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Estimating Effort and Time with Probabilities in Project Management

Dealing with probabilities is part of our daily lives. The weather forecast mentions a rain probability of 80% and we know that we should take an umbrella with us. Calculation with probabilities is difficult for us. Especially for effort and risk estimations it comes to surprising and not always directly intuitive results.

Collecting Probabilities is Easy

Imagine the following conversation between a project manager and a software developer:

Martin: "What do you think, how much time you will need to develop the functionality?"
Barbara. "Hard to say. In less than 7 days I have no chance. To be ready in 14 days, I'm one hundred percent sure."
Martin: "How many days will you need to for a 50% probability to be done?"
Barbara: "I think at 10 days, I have a 50-50 chance."
Martin: "Ok, thanks for the information."

Now, if Martin draws the result as a probability distribution, we will see a diagram like in Figure 1.
Figure 1: Cost estimation as probability distribution
He has chosen a triangular distribution, but other asymmetric distribution would be possible too. The boundary between the blue and green area represents the fiftieth percentile (P50). Between the red and blue area is the fifth percentile (P5). There is a 5% chance to be completed before that date. In the ninety-fifth percentile (P95), we have yet to be a 95 percent chance finished. This date is between the green and yellow area.

Calculating with Probability Distributions

Our project manager makes now an estimation of five work packages and calculates the sum of the individual values ​​(see Figure 2).
Figure 2: Differences between the sum of
individual values (Summe) ​​and the
actual distribution (Verteilung)
The result shows significant variations in the double-digit percentage of the sum of individual values ​​and the difference caused by the convolution of probability distribution. These differences just become evident and understandable if the resulting distribution is seen in Figure 3.
Figure 3: Resulting distribution from folding of the
five triangular distributions 

This distribution resembles a bell curve, i.e. the increase and decrease at the ends is much flatter than the original triangle distribution. This leads to a particularly large error especially at the ends of the bell curve.

Demo of a Probability Density Function Calculator

Probability Density Function Calculator - Download

Download of PCALC, the program that can do computation with probability distributions: (67,8 MB )


The deviations shown in projects estimations lead to systematic over-or underestimation of work packages. Who wants to calculate with probabilities may not add up only the individual probabilities. The only exceptions are the P0 and P100 points.

The idea of estimations with probabilities in  projects is already several years old (see further readings). However, it is not used in operational project management, a large number of projects work with simple spread sheets. One reason is the lack of tools which can calculate with probabilities.

Further Readings

  • Tom DeMarco; Controlling Software Projects: Management, Measurement, and Estimates: Management, Measurement and Estimation; Prentice Hall; 1978

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