Holiday Beauty Pageant

For the holiday season, my town decorates a tall evergreen fir tree on the main square with blue electric lights. The tree has apparently been trimmed recently and now stands pitifully thin, surprisingly crowned by two equal peaks: branches that grew out of a trunk. So, here we are, with a tall, dragonish, two-headed blue-lit tree. It looks simultaneously magical and spooky in the dark and chilly kingdom of night. While contemplating the blue color of the lights, a mathematical thought occurred to me. This is what one may call the mean of all the winter holiday decorations my townspeople use.

Mean is what we frequently call by mistake an average. In fact there are three kinds of averages: mean, mode and median. Mean is Miss Congeniality, friendly and lovable: all the sample values added up and then divided by their number. Mode is Miss Popularity: the value that occurs most frequently. Median is Miss Centered: a central sample from the line where samples are arranged by their value (from smaller to larger).

In my town the major religious affiliations of the people are: 70% Christian and 6% Jewish. So, the blue Christmas tree may be a carefully planned mean holiday ornament combining the red, white and green colors of Christmas with the blue and white colors of Hanukkah - a compromise to satisfy everyone's holiday traditions. If town clerks had used another mathematical concept, the mode (the most popular town decoration), it would have resulted in a brightly lit red-white-and-green Christmas tree, leaving some people upset.

Let's muse on different holiday statistics – the traditional food with which my townspeople are celebrating. Here, we may not want to adopt the mean of holiday food distribution: honey ham sprinkled with latkes and eggnog with 6% Manischewitz wine. If we selected the most popular food on my neighbors' celebratory tables (the mode of holiday food distribution), we'd probably enjoy turkey or ham, mashed potatoes, gingerbread and peppermints.

What about our presents? A quick survey of my street neighbors, two weeks before the Holidays, revealed the following numbers: wives spent $150, $220 and $90 each, their husbands $0, $0, $25, kids $0, $0, $0, $0, $2, $5, $10.
Mean of this distribution is:
($150 + $220 + $90 + $0 + $0 + $25 + $0 + $0 + $0 + $0 + $2 + $5 + $10) / 13 = $38.60
Mode is the most frequent entry = $0.
Median is the central number in the line: $0, $0, $0, $0, $0, $0, $2, $5, $10, $25, $90, $150, $220 that will be $2.

This appears to be a sad beauty pageant. Mean (Miss Congeniality) has a reasonable $38.60, mode (Miss Popularity) is completely broke with $0, and median (Miss Centered) is counting the last $2. Perhaps we should send our husbands shopping as Santa Claus may be in danger of being laid off.

P.S. This story has been written in December 2008, in the middle of the recession.

Top image by alaricy, distributed under the Creative Commons license.

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