3 Point Estimate: Triangular Distribution vs Beta Distribution (PERT)
The Three Point Estimate technique is used to arrive at a better estimate of the time required to complete a particular activity, work package, and can be rolled up to the entire project. Further, this technique can be used for Time as well as Cost also.
Triangular Distribution: (P + O + ML ) / 3
This is like a simple average of the three estimates. When plotted in a chart, it usually results in a sharp peak, thus the name Triangular Distribution.
For PMP exam, look for subtle clues in the exam question. If they talk about a new project, no experience, or not much reliable data available, then it is best to use Triangular distribution.
If we are doing a project for the first time, and there is no previous history of similar projects to fall back on, it may make sense to use the Triangular distribution, as there is no basis to give more weight to a particular estimate. Thus, a Triangular Distribution is more suited for judgmental data estimates. These are most likely not based on a lot of experience or expertise… but are more of a guess-timates.
Beta Distribution (PERT): (P + O + 4ML ) / 6
This is a weighted average. More weight is given to the most likely. If plotted against a chart, this beta distribution will result in an more uniform, bell shaped curve, called a normal distribution. This has been tested in several different industries, across different countries, and over different time periods, but has always turned out to be more accurate than the 3 point estimate. So this method has become the most used and popular method of choice among project managers.
And if the team is made of experts, this is a similar to previous projects, we need a more accurate estimate etc. then it is best to go for the Beta Distribution using the PERT formula of weighted average.
When do you use PERT and when do you use Triangular Distribution?
However, the Beta Distribution is preferred in cases where we have a lot of historical data, and is more useful for similar type of projects, and experts are providing this based on historical evidence & experience . If we are doing a project for the first time, and there is no previous history of similar projects to fall back on, it may make sense to use the Triangular distribution, as there is no basis to give more weight to a particular estimate. Thus, a Triangular Distribution is more suited for judgmental data estimates. These are most likely not based on a lot of experience or expertise… but are more of a guess-timates.
As with all estimates, the more accurate your estimates, the better the Triangular and Beta Distribution curves, and the better we are able to estimate the project risk and its completion on time and on cost.
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Consider an experiment in which a coin is tossed three times. There are eight possible outcomes of this experiment:
HHH HHT HTH THH HTT THT TTH TTT
where H = Head and T = Tail
The number of Heads in these outcomes are:
3, 2, 2, 2, 1, 1, 1, 0
So, the number of times we can get a head in this experiment are 0, 1, 2 and 3.
In this experiment, we can say that the "Number of Heads" is a random variable. The numbers 0, 1, 2 and 3 are the “values” of the random variable.
If we express the random variable as X, then
X = Number of Heads in three tosses of a coin
Values of X = {0, 1, 2, 3}
As you can see, we have converted the outcomes from the toss of the coin experiment into numbers.
The uniform distribution can be continuous or discrete. The roll of a dice is an example of discrete uniform distribution. The uniform distribution looks like a rectangle, as shown in the following figure, where a is the minimum value and b is the maximum value of the distribution.
The uniform distribution is used in project management to determine rough estimates (range) when very little information is available about the project, and for risk management when several risks have equal probability of occurrence.
The triangular distribution is used in Project Management, often as an approximation to the beta distribution, to estimate activity duration. Assuming a triangular distribution, the expected activity duration (mean of the distribution) can be calculated using the simple average method.
Natural events rarely have finite end points. However, Beta distribution approximates natural events quite well.
A form of beta distribution which looks like a rounded-off triangle is often used in project management to determine activity duration and cost. It models the optimistic (minimum), the pessimistic (maximum) and the most likely (mode) values quite well. The triangular distribution is considered a good approximation of the beta distribution.
The normal distribution curve is symmetrical about the mean (expected value). The curve is also known as a bell-curve because of its resemblance to the shape of a bell. The simplest form of the normal distribution is known as the Standard Normal Distribution, which has a mean of 0 and variance of 1.
Because of it’s ability to accurately portray many real world events, the normal distribution has many practical applications. Be it height, weight or income of a population, or time and cost estimation in project management, all can be modeled fairly accurately using the normal distribution.
