# Average: service stations

In Fermi problems some experienced modellers favour finding the average. They use the Goldilocks Principle where M

_{2 }

is too big and M

_{1 }

is too small. What is just right?

#### Service stations in a provincial city

Estimate the number of service stations (fuel only) that are viable for some particular provincial city that is bypassed by major highways.

The first stage is to decide what quantities (variables) are needed to address the problem. When statistics and/or other relevant information are not available we need to make assumptions to estimate quantities.

**Units of time**

What unit of time will we use? Day? Week? Month? Year?

**Vehicles owned**

What is an estimate of the number of vehicles owned by members of the relevant population?

**Travel distances**

What is an estimate of the average distance (kilometres) travelled per vehicle in the chosen time period?

This latter kind of estimate is needed time and again within Fermi problems.

For a lower estimate (M1) we might think of a pensioner driving to nearby shops a couple of times a week (say 10 km?).

For an upper estimate (M2) we want a figure that will capture high weekly use by a resident, but such that fuel is still purchased locally (say 1000 km?).

These choices are typically up for debate.

**How to find an ‘average’**

The geometric mean G = √(M

_{1}

)(M

_{2}

) is a more robust estimate that the arithmetic mean A = (M

_{1}

+ M

_{2}

)/2.

It generally gives a value within an order of magnitude of the extremes used to calculate it. In this problem, G =100 while A = 505.

Similar approaches can be used to estimate, for example, the number of fast food outlets instead of service stations. Sometimes a genuine real world problem lends itself to a Fermi type approach. Real world problems are generated from many sources – one rich source for students involves reports in newspapers and other media.