Freakonomics reports:
A reader in Norway named Christian Sørensen examined the height statistics for all players in the 2010 World Cup and found an interesting anomaly: there seemed to be unnaturally few players listed at 169, 179, and 189 centimeters and an apparent surplus of players who were 170, 180, and 190 centimeters tall (roughly 5-foot-7 inches, 5-foot-11 inches, and 6-foot-3 inches, respectively). Here's the data:
It's not costless to communicate numbers. When we compare "eighty" (6 characters) vs "seventy-nine" (12 characters) - how much information are we gaining by twice the number of characters? Do people really care about height at +-0.5 cm or is +-1 cm enough?
It's harder to communicate odd numbers ("three" vs four or two, "seven" vs "six" or "eight", "nine" vs "ten") than even ones. As language tends to follow our behaviors, people have been doing it for a long time. We remember the shorter description of a quantity.
This is my theory why we end up with more rounded numbers. This is also partially why Benford's law holds: we change the scales and measurement units as to enable us to store the numbers in our minds more economically. Compare "ninety-nine" (11 characters) with "hundred" (7c), or "nine hundred ninety-nine" (24) with "thousand" (8c).
For our advanced readers, let me give you another example. Let's say I estimate something to be 100. The fact that I said 100 implies that there is a certain amount of uncertainty in my estimate. I could have written it as 1e2, implying that the real quantity is somewhere between 50 and 150. If I said 102, I'd be implying that the real quantity is between 101 and 103. If I said 103, I'd be implying that the real quantity is between 102.5 and 103.5. If I said 50, the real quantity is probably between 40 and 60.
This way, by rounding up, I have been both economical in my expression but also been able to honestly communicate my standard error.
Eventually, increased accuracy is not always worth the increased cost of communication and memorization.
So, do you still think World Cup players are being self-aggrandizing, or are they perhaps just economical or even conscious of standard errors?
[D+1: Hal Varian points to number clustering in asset markets. Also thanks to Janne helped improve the above presentation.]
You might want to reformulate one sentence just a little; you seem to be saying that three, seven and nine are even numbers. I know you americans use a curious measurement system, but I don't think it's quite this odd (sic).
By the way, is there a significant dip at 171, 181 and 191 cm as well? That would lead credence to the idea.
Might it be because players are measured in feet and inches then converted back?
A more important question is "is there really any signal to explain?", i.e. is it possible that this pattern is just the result on random variation? Some quick playing in R reveals that it may well be - run the following code a few times to see interesting patterns that you may want to ascribe to some meaningful underlying pattern.
(I just guessed at these parameters based on the shape of original plot)
This shows up in asset markets as well, e.g., http://www.psyfitec.com/2010/02/irrational-numbers-price-clustering.html
Hadley, the prior involving rounding to round and even numbers is pretty strong - so hard to dismiss it as just random.
Fraac, this could create different rounding artifacts - ones that haven't been considered. Good point though!
It would be interesting to look at height by player position - people who play in the box have an advantage if they are tall (heading the ball), people who play in midfield don't really gain by being tall.
The point being that if strikers and defenders are showing rounding effects and midfielders are not then perhaps it is self-aggrandizing.
While this may seem a trivial example, I am curious and wonder if anybody has done a similar study on the parity between between any of the former currencies in the Euro zone and the Euro. In short, the sense is that it was very likely that items in say French Francs (1 Euro = 6.55 French Francs) would eventually be rounded up to a value in Euro that would be "natural" number-wise but would really be a jump in value. I am not sure I am stating the question correctly.
Igor.
A surprising amount is being attributed to the number of letters in the written-out English words for numbers, and the counting and lines drawn are potentially problematic.
On the first point, why count letters at all, versus numerals or syllables? "99" is shorter than "100." And why privlege English, particularly for an international comparison? If this hypothesis is valid, we should see it in other languages as well. It seems very odd to attribute much of anything to having three letters in the English word "six." If we were speaking in Japanese, would we count the number of strokes in the kanji for each number, or perhaps look at Americans writing Arabic numbers and count he strokes in those numerals?
On the counting point, some of the examples seem arbitrary. Why are we comparing seven to six and not to eight? It has the same number of letters as eight. See also 11 & 12, 13 & 14, 15 & 16... Also, "ninety-nine" and "one hundred" have the same number of characters; maybe the writer actually says "hundred" more often than "one hundred," but dropping the "one " from the number seems odd in my dialect.
As Janne suggests, we see a dip on each side of the intuitive rounding numbers. Note also on the graph that we see spikes on the fives as well. 5s and 10s are pretty natural counting numbers for folks with hands like ours.
Is it not possible that players see taller as better, and thus there is some emphathetic rounding up for players who are just short of the next 10s place on the centimeters scale? My husband is 6'4", but was always reported as 6'5" for basketball, just because taller seems more intimidating.
Compare 99 to 100 in french: "quatre vingt dixe neuf" is certainly longer than "cent"
German: neunundneunzig vs hundert
Spanish: noventa y nueve vs. ciento
So I doubt that the english character count is the only language where this plays an important role. In fact the fundamental problem is a place-value number system, which means that numbers with many non zero digits will generally have many words in their textual description.