Averages are so simple, aren't they?

As I drove down the M3 today, I noticed a speed limit sign saying '50', closely followed by another sign saying 'Average Speed Check'. But are averages—or more precisely, the correct methods for calculating them—widely understood?

For example, if I drive for twelve miles at 40 mph, then another twelve miles at 60 mph, will I be breaking the average speed limit? The answer is No, because my average speed is 48 mph. {The formula for average speed is total distance divided by total time. Total distance is 24 miles; total time is 0.3 hours (for the first bit) plus 0.2 hours (for the second bit), i.e. 0.5 hours.}

But how many people can do the calculation in their head while driving?

Not that I would advocate Jeremy Clarkson's idea for defying average speed limits—i.e. bomb up the motorway at a crazy speed, then make up the total time by having a meal at a service station.

Another area where averages aren't quite what they might seem is in currency conversion. If someone asks you what was the average dollar:euro rate in 2007, you need to be aware that one isn't the precise inverse of the other. Check it out on oanda.com:

• The average rate from dollars into euros in 2007 was 0.73082
  
• But the average rate from euros into dollars in 2007 was 1.37074, the inverse of which is 0.72953.
  

The difference isn't due to rounding error. You can demonstrate it to yourself with a simple example—suppose for the first six months of the year, the euro was worth two dollars, and that for the second six months, the euro was worth one dollar:

• Thus if I trade one euro in the first six months, I get two dollars in exchange; and then I get a single dollar for my second euro in the second six months. So I get a total of three dollars for my two euros: average rate is €1=$1.5.
  
• But if I go the other way, and trade one dollar in the first six months, I get €0.5 in exchange. for my second dollar, traded in the second six months, I get €1. Thus $2=€1.5, i.e. €1=$1.333.
  

So the average rate depends on how much you exchange. It's an important point for the intelligence analyst calculating revenues overseas. And the difference between the two averages, however small it may seem, enables the wily trader to pull a fast one over the unsuspecting customer.