# Three-point estimation

The three-point estimation technique is used in management and information systems applications for the construction of an approximate probability distribution representing the outcome of future events, based on very limited information. While the distribution used for the approximation might be a normal distribution, this is not always so and, for example a triangular distribution might be used, depending on the application.,[1]

In three-point estimation, three figures are produced initially for every distribution that is required, based on prior experience or best-guesses:

• a = the best-case estimate
• m = the most likely estimate
• b = the worst-case estimate.

These are then combined to yield either a full probability distribution, for later combination with distributions obtained similarly for other variables, or summary descriptors of the distribution, such as the mean, standard deviation or percentage points of the distribution. The accuracy attributed to the results derived can be no better than the accuracy inherent in the 3 initial points, and there are clear dangers in using an assumed form for an underlying distribution that itself has little basis.

## Estimation

Based on the assumption (possibly unwarranted[citation needed]) that a double-triangular distribution governs the data, several estimates are possible. These values are used to calculate an E value for the estimate and a standard deviation (SD) as L-estimators, where:

E = (a + 4m + b) / 6
SD = (b − a) / 6

E is a weighted average which takes into account both the most optimistic and most pessimistic estimates provided. SD measures the variability or uncertainty in the estimate. In Project Evaluation and Review Techniques (PERT) the three values are used to fit a Beta distribution for Monte Carlo simulations.[citation needed]

The triangular distribution is also commonly used. It differs from the double-triangular by its simple triangular shape and the mode does not have to coincide with the median. The mean (expectation) is then:

E = (a + m + b) / 3.

In some applications,[1] the triangular distribution is used directly as an estimated probability distribution, rather than for the derivation of estimated statistics.

## Project management

To produce a project estimate the project manager:

• Decomposes the project into a list of estimable tasks, i.e. a work breakdown structure
• Estimates the E value and SD for each task.
• Calculates the E value for the total project work as E (Project Work) = Σ E (Task)
• Calculates the SD value for the total project work as SD (Project Work) = √(Σ SD (Task) 2)

The E and SD values are then used to convert the project estimates to confidence levels as follows:

• Confidence level in E value +/- SD is approximately 68%
• Confidence level in E value +/- 1.645 × SD is approximately 90%
• Confidence level in E value +/- 2 × SD is approximately 95%
• Confidence level in E value +/- 3 × SD is approximately 99.7%
• Information Systems typically use the 95% confidence level, i.e. E Value + 2 × SD, for all project and task estimates.[2]

These confidence level estimates assume that the data from all of the tasks combine to be approximately normal (see asymptotic normality). Typically, there would need to be 20–30 tasks for this to be reasonable, and each of the estimates E for the individual tasks would have to be unbiased.