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.
- 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. 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.
- Belief that a sample's average must equal its expected value. For example, Daily Show host Jon Stewart joked that out of ten Republican candidates for president, "the law of averages says one of these guys is a little Barney in the Franks." Even if 10% of the population is homosexual, there is no guarantee that exactly one member in a group of ten must be homosexual. Similarly, 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 the law of averages is simply named in place of 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 time each team would gradually balance out to have the same amount of wins and losses, regardless of how skilled or how bad one team might be. Or, in the game of beer pong, the law of averages states that both partners could be expected to make the same amount of cups as the night went on, regardless of one partner's skill level.
- Grinstead and Snell Introduction to Probability is a Dartmouth College public-access text.