The normal distribution’s end points come very close to the horizontal axis but never actually touch it. This is unlike the beta distribution, whose end points touch the horizontal axis i.e. have a zero probability.
Trivia: If both Variance and Standard Deviation give a measure of the spread of values of a random variable, then why do we need 2 variables to tell us the same thing?
The y-axis of the distribution curve gives the probability of occurrence of any value of a random variable. The ratio of the area under the curve between any two points a, b to the total area under the curve, gives the probability that the value of the random variable will lie between a and b. This is where standard deviation comes into play.
According to statistics, for the normal distribution, 68.2% values of a random variable fall within 1 standard deviation of the mean, 95.5% within 2 standard deviations of the mean and 99.7% within 3 standard deviations.
In other words, there’s a 68.2% probability that the value of a random variable will lie between the range [μ-σ, μ+σ], 95.5% probability to be within [μ-2σ, μ+2σ] and 99.7% probability to be within [μ-3σ, μ+3σ].
The area under the curve in the range [μ-σ, μ+σ] is 68.2% of the total area under the curve. This area is shown in blue. Similarly 95.5% area is covered in [μ-2σ, μ+2σ] range, and is the sum of the blue and brown colored regions. And lastly, 99.7% area is covered in [μ-3σ, μ+3σ] range and is the total area shown in blue, brown and green.
Let’s understand this with an example.
Say we have a set of coins with denominations of 1 to 999. The set of coins is the population and the denomination of each coin is a random variable. The mean of this entire population is 500. From this population, we select a sample of 50 coins, and take a mean of their denomination. It is possible that the mean of those 50 coins is less than 500, equal to 500, or even more than 500. Now if you repeatedly draw 50 coins from the set, calculate their mean, and plot the mean vs frequency graph, the resulting distribution will be close to normal. In general, the larger the sample size, the closer the resulting distribution will be to normal.
Let’s put this in project management perspective. We learned in the previous articles that the basic assumptions in PERT are that the individual activity durations are random variables and that they follow the beta distribution. Consider the individual activity durations as the random variables and the duration of activities on the critical path as the sample. In order to calculate the total project duration, we add up the PERT estimates of the activities on the critical path. According to CLT:
The Three Point Estimate technique is used to arrive at a better estimate of the time required to complete a particular activity, work package, and can be rolled up to the entire project. Further, this technique can be used for Time as well as Cost also.
Triangular Distribution: (P + O + ML ) / 3
This is like a simple average of the three estimates. When plotted in a chart, it usually results in a sharp peak, thus the name Triangular Distribution.
For PMP exam, look for subtle clues in the exam question. If they talk about a new project, no experience, or not much reliable data available, then it is best to use Triangular distribution.
If we are doing a project for the first time, and there is no previous history of similar projects to fall back on, it may make sense to use the Triangular distribution, as there is no basis to give more weight to a particular estimate. Thus, a Triangular Distribution is more suited for judgmental data estimates. These are most likely not based on a lot of experience or expertise… but are more of a guess-timates.
This is a weighted average. More weight is given to the most likely. If plotted against a chart, this beta distribution will result in an more uniform, bell shaped curve, called a normal distribution. This has been tested in several different industries, across different countries, and over different time periods, but has always turned out to be more accurate than the 3 point estimate. So this method has become the most used and popular method of choice among project managers.
And if the team is made of experts, this is a similar to previous projects, we need a more accurate estimate etc. then it is best to go for the Beta Distribution using the PERT formula of weighted average.
When do you use PERT and when do you use Triangular Distribution?
However, the Beta Distribution is preferred in cases where we have a lot of historical data, and is more useful for similar type of projects, and experts are providing this based on historical evidence & experience . If we are doing a project for the first time, and there is no previous history of similar projects to fall back on, it may make sense to use the Triangular distribution, as there is no basis to give more weight to a particular estimate. Thus, a Triangular Distribution is more suited for judgmental data estimates. These are most likely not based on a lot of experience or expertise… but are more of a guess-timates.
