Thursday, March 25, 2010

Can most people be above average?

Can there be truth to the Lake Wobegon "effect": "where all the children are above average"?

The short (and correct) answer is no.

But could there be a town where "the number of above-average children, exceeds the number of below-average children"?

Quite certainly possible!

Without trying to sound like a lawyer, it all hinges on the definition of the word "average". There are at least three commonly used measures of the average or "central tendency" of a bunch of data.

The mean, median, and the mode.

The "mean" is the "average" you are used to, where you add up all the numbers and then divide by the number of data points. When you see somebody's batting average, you are considering the mean:

total number of runs/total number of "outs"

It is quite possible that the number of data-points greater than the mean is larger than half (and vice versa). A common, but interesting example, is the fact that most US mutual funds (70%) under-perform the average benchmark (or passive index funds).

The "median" is the "middle" value in the list of numbers.

To find the median, you sort your numbers in ascending (or descending) order, and make a list. You then pick the guy in the middle of the list. A common example of this measure are "median incomes", which is really a fancy way of saying that you make everybody from Bill Gates to the poorest person on earth stand in order, in one straight line. The income of the guy at the center of the line is the median income.

Thus, if by average you mean median (no pun intended), then there are exactly as many people above average, as they are below it.

Ergo, no Lake Woebegone effect.

The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list. It is also non-unique. There can be two or more modes. Again, you can easily have more values above the mode, than below it.

In summary, if average is defined as the mean or the mode, then the "weak form" of the Lake Wobegon Effect can certainly be true.

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