Project Estimation and PERT (Part 5): So far we have learned around the basics that PERT, just how PERT is provided to estimate activity completion time, and also how the PERT formula was derived. We know that a job manager has to wear number of hats. In this article, we are going to wear the probabilist and also statistician hats and review the simple concepts of probability and also statistics pertinent to task estimation. The is absolutely “out the scope” as far as the PMP exam is concerned. But if you ever wanted to discover the basics of arbitrarily variables, distributions, mean, variance, typical deviation etc., without sacrificing a the majority of precious time, climate this article is just for you. I think the will give you the necessary foundation for understanding the concepts of project estimation and also bring girlfriend one step closer to implementing theory into practice.

Random variable

The term random variable is a misnomer. Random variable is actually no a variable, but a function. The outcome of every occasion may not constantly be a number. Because that example, as soon as you toss a coin, the result is a Head or a Tail. Random variable is a role which assigns a unique numerical worth to every possible outcome of an event. For example, in a toss the a coin, exactly how do we express this outcomes in a number? stop see.

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Consider an experiment in which a coin is tossed three times. There room eight feasible outcomes of this experiment:


where H = Head and also T = Tail

The number of Heads in this outcomes are:

3, 2, 2, 2, 1, 1, 1, 0

So, the variety of times us can obtain a head in this experiment are 0, 1, 2 and also 3.

In this experiment, we have the right to say the the “Number that Heads” is a random variable. The number 0, 1, 2 and also 3 space the “values” that the random variable.

If us express the arbitrarily variable as X, then

X = number of Heads in 3 tosses of a coin values of X = 0, 1, 2, 3

As you deserve to see, we have converted the outcomes indigenous the toss that the coin experiment right into numbers.

Types of random Variables

Random variables space of 2 types:

: A arbitrarily variable through infinite number of values. Examples: The lifespan of people, and also that of your LCD TVs and mobile phone batteries are examples of constant random variables. They have actually an infinite variety of possible values. Similarly, the variety of miles a vehicle will run prior to being scrapped, the length of a telephone conversation, the amount of rainfall in a season also fall in the same category. The term of an activity on a task is also an instance of a consistent random variable. Discrete: A arbitrarily variable v countable variety of distinct values. Examples: The score of college student in an examination is a discrete random variable and so is the variety of students that fail the exam. The experiment of three tosses the a coin (described above) is additionally an instance of discrete arbitrarily variable, with distinctive values 0, 1, 2 and 3.Distribution

If us plot a graph v the values of the random variable X on the x-axis and the probability of event of the values on the y-axis, climate the plot is well-known as a Distribution.

Probability Distribution function (PDF)

The (mathematical) function that describes the shape of the circulation is well-known as the Probability Distribution duty (PDF).

Common Probability Distributions

Some the the common distribution patterns room Uniform Distribution, Beta Distribution, triangular Distribution and also Normal Distribution.

Uniform Distribution

In the uniform distribution, each value of the random variable has equal probability the occurrence. For example, in a role of a dice, each value (1 come 6) has actually an same probability. The random variable is the score ~ above each role of the dice, and the values are 1 come 6. Each value has an same probability of 1/6.

The uniform circulation can be continuous or discrete. The roll of a dice is an instance of discrete uniform distribution. The uniform distribution looks like a rectangle, as shown in the complying with figure, whereby a is the minimum value and also b is the maximum value of the distribution.


The uniform circulation is used in project monitoring to recognize rough estimates (range) once very small information is available about the project, and also for risk management when several risks have equal probability the occurrence.

Triangular Distribution

The triangular distribution is a consistent probability distribution with a minimum value, a mode (most most likely value), and a preferably value. The triangular circulation differs indigenous the uniform distribution in that, the probability of the values of the arbitrarily variable are not the same. The probability the the minimum, a and maximum value, b is zero, and the probability the the setting value, c is the greatest for the entire distribution.


The triangular circulation is offered in task Management, regularly as one approximation come the beta distribution, to estimate activity duration. Assuming a triangle distribution, the expected activity duration (mean the the distribution) have the right to be calculated making use of the basic average method.

Beta Distribution

The beta distribution is established by 4 parameters:

a - minimum valueb - best valueα - shape parameterβ - form parameter

Where a and also b space finite numbers.

Natural occasions rarely have actually finite finish points. However, Beta circulation approximates natural events quite well.


A kind of beta circulation which looks choose a rounded-off triangle is regularly used in project administration to determine activity duration and also cost. The models the optimistic (minimum), the cynical (maximum) and also the most likely (mode) values quite well. The triangular circulation is thought about a an excellent approximation that the beta distribution.

Normal Distribution

A arbitrarily variable which have the right to take any type of value native -∞ come +∞, is claimed to follow a typical Distribution. The normal distribution models organic events an extremely well. In practice, the normal circulation is additionally used to design distributions v non-negative values. For example, the height of adults in a nation is taken into consideration to monitor a typical distribution, also though the height of a person deserve to never be a an unfavorable number.


