Statistics are a very important part of data analytics. Statistical techniques are used to extract insights and meaningful information from data. A good understanding of statistics is essential for any data analyst to succeed. In this blog, let’s discuss how statistics are used in project management to achieve the best outcome.

## Introduction

Scheduling is one of the most important aspects of project management. Projects involve tasks with dependencies and a required duration. Often, the tasks either require more or less time than initially planned to be finished. How does a project manager overcome this? The answer is to use statistics to estimate the duration of the task.

## Estimating Time Using Beta Distribution

For a task, a project manager can inquire with the team that is working for them and the expectations set by upper management that would help them derive. Let us consider the following:

- the minimum time required for the task to finish. (α)
- the maximum time required for the task to finish. (β)
- the time required for the task to finish. (m)

### Beta Distribution

Beta distribution in statistics has two shape parameters, alpha, and beta, which allow it to be useful to model a wide range of phenomena. In project management, the Beta distribution is particularly useful for estimating the time required for completion. This allows project managers to estimate completion duration, identify areas of risk and uncertainty, and make informed decisions about resource allocation and scheduling more accurately.

### Calculating Expected Duration of Tasks Using Beta Distribution

To estimate the time required to calculate the duration of a task the three values mentioned above α, β, and m will be used.

- Expected Time (E) = (α+4m+β)/6
- Variance (V) = 〖((β-α )/6)〗^2
- Standard Deviation (σ) = (β-α )/6

For a task the estimated time of completion = E ± σ

### Calculating Expected Duration for Chain of Tasks Using Beta Distribution

For a chain of tasks, the estimated time required to calculate the duration of the path of the three values α, β, and m for each task will be used.

- Expected time of the path (E) – Sum of expected times of all tasks in the path
- The variance of the path (V) – Sum of Variances of each task in the path
- Standard Deviation of the path (σ) – Square root of the sum of Variances of the path (2)

The expected duration of the path = E ± σ

### Calculating Probability for Completion

Probability is an important tool in project management to determine the chance of a task or chain of tasks getting completed for their durations.

To calculate the probability of a task, a Z score should be found through which the probability can be found.

Z = (Desired Duration – Expected Duration) / σ

After calculating the Z score, probability can be found using the Z score table or a calculator to get the p% value.

In a project, with multiple paths, the probability of completing the project in a desired duration will be based on the critical path. If there are multiple critical paths, then the probability is the product of the probabilities of the two paths.

**Note: If the desired duration is less than or equal to paths that are not the critical path, these non-critical paths should also be considered while calculating the probability of finishing the project in the desired duration.**

### Example project

Let’s take a very simple project which has the following WBS:

Task | Predecessor | Duration | Minimum Duration | Maximum Duration |

A | 5 | |||

B | 4 | 3 | 7 | |

C | B | 6 | 4 | 7 |

D | A | 3 | ||

E | 9 | 8 | 16 |

If we construct the AON diagram for the following WBS we just use the most likely duration, we get the following diagram with the critical path marked in red. The project will be finished in 10 days.

#### AON diagram

But some of the tasks have a minimum duration and maximum duration. Now to find the time estimates for the tasks that have uncertain duration we calculate using the formulas mentioned above to find E, V, and σ.

Task | Predecessor | Duration | Minimum Duration | Maximum Duration | E | V | σ |

A | 5 | 5.00 | 0.00 | 0.00 | |||

B | 4 | 3 | 7 | 4.00 | 0.44 | 0.67 | |

C | B | 6 | 4 | 7 | 5.83 | 0.25 | 0.50 |

D | A | 3 | 3.00 | 0.00 | 0.00 | ||

E | 9 | 8 | 16 | 10.00 | 1.78 | 1.33 |

Now to calculate the time estimates for the paths, we need to add individual variances of the tasks in the path and the square root of that will give us the standard deviation. The sum of the time estimates will give us the time estimate of the path.

**For path B – C:**

The variance of the path = Variance of task B + Variance of task C

V = 0.44 + 0.25 = 0.69

Standard deviation = Square root of Variance of the path

σ = sqrt(0.69)= 0.83

Time estimate = Sum of time estimates of the tasks in the path

E = 4 + 5.83 = 9.83

Time for the path = E ± σ = 9.83 ± 0.83

**For Path E:**

Since it is the only task in the path,

V = 1.78

σ = 1.33

E = 10

Time for the path = E ± σ = 10 ± 1.33

Now if we construct the new AON, it will look like this:

#### New AON diagram

Now if we wanted to find the probability of the project finishing in let’s say 12 days:

Critical path –> Start – E – End

The standard deviation of the path = 1.33

**Z = (Desired Completion – Critical Path Duration) / Standard Deviation of the Critical Path**

So, Z = 12 – 10 /1.33

Z = 1.50

The probability with Z = 1.50 is 93%. So the probability of the project finishing in 12 days is 93%.

Now let’s look at a trickier problem, i.e., the probability of the project being finished in 9 days.

If the project is to be finished in 9 days, there will be two critical paths to consider as B – C also has a time estimate greater than 9.

**For critical Path: Start – E – End:**

The standard deviation of the path = 1.33

**Z = (Desired Completion – Critical Path Duration) / Standard Deviation of the Critical Path**

So, Z = 9 – 10 /1.33

Z = -0.75

The probability with Z = -0.75 is 22%

**For critical path Start – B – C – End:**

The standard deviation of the path = 0.83

**Z = (Desired Completion – Critical Path Duration) / Standard Deviation of the Critical Path**

So, Z = 9 – 9.83 /0.83

Z = -1

The probability with Z = -1 is 16%

The probability of the project finishing in 9 days is the product of the probabilities of the two parallel paths.

Probability of the project = 22% x 16% = 3%

The probability of the project finishing in 9 days is 3%.

## Conclusion

Statistics is a powerful tool that can be used in a wide variety of fields. In project management, it helps the project manager to determine the probabilities and estimates of completion of the project to minimize risk and make informed decisions on resource allocation and scheduling.

Before signing off, I want to take a moment to thank **Abhijit Paul** for sharing his expertise and insights with us. I am sure our readers have gained a lot from your knowledge.

Keep reading on **data Demystify** to learn more about such information.