UPDATE

Apparently this was a completely wrong interpretation from my side; I mixed up several things, probably because I had a hammer and was looking for a nail, i.e. matching √ Π to something I envisioned, so I was starting from the solution and working my way back.

Here is an excerpt from a hacker news comment that did not sugarcoat it, but was very valuable to me:

After this comment I decided to do some digging, so I reached out to someone who I consider an absolute math wizard: Steven De Keninck

without context I can only observe that its trivially equal to "Est = hpEst * PI^{dims/2}" - I would venture to say few mathematicians would stop before that simplification. Also PI as base for the exponential feels very arbitrary. I'll go with option 2.

— Steven De Keninck (@enkimute) March 4, 2021

… Which is a very polite answer that you can interpret as “your idea has nothing to do with math, you might as well go for numerology”.

Please consider this part as one of my not so successful thought experiments ;).

In some cases you need to provide an estimate in a context you do not know yet.

For a lot of those cases, I used to think about the happy path, and then applied a multiplier of 3. I had no idea why this worked, but it worked reasonably well…

Remark:

People described this approach before (multiply with √ Π, Π or Π²), but I never found any scientific reasoning behind it.

Because it had not been properly named, I baptized it ‘‘Gaussian estimates’’.

If this approach already has a well-defined name - which seems very likely, as my reasoning seems quite obvious after the fact - please let me know!

So let’s try to figure out why 3 (or Π) might be a good multiplier in some cases:

About uncertainty

In a future context you have the following knowns and unknowns:

• The known knowns: what you know you know;
• The known unknowns: what you know you don’t know;
• The unknown unknowns: what you don’t know you don’t know.

Some examples of known knowns:

• We will have to integrate with systems a, b & c;
• The functionality need to cover e, f & g;
• Our SLA will need to be h.

Some examples of known unknowns:

• How responsive is the customer;
• How well defined is the scope;
• Do we have any constraints we need to take into account.

Next to this, there will be unknown unknowns: aspects that we do not know we do not know, for example

• Person x might provide subpar cooperation;
• Person y might get ill and not be available for a while;
• There might be a history with technology z in the past, which results in a lack of trust.

We can factor in the known unknowns in the constraints and assumptions of our contract, but what about the unknown unknowns?

A little statistics primer

My approach to factoring in the unknown unknowns is rooted in statistics.

There are 3 kinds of lies: lies, damned lies & statistics - Mark Twain

First we will take a look at 2 different statistical distributions:

We have something called a Standard Uniform Distribution:

• mean = 1;
• median = 1;
• standard deviation = 0;
• area = 1.

Basically this just means that f(x) = 1.

We also have something called the Gaussian Distribution or normal distribution:

• mean = 1;
• median = 1;
• standard deviation = σ (sigma);
• area = √ Π.

The most important aspect to Gaussian distribution is the relationship with randomness; here is an excerpt from wikipedia:

Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.

The ‘‘Gaussian multiplier’’

Imagine we are estimating a project where the requirements are very well defined and the team is operating smoothly; the only unknown you have is not knowing the integration API well.

We ask to estimate the happy path, broken down in fixed units of time (one day). This would imply your tasks would have a standard uniform distribution.

As everything else is routine, only the things concerning the integration API will have uncertainty included…

In reality, when executing these integration tasks, some will take a lot longer, and some will be shorter. Because we do not know what the actual distribution would be, we fall back to the distribution that occurs naturally in most of these cases: the Gaussian distribution. So converting from a standard uniform distribution to a Gaussian distribution should help.

To convert from one to the other, we divide by the area of the standard uniform distribution (1), and multiply with the area of the Gaussian distribution (√ Π). Note that all tasks still have a mean and median of 1, but we introduced the standard deviation.

Now, let’s assume you also need to interact with another organization that has control over the API for every task, and that you will estimate this with an overhead multiplier of 30% if everything goes well (so estimates should be multiplied by 1.3). As you have not worked with the provider before, your estimate will probably not have a uniform distribution but a Gaussian distribution, so multiply with (√ Π) again.

So the actual estimate for our integration task would be:

happy path * 1.3 * (√ Π) * (√ Π)

= happy path * 1.3 * Π

A more pragmatic approach to the ‘‘Gaussian multiplier’’.

The above explanation contains the gist of the idea, but requires you to have a very good grasp of what an uncertainty dimension is:

• Time;
• Scope;
• Cost;
• … .

In order to make this a bit more digestible, I created a pragmatic heuristic that should be good enough for most cases. Without further ado, I present to you my suggested model:

This should provide you a pragmatic, but scientifically sound approach to doing estimates for future contexts.

I did not bother to go beyond Π², because the probability of your estimate would be too low beyond that point…