How can most of the children be above average?

To paraphrase Garrison Keillor's description of Lake Wobegon, that is.

During a morning trip to McDonald's - a place I boycotted and managed to avoid for six years, but now occasionally visit for a breakfast sandwich and shake disguised as coffee - I found the following bit of info from the delightful "Coffee News" rag:

"The IQ of the average person is 109."

This is mind-boggling, no matter the mind.

The average IQ is intended to be 100, with the standard deviation of about 15. That's how it's set up. You can already see the fallacy of the 109 average, but let me give a little more explanation to IQ scores.

When used with children, older IQ tests would measure the child's intellectual age, then compare it to his/her physical age, for the IQ. In other words, if an eight year-old child had a measured intellectual age of 6, the IQ would be 75, since 6 is 75% of 8. If that child's intellectual age was 10, the corresponding IQ would 125. An IQ of 100 meant that the child was right where we expected him/her to be, right on schedule.

This system of measuring IQ has its pitfalls, of course. Only a certain range of ages will work within the system. Funny to imagine a two day-old baby being saddled with an IQ of 50 because they didn't learn anything yesterday, or a guy my age having an IQ of 200 because he's thinking like a 78 year-old.

IQ measurement is now based on a bell curve, where a subject's measured score on an IQ test is compared against other registered scores from that age group. The subject's place on the bell curve is found and the IQ score is then obtained. I don't know how age groups are determined, but I imagine that most adults are in the same group.

It'll be hard for me to better explain how bell curves and standardized scores work without drawings here, but I'll give it a try.

Imagine that your IQ is one standard deviation above the average IQ score of people your age. On a bell curve, this would be the point on the right side where it changes concavity -  where the shape of the curve, instead of curving down, flattens and is about to start curving up. (Very cool that a bell curve's inflection points are exactly one standard deviation from the mean. Calculus and statistics have their Promissory Point here!) According to statistics, you would have a standardized score, or z-score, of +1. You would have an IQ of 115, since the average IQ score plus one standard deviation of it is 100 + 15. Your percentile, or percentage of people with scores at or below your score, would be 84.1%. (Finding the percentile requires some calculation and a normal distribution table or one of them fancy graphing calculators.)

84th percentile? That's pretty high, amongst the top 16% in your age group, and a likely candidate for being labeled "smart." Mensa, known as the High IQ Society, allows membership if candidates measure in the top two percent of any standardized (and recognized) IQ test. If only they would ask for a z-score of +2 or above instead, to appease us math geeks!

Back to the "Coffee News" meme. An IQ score of 109 would translate into a standardized score of +0.6, since 9 is six-tenths of 15. The percentile of someone with an IQ of 109 is 72.6%, meaning that such an IQ ranks in the top 27.4% of all people in that age group. Maybe not Mensa material, but definitely someone "ahead of the curve." Someone considered above average, to be sure, as an "average" IQ would correspond to a 50th percentile.

So how can the claim be true? How can the average IQ score have a percentile higher than 50? Since the bell curve is symmetric, and the median (middle score) is the same as the mean, how can more than 50% of the populace be above average?

Is my McDonald's within the Lake Wobegon boundaries?