HomeTechniques and Tips@RISK DistributionsSpecifying Distributions in Terms of Desired Mean and Standard Deviation

# 3.20. Specifying Distributions in Terms of Desired Mean and Standard Deviation

Applies to:
@RISK 4.x–7.x
RISKOPTIMIZER 1.x–5.x

I want a BetaGeneral distribution with a given mean and standard deviation. What α1 and α2 (alpha1 and alpha2) should I enter in the Define Distribution window?

Can I do this if I know other statistics, such as the mode and the variance? Can I do this for other types of distribution?

Let's take the easiest alternatives first.

If you have @RISK 5.5.0 or newer, the JohnsonMoments distribution is available.  It lets you specify mean, standard deviation, skewness, and kurtosis, and it comes up with an appropriate distribution shape for those parameters.

If you have a particular distribution in mind and you want to target percentiles (including the median), you may be able to use a form of the distribution that specifies percentiles in place of one or more parameters. In Define Distribution select one of the distributions from the Alt. Parameters tab in the Define Distribution dialog.  @RISK will then calculate the needed parameters automatically.

If those alternatives don't meet your needs, you may be able to solve for the distribution parameters that give the desired mean and standard deviation (or other statistics) for your desired distribution.  It all depends on whether closed forms exist for your desired statistics. (A closed form, for this purpose, is an algebraic formula that can be implemented in Excel.  If you're targeting standard deviation, use the square root of the variance formula.)

There are two places to look for these closed forms.  In @RISK 5.x and newer, and in RISKOptimizer 5.x, look in the product's help topic for the particular distribution function that is of interest.  In @RISK 4.x or RISKOptimizer 1.x, click the Windows Start button, then Programs or All Programs, then Palisade DecisionTools, then Online Manuals, then Distribution Function Summary.

Example:

The attached example shows how to find alpha1 and alpha2 for a BetaGeneral that give a desired mean and standard deviation.  Please download FindDistributionParams.xls and open it in RISKOptimizer or in Excel. All constraints and options are already set as needed.

The green cells are the desired statistics for the distribution, here minimum, maximum, mean, and standard deviation. The red cells are the adjustable cells for Solver or RISKOptimizer; they're arbitrarily set to 1 at the start. The purple cells are the formulas for mean and standard deviation, in terms of the adjustable red cells.

As you see, when α1 = α2 = 1 in a BetaGeneral distribution, the mean is 57.5 and the standard deviation is 24.5. These deviate from the desired values by a total of 27.04 units, the "error to minimize" in blue. RISKOptimizer or Solver is given the blue cell as the target to minimize.

When you run RISKOptimizer or Solver, it adjusts the red α1 and α2 until it converges on parameters that give the desired mean and standard deviation, or as close as possible to them.

Variations on the example:

If you want to target different statistics, such as kurtosis and mode or skewness and mean, change the captions A21:A24 and the formulas E27 and H27.

If you're interested in a different distribution, you may need to change the captions D21:D22 in addition to the above, and you may also need to edit the constraints in RISKOptimizer or Solver. (In a BetaGeneral distribution, α1 and α2 must be positive, but parameters for many other distributions have different constraints.)  If the distribution has three parameters or more, insert the additional parameters and add appropriate RISKOptimizer or Solver constraints.

Additional keywords: RiskJohnsonMoments distribution, Johnson Moments

Last edited: 2015-06-19