As with all estimates, the more accurate your estimates, the better the Triangular and Beta Distribution curves, and the better we are able to estimate the project risk and its completion on time and on cost.
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Random variable
The term random variable is a misnomer. Random variable is actually not a variable, but a function. The outcome of every event may not always be a number. For example, when you toss a coin, the outcome is a Head or a Tail. Random variable is a function which assigns a unique numerical value to every possible outcome of an event. For example, in a toss of a coin, how do we express these outcomes in a number? Let's see.Consider an experiment in which a coin is tossed three times. There are eight possible outcomes of this experiment:
HHH HHT HTH THH HTT THT TTH TTT
where H = Head and T = Tail
The number of Heads in these outcomes are:
3, 2, 2, 2, 1, 1, 1, 0
So, the number of times we can get a head in this experiment are 0, 1, 2 and 3.
In this experiment, we can say that the "Number of Heads" is a random variable. The numbers 0, 1, 2 and 3 are the “values” of the random variable.
If we express the random variable as X, then
X = Number of Heads in three tosses of a coin
Values of X = {0, 1, 2, 3}
As you can see, we have converted the outcomes from the toss of the coin experiment into numbers.
Types of Random Variables
Random variables are of 2 types:- Continuous: A random variable with infinite number of values.
Examples:
The lifespan of people, and that of their LCD TVs and mobile phone batteries are examples of continuous random variables. They have an infinite number of possible values. Similarly, the number of miles a car will run before being scrapped, the length of a telephone conversation, the amount of rainfall in a season also fall in the same category.
The duration of an activity on a project is also an example of a continuous random variable. - Discrete: A random variable with countable number of distinct values.
Examples:
The score of students in an examination is a discrete random variable and so is the number of students who fail the exam. The experiment of three tosses of a coin (described above) is also an example of discrete random variable, with distinct values 0, 1, 2 and 3.
Distribution
If we plot a graph with the values of the random variable X on the x-axis and the probability of occurrence of the values on the y-axis, then the plot is known as a Distribution.Probability Distribution Function (PDF)
The (mathematical) function that describes the shape of the Distribution is known as the Probability Distribution Function (PDF).Common Probability Distributions
Some of the common distribution patterns are Uniform Distribution, Beta Distribution, Triangular Distribution and Normal Distribution.Uniform Distribution
In the uniform distribution, each value of the random variable has equal probability of occurrence. For example, in a roll of a dice, each value (1 to 6) has an equal probability. The random variable is the score on each roll of the dice, and the values are 1 to 6. Each value has an equal probability of 1/6.The uniform distribution can be continuous or discrete. The roll of a dice is an example of discrete uniform distribution. The uniform distribution looks like a rectangle, as shown in the following figure, where a is the minimum value and b is the maximum value of the distribution.
The uniform distribution is used in project management to determine rough estimates (range) when very little information is available about the project, and for risk management when several risks have equal probability of occurrence.
Triangular Distribution
The triangular distribution is a continuous probability distribution with a minimum value, a mode (most likely value), and a maximum value. The triangular distribution differs from the uniform distribution in that, the probability of the values of the random variable are not the same. The probability of the minimum, a and maximum value, b is zero, and the probability of the mode value, c is the highest for the entire distribution.
The triangular distribution is used in Project Management, often as an approximation to the beta distribution, to estimate activity duration. Assuming a triangular distribution, the expected activity duration (mean of the distribution) can be calculated using the simple average method.
Beta Distribution
The beta distribution is determined by 4 parameters:- a - minimum value
- b - maximum value
- α - shape parameter
- β - shape parameter
Natural events rarely have finite end points. However, Beta distribution approximates natural events quite well.
A form of beta distribution which looks like a rounded-off triangle is often used in project management to determine activity duration and cost. It models the optimistic (minimum), the pessimistic (maximum) and the most likely (mode) values quite well. The triangular distribution is considered a good approximation of the beta distribution.
