# Which distribution for modelling duration of tasks?

Recently I was present with a task estimation technique. Instead of letting people rate a task for x - amount of hours, I let them discretize tasks into discrete sizes like small / medium / large / x-large. (The tasks are planning poker from scrum for people aware of this.)

After a bit of tracking we should be able to estimate the duration based upon historical data (i.e. statistics). I have been given a set of sample data (real data, but not from my work) and would like see there actually can be made any distribution which fit this. (Of course, I would need to recalculate this for my own tasks.)

The distributions from the presentations look like this:

It seem like some right skewed distribution. From University, I recall working with a distribution looking like this, but I am not certain about the name. I was hoping for something parametric where I can derive some simple parameters form the data (like mean and variance for a normal distribution)

I have made a histogram of the data:

(I know there are too few data.)

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Not sure if this is useful, but survival/event history/time before failure models are used to answer the dual questions of whether and when an event occurs. By asking about duration are you trying to answer whether and when a task is completed? If so a survival/event history/time before failure model might be useful. –  Alexis Jul 27 at 15:55

Some choices include Weibull, Gamma (including exponential), and lognormal distributions, possibly with a shift-parameter if there's a non-zero minimum possible time. (However from your diagram it looks like there's also potentially a discreteness issue.)

If the presentation drawing is reasonably accurate, a shift-parameter will probably be required.

If there's a tendency for the times to be highly skew, log-logistic, inverse Gaussian or Pareto might be considered. (It doesn't look to be the case here though.)

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Thanks for suggestions! Regarding the discreteness issue: I choose some bins for the histogram 1-5, 5-10... and etc. –  Steffen Villadsen Jul 28 at 20:57

You might want to try an Erlang distribution. Here are a couple of excerpts from the page:

The distribution is a continuous distribution, which has a positive value for all real numbers greater than zero, and is given by two parameters: the shape $k$, which is a positive integer, and the rate $\lambda$, which is a positive real number

Events that occur independently with some average rate are modeled with a Poisson process. The waiting times between $k$ occurrences of the event are Erlang distributed.

I am not sure if there is a good intuitive explanation about why Erlang would fit here but I'll give it a shot: if you think of each task consisting of smaller subtasks, then you can model this in three ways:

1. Each subtask takes about same amount of time (so using the notation of the Wikipedia page, the $\mu$ is the same for all tasks but the larger estimates have higher number of subtasks (the $k$ in the Erlang formula).

2. The converse: each size task has the same number of subtasks but each subtask is longer.

3. A combination of (1) and (2).

Just be careful in fitting that you have enough historical data to build a decent model without overfitting and kudos to you for applying statistics to Scrum project estimation!

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