# Different kinds of Average

Saturday, September 22, 2007

## On the average . . .

The pointy-haired Überboss - capo di tutti capi - marched into the employees' union meeting and point blank stated "Half of you are under average!" . This got the union guys protesting and spluttering before they'd had time to think (i.e. their usual mode is protest then think later about what they're saying ;-)

So I butted in with "Correct, that's one definition of average ;-)" and then went on to point out "But what's worse is that the pay of the average employee is less than the average pay! You're not paying us fairly!" This of course got the shop stewards and unionised mob into more of a misty red rage. But it's true, here's an example :-

The company has 8 blue-collar workers on 10€/hour and 7 white-collar guys on 20€/hour and 5 bosses on 100€/hour. So the average salary is (8*10 + 7*20 + 5*100)/(8+7+5) = 720/20 = 36€ per hour. This kind of average is called the mean.
Note that most people (15 out of 20) earn less than the mean.

Now certainly the 5 bosses wouldn't (want to) be described as an average employee, and since there are 8 blue-collar workers as opposed to 7 white-collar ones, the average employee is a blue-collar worker. And 10€ are less than 36€. This kind of average is called the mode, i.e. the class of employees having the most members.

Another way of finding the average employee is to sort the employees by pay, first the eight 10€/hour guys, then the seven 20€/hour guys, and finally the five 100€/hour bosses. That's twenty people in total. So the average guy - the one in the middle - is the 10th one, a 20€/hour white-collar guy. And 20€ are still less than 36€. This type of average, the middle member of an ordered set, is called the median.

So all I'd said was that the median and/or the mode were less than the mean. And this is always so when the distribution is skewed like this; it's nothing for the union guys or the pointy-haired boss to get heated about. In fact if you look at a company like Microsoft, the distribution is almost L-shaped, due to Bill Gates earning an obscene amount of money by seriously overcharging us for his distinctly average products :-(

And the moral of today's tale? When someone says "average" make sure you know which one he means: median, mode, or mean. And why he chose to use that one!

Now go visit my blog please, or look at other interesting maths stuff :-)

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