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Location: Georgia Southern University, Georgia, United States

Tuesday, October 04, 2005

Reversion to the mean? - Critique of 9/28/05 WSJ article

As mentioned in our first post on this site, The Wall Street Journal is our favorite newspaper. Nevertheless we will continue to point out when its writers commit a faux pas. The September 28, 2005 Common Sense article on Energy Prices asserts in its discussion of reversion to the mean that "The classic example is a series of coin tosses. If a coin comes up tails 90 times out of the first 100 tosses, look for heads to make a comeback over the next 100."

A coin toss is a classical example, but not of reversion to the mean. We refer to another of our favorite sites, Wikipedia.org, for a discussion of the "Law of Averages":
"There are common ways to misunderstand and misapply the law of large numbers:

* "If I flip this coin 1000 times, I will get 500 heads results." False. While we expect approximately 500 heads, it is not the case that we will always get exactly 500 heads results. If the coin is fair the chance of getting exactly 500 heads is about 2.52%. Similarly, getting 520 heads results is not conclusive proof that the coin's true probability of getting heads on a single flip is .52

* "I just got 5 tails in a row. My chances of getting heads must be very good now." False. It was unlikely at the beginning that you would get six tails in a row, but the probability of six tails was the same as five tails followed by a head: 1/64. Looking forward after the fifth toss, these probabilities are still equal. The only difference is that there are no other possibilities, so the probability of either outcome is 1/2."
As the last paragraph makes clear, after tossing a 'fair coin' 100 hundred times, the probability that 90 will be tails is slim indeed. But even if that turns out to be the case, over the next 100 flips of the coin we still expect that approximately 50 will be heads. Granted if 50 of the next 100 tosses are heads then indeed "heads has made a comeback" from 10 out of 100 to 60 out of 200. Nevertheless, as noted in the "Law of Averages":
"However, it is important that while the average will move closer to the underlying probability, in absolute terms deviation from the expected value will increase. For example, after 1000 coin flips, we might see 520 heads. After 10,000 flips, we might then see 5096 heads. The average has now moved closer to the underlying .5, from .52 to .5096. However, the absolute deviation from the expected number of heads has gone up from 20 to 96."
Note in our example, the absolute deviation from the expected value has remained unchanged.

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