Beta Distribution

The Beta distribution models events which are constrained to take place within an interval defined by a minimum and maximum value.  For this reason, the Beta distribution is used extensively in PERT, CPM and other project planning/control systems to describe the time to completion of a task.  The section below describes the use of the PERT Approximations.

Parameter	Description	Characteristics
Min	Minimum Value	A float > -∞ and < ∞
Max	Maximum Value	A float > -∞ and < ∞
shape_a (a)	Shape Factor	A float > 0
shape_b (b)	Shape Factor	A float > 0

Range

The range of this statistical distribution is from min to max.

Profiles

Functions

The function Β_i is the Incomplete Beta Function and Β is the Beta Function.

In the probability density function, the role of the Beta function is to normalise the function, such that the area under the curve is one.

Properties

When shape_a and shape_b are both equal to one, the beta distribution becomes equivalent to a uniform distribution, which does not have modal value.  When both shape_a and shape_b are both less than one, the distribution becomes bimodal, with modal values at the minimum and maximum value.

The median is derived using numerical methods.

Spreadsheets

MS Excel has two statistical functions for the Beta distribution.  BetaDist returns the cumulative probability function (i.e. F(x)).  The function takes the following arguments:

BetaDist(x,shape_a,shape_b,min,max)

The Min and Max values are optional.  If used both min and max values must be supplied or the function return an error.  If they are not specified, the return values is based on a standard beta distribution where the min and max values are 0 and 1 respectively.

The BetaInv function is the inverse of BetaDist and returns the value of x for a given value of cumulative probability (i.e. G(p)).  The function takes the following arguments:

BetaInv(p,shape_a,shape_b,min,max)

The min and max arguments behave the same interpretation as for BetaDist and p should be in the range 0<p<1.

The PERT Approximations

The project management community has evolved approximations for the mean and standard deviation of a Beta distribution which allow it to be handled with two parameters, rather than four. The process for modelling a task for PERT or similar analysis using these approximations is described below:

Step 1

Acquire estimates for the minimum, modal (most likely) and maximum time to completion, the figures for this example are:

Min	10.0 programmer days
Mode	13.5 programmer days
max	20.0 programmer days

Step 2

Using the three point estimation formula (sometimes referred to as  the "PERT Approximations"), estimates of the mean and standard deviation can be obtained:

The values for the example are:

mean	14.00
stdev	1.67

Step 3

Use the matching moment equations in which the minimum and maximum values are known to calculate shape factors which are consistent with the mean and standard deviation.

The shape factors for the example are:

shape_a	3.05
shape_b	4.57

The graph and table show the distribution and some of its parameters, an equivalent triangular distribution, which is an alternative solution is included for comparison:

	Beta	Triangular
Mean	14.00	14.50
Mode	13.65	13.5
Standard Deviation	1.67	2.07
Q1 (25%)	12.75	12.96
Q2 (50% - Median)	13.91	14.30
Q3 (75%)	15.17	15.97

The mean, mode and standard deviation in the above table are derived from the minimum, maximum and shape factors which resulted from the use of the PERT approximations.

Page Updated 18-Jan-2009