law of averages

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The law of averages is a lay term used to express a belief that outcomes of a random event shall "even out" within a small sample.

As invoked in everyday life, the "law" usually reflects bad statistics or wishful thinking rather than any mathematical principle. While there is a real theorem that a random variable will reflect its underlying probability over a very large sample, the law of averages typically assumes that unnatural short-term "balance" must occur.

Examples

• Belief that an event is "due" to happen: For example, "The roulette wheel has landed on red three consecutive times. The law of averages says it's due to land on black!" Of course, the wheel has no memory and its probabilities do not change according to past results. So even if the wheel has landed on red 10 consecutive times the probability that the next roll will be black is still 47.6% (it would be exactly 50% if there were no green zero). Similarly, there is no statistical basis for the belief that a losing sports team is due to win a game or that lottery numbers which haven't appeared recently are due to appear soon. This sort of belief is called the gambler's fallacy.

• Belief that a sample's average must equal its expected value. For example, if one flips a fair coin 100 times, there is only an 8% chance that there will be exactly 50 heads.

• Belief that a rare occurrence will happen given enough time: For example, "If I send my résumé to enough places, the law of averages says that someone will eventually hire me." This may actually be true assuming nonzero probabilities and that the number of tries is really large enough; the law of averages is then simply the Law of Large Numbers.

• Belief that over time, statistics must accumulate to gradual even amount, regardless of the actual scenario. For example, the law of averages would expect that in a football league of ten teams, over a period of time each team would gradually balance out to have the same amount of wins and losses, regardless of how skilled or how unskilled any one team might be.

See also

• Law of large numbers
• Gambler's fallacy

References

• Grinstead and Snell Introduction to Probability [1] is a Dartmouth College public-access text.

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