# Statistical regularity

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Statistical regularity

Statistical regularity is a notion in statistics and probability theory that random events exhibit regularity when repeated enough times or that enough sufficiently similar random events exhibit regularity. It is an umbrella term that covers the law of large numbers, all central limit theorems and ergodic theorems.

If one throws a die once, it is difficult to predict the outcome, but if we repeat this experiment many times, we will see that the number of times each result occurs divided by the number of throws will eventually stabilize towards a specific value.

Repeating a series of trials will produce similar, but not identical, results for each series: the average, the standard deviation and other distributional characteristics will be around the same for each series of trials.

The notion is used in games of chance, demographic statistics, quality control of a manufacturing process, and in many other parts of our lives.

Observations of this phenomenon provided the initial motivation for the concept of what is now known as frequency probability.

This phenomenon should not be confused with the Gambler's fallacy, it only concerns regularity in the (possibly very) long run.

References

* Leon-Garcia, Albert (1994) "Probability and Random Processes for Electrical Engineering" (2nd edition), Prentice Hall
* Whitt, Ward (2002) "Stochastic-Process Limits, An Introduction to Stochastic-Process Limits and their Application to Queues", Chapter 1: Experiencing Statistical Regularity, [http://www.columbia.edu/~ww2040/book.html link to selected chapters]

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