# Project Management Mathematics IV- Monitoring & Control

This is in continuation of the article Project Management Mathematics III- Monitoring & Control.

Continuous probability distributions (particularly beta and triangular distributions) are commonly used in Perform Quantitative Risk Analysis. According to the PMBOK® Guide, continuous probability distributions include normal, lognormal, triangular, beta, and uniform distributions. Distributions are graphically displayed and represent both the probability and time or cost elements.

This article will cover the following continuous distributions.

- Triangular and Beta distribution
- Uniform distribution
- Normal or Bell distribution

## Triangular and Beta distribution

Triangular distributions use estimates based on the three-point estimate (the pessimistic, most likely and optimistic values). This means that during your interviews, you’ll gather these pieces of information from your experts. Then you’ll use them to quantify risk for each WBS element.

Depending on the assumed distribution of values within the range of three estimates the expected duration, E (in three point estimate) can be calculated using the following formula.

*Triangular distribution*

Expected Duration (E) = (O+M+P)/3

*Beta Distribution*

Expected Duration (E) = (O+4*M+P)/6

Where O=Optimistic duration; M= Most likely duration; P=Pessimistic duration

The data shown in the figure, on the left is Beta distribution and on the right is Triangular distribution, shows distribution determined by two shapes parameter. X axis represents the possible values of time or cost and Y axes represent Probability.

Example :

If P=26, M=22 and O=14 ,find Expected duration and range of the project

** Assuming Triangular distribution**,

The expected duration of a project comes out to be 21.33 days and

the range= (P – O)/6 = +-2 days

So the project can take from 19.33 to 23.33 days for completion

*Assuming Beta distribution,*

The expected duration of a project comes out to be 20.66 days and

the range= (P – O)/6 = +-2 days

So the project can take from 18.66 to 22.66 days for completion

Duration estimates based on three points with an assumed distribution provide an expected duration and clarify the range of uncertainty around the expected duration

## Uniform distribution

A Uniform distribution assigns equal probability to all values between its minimum and maximum. It is also known as a rectangular distribution. The uniform distribution lies between two values on the x-axis. The total area is equal to 1.0 or 100% within the rectangle. Most statistical software programs have capability to solve for probabilities for the uniform distribution.

In this graph Area of a rectangle=length * height and total area equals to 1 so putting the values length = X2- X1 and Area =1, the height or the value of Y can be calculated which equal to Area/length or 1/(X2-X1)

*Normal or Bell distribution*

Data can be distributed in different ways; it can spread out more on the right or more on the left or jumbled. But there are many cases where the data tends to be around a central value with no bias left or right and it gets close to a Bell shape distribution called “Normal distribution”

The Normal Distribution has:

- mean = median = mode
- symmetry about the center
- 50% of values less than the mean
- and 50% greater than the mean

In the above figure standard deviation is denoted by symbol sigma.

- 1 sigma= 68.26%
- 2 sigma=95.46%
- 3 sigma=99.73%

A 68.3 percent probability is calculated using plus or minus one standard deviation, a 95.4 percent probability uses plus or minus two standard deviations, and a 99.7 percent probability uses plus or minus three standard deviations.

Normal and lognormal distributions use mean and standard deviations to quantify risk, which also require gathering the optimistic, most likely, and pessimistic estimates. Discrete distributions represent possible scenarios in a decision tree.

**Questions & Answers**

- If the expected value is of an activity is 110 and the standard deviation is 12, which of the following is true?
- A. There is approximately a 99 percent chance of completing this activity in 90 to 124 days.
- B. There is approximately a 68 percent chance of completing this activity in 98 to 122 days.
- C. There is approximately a 95 percent chance of completing this activity in 86 to 122 days.
- D. There is approximately a 75 percent chance of completing this activity in 90 to 124 days.

Correct Answer: B. A 68 percent probability is calculated using plus or minus one standard deviation, a 95 percent probability uses plus or minus two standard deviations, and a 99 percent probability uses plus or minus three standard deviations.

- If you know the expected value is 600 and the standard deviation is 14, you can say with approximately a 95 percent confidence rating which of the following?
- A. The activity will take from 586 to 614 days.
- B. The activity will take from 558 to 636 days.
- C. The activity will take from 568 to 606 days.
- D. The activity will take from 572 to 628 days.

Correct Answer: D. There is a 95 percent probability that the work will finish within plus or minus two standard deviations. The expected value is 600, and the standard deviation times 2 is 24, so the activity will take from 572 to 628 days.

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