Normal Distribution
A random variable which can take any value from -∞ to +∞, is said to follow a Normal Distribution. The normal distribution models natural events very well. In practice, the normal distribution is also used to model distributions with non-negative values. For example, the height of adults in a country is considered to follow a normal distribution, even though the height of a person can never be a negative number.
The normal distribution curve is symmetrical about the mean (expected value). The curve is also known as a bell-curve because of its resemblance to the shape of a bell. The simplest form of the normal distribution is known as the Standard Normal Distribution, which has a mean of 0 and variance of 1.
Because of it’s ability to accurately portray many real world events, the normal distribution has many practical applications. Be it height, weight or income of a population, or time and cost estimation in project management, all can be modeled fairly accurately using the normal distribution.
The normal distribution’s end points come very close to the horizontal axis but never actually touch it. This is unlike the beta distribution, whose end points touch the horizontal axis i.e. have a zero probability.
Expected Value or Mean (μ)
Expected value of a random variable is it’s mean or average value. This expected value is used in project management to represent the expected activity duration (or PERT estimate).Variance (σ ^ 2)
Variance of a random variable is a measure of its spread. Variance is always non-negative. A small variance indicates that the values of the random variable are placed close to the mean, whereas a large variances indicates that the values are spread away from the mean.Standard Deviation (σ)
Standard deviation is the square root of variance and is also always non-negative. It also gives a measure of the dispersion of values of a random variable.Trivia: If both Variance and Standard Deviation give a measure of the spread of values of a random variable, then why do we need 2 variables to tell us the same thing?
The y-axis of the distribution curve gives the probability of occurrence of any value of a random variable. The ratio of the area under the curve between any two points a, b to the total area under the curve, gives the probability that the value of the random variable will lie between a and b. This is where standard deviation comes into play.
According to statistics, for the normal distribution, 68.2% values of a random variable fall within 1 standard deviation of the mean, 95.5% within 2 standard deviations of the mean and 99.7% within 3 standard deviations.
In other words, there’s a 68.2% probability that the value of a random variable will lie between the range [μ-σ, μ+σ], 95.5% probability to be within [μ-2σ, μ+2σ] and 99.7% probability to be within [μ-3σ, μ+3σ].
The area under the curve in the range [μ-σ, μ+σ] is 68.2% of the total area under the curve. This area is shown in blue. Similarly 95.5% area is covered in [μ-2σ, μ+2σ] range, and is the sum of the blue and brown colored regions. And lastly, 99.7% area is covered in [μ-3σ, μ+3σ] range and is the total area shown in blue, brown and green.
Central Limit Theorem
Central Limit Theorem (CLT) says that the mean of a large sample of independent random variables, each having a finite mean and variance, will be normally distributed. For CLT to be applicable, certain requirements should be met. The sample size should be fairly large (more than 30), the random variables should be independent of each other and should have the same type of distribution.Let’s understand this with an example.
Say we have a set of coins with denominations of 1 to 999. The set of coins is the population and the denomination of each coin is a random variable. The mean of this entire population is 500. From this population, we select a sample of 50 coins, and take a mean of their denomination. It is possible that the mean of those 50 coins is less than 500, equal to 500, or even more than 500. Now if you repeatedly draw 50 coins from the set, calculate their mean, and plot the mean vs frequency graph, the resulting distribution will be close to normal. In general, the larger the sample size, the closer the resulting distribution will be to normal.
Let’s put this in project management perspective. We learned in the previous articles that the basic assumptions in PERT are that the individual activity durations are random variables and that they follow the beta distribution. Consider the individual activity durations as the random variables and the duration of activities on the critical path as the sample. In order to calculate the total project duration, we add up the PERT estimates of the activities on the critical path. According to CLT:
The total project duration is assumed to follow a normal distribution and the variance in project duration can be calculated by summing up the variances in the durations of activities on the critical path.In real world project management, the conditions under which CLT is applicable, are not always met. For instance, the activities on a project, and hence their durations, are not always independent of each other. The critical path may not have 30 activities. We'll discuss these issues in a follow-up article on the limitations of PERT.
