give an incorrect result.
Common Use Cases
- Analysts estimating the probability that an asset price, claim size, or income falls above or below a specific threshold.
- Reliability engineers modeling failure times or particle sizes known to follow a log-normal pattern.
- Students and researchers working through log-normal distribution problems who want to check a hand calculation.