Applies to:
@RISK, all releases
RISKOPTIMIZER, all releases

 

The RiskLognorm() Function Doesn’t Match Log-Normal Formulas in Textbooks

Question
When I enter μ = 4 and σ = 2 in RiskLognorm(), I expect the simulated distribution to have a mean of 54.6 and a standard deviation of 2953.5, as described in statistics textbooks.

Instead, the simulation produces a mean close to 4 and a standard deviation close to 2. What’s wrong?

Answer
Nothing is wrong. @RISK and RISKOptimizer provide two different log-normal distributions, which differ only in how their parameters are defined.

The Two Log-Normal Functions in @RISK

1. RiskLognorm(μ, σ) — Mean/Standard Deviation Form

In RiskLognorm(), the values you enter for μ and σ are the actual mean and actual standard deviation of the distribution (subject to normal sampling variation).

For RiskLognorm(μ, σ):

• Actual mean = μ

• Actual standard deviation = σ

Use this function when you already know the desired mean and standard deviation of the distribution.

This is why entering μ = 4 and σ = 2 results in a distribution with mean ≈ 4 and standard deviation ≈ 2.

---

2. RiskLognorm2(μ, σ) — Textbook (Log-Space) Form

RiskLognorm2() follows the parameterization commonly used in statistics textbooks, where μ and σ are the mean and standard deviation of the underlying normal distribution in log space.

For RiskLognorm2(μ, σ):

• Actual mean
  [
  \exp\left(\mu + \frac{\sigma^2}{2}\right)
  ]

• Actual standard deviation
  [
  \exp\left(\mu + \frac{\sigma^2}{2}\right)\sqrt{\exp(\sigma^2) - 1}
  ]

With μ = 4 and σ = 2:

• Actual mean ≈ 54.6

• Actual standard deviation ≈ 2953.5

Use this function when your parameters come from textbooks or theoretical formulas for the log-normal distribution.

Using Geometric Mean and Geometric Standard Deviation

If you know the geometric mean and geometric standard deviation of a log-normal distribution:

• Use RiskLognorm2()

• Set:

  • μ = ln(geometric mean)

  • σ = ln(geometric standard deviation)

For more details, see Geometric Mean and Geometric SD in Log-normal.

---

Summary

If you know…	Use
Actual mean and standard deviation	RiskLognorm()
Log-space parameters from textbooks	RiskLognorm2()
Geometric mean and geometric SD	RiskLognorm2() with logged inputs

 

Attachments:
LogNormal.xlsx