ITTO – Project Management Mathematics – Closing

Tools & Techniques – Project Closing

After reading through all the articles relating to all phases and areas of PMP, we are into the last article in the series on tools and techniques used in project closing phase.

By definition every project has a definite start and end dates. This is the time or phase in project life cycle where each activity that was started prior of ainitiating the project will be closed before the projectclosure. It includes releasing the resources, closing all agreements and contracts with the vendors, financial settlements are done, deliverables are signed off, project artifacts are archived, and to close the lessons learnt document besides the one gone through in the article dedicated to project closing phase earlier.One such thing what a project management does before calling the project done is closing the project’s performance measurement. This is the area of our current interest in this article.

What are we speaking about here? We are speaking about Measures of central tendency and dispersion of data. What! STATISTICS? Yes, we are speaking about statistics here. We use statistical methods to measure the projects performance before closing it.  We will be discussing on Mean, Median, Mode, Standard deviation, Variance and Range.

Measures of Central Tendency: Mean

Measures of central tendency give us the ways to describe the data or characterize the data when given a dataset. For our discussion here, assume that data is well organized and understood. How do we describe it? A well-known and a simple way to describe such data are to find out its mean. For PMP exams kindly remember it is only arithmetic mean that you may find questions on. I pretty well know that everyone of us will readily say what a mean is. It is sum of the quantities divided by number of quantities. I don’t want to use mathematical symbols or formulae instead I want to use plain English. So, our definition for mean is “Sum of all elements divided by number of elements.”

Let us see the usage of it through a question:

Question: During the risk planning, it has been identified that there were 3 most probable risks with a respective estimated impact of 45,000 INR, 70,000 INR and 33, 000 INR if they occur during project execution. Considering the given information what is the average risk impact that could occur to the project?

Average = Sum of all the elements / number of elements.

So, apply it.

Average risk impact = 45000+70000+33000 / 3 → 148000 / 3 →49333.33 INR.

That explains the mean and its usage. Similarly mean is used at many places, but it’s straight and simple.


Measures of Central Tendency: Median

One situation where mean is not recommended is when we have unusual extreme data when compared with rest of the data in the data set. Let us take an example, marks of 10 students. Let them be 34, 39, 48, 42, 32, 18, 55, 38, 51, 66, and 148. If you observe the given marks, 9 of them are in one range and one last thing is at extreme range. Mean is not recommended in such scenarios, but it is recommended to go for median as a measure of central tendency.

While computing median, first thing you have to do is to arrange the data ascending order. Once the data is arranged in ascending order then select the middle element in the arrangement and consider it as median. This works fine when the number of elements is in ODD number, but if they are in EVEN number, then what is the way? In such case where the number of elements is in even count, you have to take the middle two elements and find the average of them. Result is median.

Let us take an example:

12 22 32 41 52 57 61 65 67 70 80

Above data is arranged in ascending order and they are in ODD count. So, we have learnt that if the count is in ODD number then take exactly the middle element. So the middle element is 57 and it is the median.

12 22 32 41 52 57 61 65 67 70

Above data is in ascending order and the count is an EVEN number. We have learnt that if the count is even then take the middle two elements and average them. Middle two elements are 52 and 57. Their sum is 109 and the median is 109/2 = 54.5.

Measures of central tendency: MODE

A value which is seen more number of times in a given data set is the MODE of that data distribution. It is very straight and simple.


14, 33, 14, 23, 56, 45, 14, 98, 76, 14, 44, 54.
What is the mode in the above distribution? It is pretty simple. Which number is seen more times in the given data set? YES, it is 14 so MODE = 14. Now you may get a doubt. What if more than two values occurs for the same number of time? Then such data distribution will not have any mode and this is same if every element is existing only once in the distribution. That means there is no value which is occurring more than once. Even in that situation there won’t be any MODE value for that distribution.

With that we are done with measures of central tendency. Now let us start looking into Data dispersion.

Data dispersion:Range

Let me put it simple, RANGE = Largest Number – Smallest Number. (Different between two extremes in a distribution). Let us look at an example now.


In the above distribution smallest number is 10 and largest number is 123, so the range is 123-10 = 113.

I know, at the first look itself you might be getting a little doubtful with the usage of range, but sorry, scope of this document is to introduce you to range but not to discuss about pros and cons.

Data dispersion: Variance

How do we find variance for a given distribution? Do the following:-

  1. (Mean – element) 2 →subtract every value in the distribution from its mean and square the result.
  2. Add all the values you got from subtracting from mean and squaring.
  3. And divide the result you got in step 2 above by number of values in the original distribution. That means find the average.

Data Dispersion: Standard Deviation

Forget the theory; let me put this also simple. Standard deviation is square root of variance. So, remember that to calculate standard deviation, first you need to calculate variance. When we are speaking about activity estimations (estimated time to complete an activity on a project), we will be using PERT method to calculate the expected duration of the project or activity. Estimates are provided as P (pessimistic), O (optimistic) and M (Most likely). When you are given estimates for all the activities of a project and asked to find standard deviation the,

  • Computer the variance of each estimated activity with formula ((p-o)/6)2
  • Add all individual variance to get one total variance for the project.
  • Get the square root of the project variance which is standard deviation we are looking for.

Questions & Answers

  1. Estimates to complete a task is given as Optimistic = 11 days, pessimistic = 29 days and most likely = 22 days. Calculate SD of the given schedule.
    • a. 3 days
    • b. 6.5 days
    • c. 1.333 days
    • d. 20.6 days

    Correct Answer: a [ SD = p-O/6]

  2. An activity in your project is estimated as optimistic 10 days, pessimistic 30 days and most likely is 20 days. Calculate the schedule estimate variance.
    • a. 2.7556
    • b. 27.556
    • c. 11.11
    • d. 2.75

    Correct Answer: c [Variance = (P-O/6)2]


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About Sparsh Goyal

A passionate IT professional, Sparsh Goyal boasts of 4.3+ years of experience. He has worked for various projects under AWS, Google Cloud Platform, Spring Boot, Python, Microservices, RESTful, RESTFUL APIs/SOAP, Scripting, Shell and JAVA. He is also working towards gaining proficiency in Oracle Cloud PaaS, DevOps, SaaS and Docker/Kubernetes. His primary and secondary skills validate his relentless pursuits of expanding his horizon and developing more as an IT person. He boasts of the following certifications: *Google Professional Cloud Security Engineer. *AWS Cloud Solutions Architect Associate. *Oracle certified JAVA programmer.
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