The normal circulation curve is symmetrical about the mean (expected value). The curve is likewise known together a bell-curve due to the fact that of its same to the shape of a bell. The simplest kind of the normal circulation is known as the typical Normal Distribution, which has a mean of 0 and also variance that 1.

Because of it’s capacity to accurately portray countless real human being events, the normal distribution has countless practical applications. It is in it height, weight or income of a population, or time and cost estimate in project management, all have the right to be modeled relatively accurately making use of the common distribution.

The normal distribution’s finish points come very close come the horizontal axis however never actually touch it. This is unequal the beta distribution, whose end points touch the horizontal axis i.e. Have a zero probability.

Expected worth or mean (μ)

Expected worth of a arbitrarily variable is it’s mean or average value. This supposed value is used in project management to stand for the expected activity duration (or PERT estimate).

Variance (σ ^ 2)

Variance of a random variable is a measure up of that spread. Variance is always non-negative. A tiny variance shows that the values of the random variable are inserted close come the mean, vice versa, a huge variances suggests that the values are spread far from the mean.

Standard Deviation (σ)

Standard deviation is the square source of variance and also is additionally always non-negative. It likewise gives a measure of the dispersion of values of a random variable.

Trivia: If both Variance and Standard Deviation give a measure up of the spread out of values of a random variable, then why carry out we require 2 variables come tell us the very same thing?

The y-axis of the circulation curve offers the probability of incident of any value of a random variable. The proportion of the area under the curve between any type of two point out a, b come the full area under the curve, provides the probability the the worth of the arbitrarily variable will lie in between a and b. This is where traditional deviation comes into play.

According to statistics, because that the typical distribution, 68.2% values of a random variable autumn within 1 standard deviation of the mean, 95.5% in ~ 2 standard deviations of the mean and 99.7% within 3 standard deviations.

In other words, yes sir a 68.2% probability the the value of a arbitrarily variable will lie in between the selection <μ-σ, μ+σ>, 95.5% probability come be within <μ-2σ, μ+2σ> and 99.7% probability to be in ~ <μ-3σ, μ+3σ>.


The area under the curve in the variety <μ-σ, μ+σ> is 68.2% of the full area under the curve. This area is displayed in blue. Similarly 95.5% area is covered in <μ-2σ, μ+2σ> range, and is the sum of the blue and also brown colored regions. And also lastly, 99.7% area is covered in <μ-3σ, μ+3σ> selection and is the full area shown in blue, brown and also green.

Central limit Theorem

Central border Theorem (CLT) states that the average of a large sample that independent random variables, each having a finite mean and variance, will certainly be generally distributed. Because that CLT to be applicable, certain requirements need to be met. The sample size should be fairly big (more 보다 30), the random variables have to be independent of each other and should have the same form of distribution.

Let’s recognize this v an example.

Say we have actually a set of coins with denominations of 1 to 999. The set of coins is the populace and the denomination of every coin is a random variable. The mean of this entire population is 500. Indigenous this population, we pick a sample of 50 coins, and also take a median of their denomination. It is possible that the typical of those 50 coins is less than 500, equal to 500, or even more than 500. Currently if friend repeatedly attract 50 coins from the set, calculate your mean, and also plot the mean vs frequency graph, the resulting circulation will be close come normal. In general, the larger the sample size, the closer the resulting circulation will be to normal.

Let’s put this in project management perspective. We learned in the previous articles that the simple assumptions in PERT are that the individual activity durations are random variables and also that they follow the beta distribution. Think about the individual task durations together the random variables and the term of tasks on the an essential path together the sample. In bespeak to calculation the total project duration, we include up the PERT approximates of the activities on the vital path. Follow to CLT:

The complete project term is assumed to monitor a normal distribution and the variance in task duration have the right to be calculate by summing increase the variances in the durations of activities on the vital path.

In real civilization project management, the problems under which CLT is applicable, are not constantly met. For instance, the tasks on a project, and also hence your durations, space not constantly independent of each other. The critical path may not have actually 30 activities. We’ll comment on these issues in a follow-up write-up on the limitations of PERT.

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With this us have pertained to the finish of this post. Together you know, this object is a study in itself, and I cannot maybe cover the in sufficient detail. I’ve tried my ideal to summarize lot of details in a concise manner. Please usage the comment section below to permit me recognize whether you discovered it useful.

8-part series on project Estimation and PERT

Get Intimate with PERTThree allude Estimate - The strength of 3 in project EstimationWhat is PERT?The magical PERT FormulaProbability and Statistics in Project monitoring (you space here)PERT and also CPM obtain CozyPMP Quiz contest - task Duration EstimatesStandard Deviation and Project term Estimates

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